Percolation on Transitive Graphs Citation Easo, Philip (2025) Percolation on Transitive Graphs. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/899s-pe86. https://resolver.caltech.edu/CaltechTHESIS:05302025-191944015 Abstract Percolation on a transitive graph is an idealized mathematical model for a homogeneous system undergoing a phase transition. We will investigate how the geometry of an infinite transitive graph determines whether percolation undergoes a phase transition, and if so, at what critical point. Building on these ideas, we will develop a new theory of percolation on finite transitive graphs. This theory unifies the percolation phase transition on infinite transitive graphs with the giant-cluster phase transition in the celebrated Erdős-Rényi model from combinatorics. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Percolation Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2025. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2023. Thesis Availability: Public (worldwide access) Research Advisor(s): Hutchcroft, Tom Thesis Committee: Tamuz, Omer (chair) Hutchcroft, Tom Zhang, Lingfu Conlon, David Defense Date: 28 May 2025 Non-Caltech Author Email: philipeaso (AT) gmail.com Record Number: CaltechTHESIS:05302025-191944015 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05302025-191944015 DOI: 10.7907/899s-pe86 ORCID: Author ORCID Easo, Philip 0000-0002-5606-3727 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17315 Collection: CaltechTHESIS Deposited By: Philip Easo Deposited On: 04 Jun 2025 18:29 Last Modified: 17 Jun 2025 18:37 Thesis Files PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Percolation on transitive graphs Thesis by Philip Easo In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended May 28, 2025 © 2025 Philip Easo ORCID: 0000-0002-5606-3727 All rights reserved ii ACKNOWLEDGEMENTS iii I am deeply grateful to my advisor Tom Hutchcroft for introducing me to this beautiful area of research, for sharing his ideas and encouraging mine, for bringing me to Caltech, and for much more — beyond mathematics — which he has taught me by way of example. I am also deeply grateful to my partner Henrietta Coales for her constant support, for keeping me grounded, and for joining me on this adventure in California.1 1 I also acknowledge that the name “sandcastle” of the object introduced in Chapter 5 was her suggestion. ABSTRACT iv Percolation on a transitive graph is an idealized mathematical model for a homogeneous system undergoing a phase transition. We will investigate how the geometry of an infinite transitive graph determines whether percolation undergoes a phase transition, and if so, at what critical point. Building on these ideas, we will develop a new theory of percolation on finite transitive graphs. This theory unifies the percolation phase transition on infinite transitive graphs with the giant-cluster phase transition in the celebrated Erdős-Rényi model from combinatorics. Chapter 1 1 INTRODUCTION Phase transitions appear in our lives in both obvious and subtle ways. The most vivid examples come from physics: as temperature slowly increases past certain critical points, ice suddenly melts and magnets lose their magnetism. The example that received the most news coverage during the recent Covid pandemic concerned the so-called 𝑅0 -value: whether 𝑅0 < 1 or 𝑅0 > 1 would predict whether the number of infected individuals would decay or grow exponentially. More hidden phase transitions are present in the formation of traffic jams and the average-case computing time required by an algorithm. Phase transitions are in fact a very typical occurence anytime one assembles many tiny identical components (picture: atoms arranged in a lattice) in which each component is only allowed to interact with its neighbouring components and one varies this “local” interaction strength. Percolation is the mathematician’s caricature of this setup. 1.1 Percolation We will model an assembly of tiny components by a graph 𝐺 = (𝑉, 𝐸). We will always assume that graphs are connected, undirected, simple, have countably many vertices, and that each vertex has at most finitely many neighbours. To model that all of the tiny components are identical, we will require that 𝐺 is (vertex-)transitive, meaning that for all vertices 𝑢 and 𝑣, there exists an automorphism 𝜙 of 𝐺 such that 𝜙(𝑢) = 𝑣. This includes, for example, the usual Eucliean lattices Z𝑑 , regular trees, and more generally, any Cayley graph of a finitely generated group. Fix a parameter 𝑝 ∈ [0, 1], which will be our “local” interaction strength. Build a random spanning subgraph 𝜔 ⊆ 𝐺 by independently choosing, for each edge 𝑒 ∈ 𝐸, whether to include 𝑒 (with probability 𝑝) or delete 𝑒 (with probability 1 − 𝑝). (See figure 1.1.) The law of 𝜔 is called (Bernoulli bond) percolation. We denote this law by P 𝑝 . To find the phase transition in this model, track the clusters (i.e. connected components) of 𝜔 while varying 𝑝. For simplicity, let us start by assuming that 𝐺 is infinite. By Kolmogorov’s 0-1 law and the monotonicity of this model with respect to 𝑝, there is always some critical parameter 𝑝 𝑐 (𝐺) ∈ [0, 1] such that    0,  P 𝑝 (𝜔 contains an infinite cluster) =   1,  if 𝑝 < 𝑝 𝑐 if 𝑝 > 𝑝 𝑐 . From a utilitarian perspective, a good reason to study percolation is that this toy model provides a testing ground for new techniques, which often trickle down to more intricate, physically-relevant models some years later. From an aesthetic perspective, percolation provides many simple-to-state yet challenging-to-solve problems that require mathematicians to exercise their creativity. This is best-illustrated by the most notorious open problem in the area: prove that when 𝐺 is the Euclidean lattice Z3 , P 𝑝 𝑐 (𝜔 contains an infinite cluster) = 0. This model was first introduced by Broadbent and Hammersley in 1957 to model the flow of a fluid through a porous medium. In their setup, the graph 𝐺 is a Euclidean lattice like Z𝑑 , and indeed, most work on percolation has traditionally taken 𝐺 to be such a lattice or to be a tree. The scope widened in 1996, when Benjamini and Schramm launched an influential research programme to systematically study percolation on general infinite transitive graphs. This programme is the context in which the present thesis should be understood. None of the results in this thesis are new for Euclidean lattices or trees, nor are we concerned with any other particular transitive graph; our goal has been to understand basic features of the percolation phase transition that hold for all transitive graphs. Figure 1.1: Example of a graph 𝐺 (left) and a spanning subgraph 𝜔 (right). 1.2 Transitive graphs In this section, we will take a tour of the zoo that is the space of transitive graphs. Given a transitive graph 𝐺 = (𝑉, 𝐸), we will always write 𝑜 to denote an arbitrary vertex. (Statements will always hold independently of the choice of 𝑜 because 𝐺 is transitive.) We write dist = dist𝐺 to denote the graph metric on 𝐺. Given 𝑛 ≥ 1 and 𝑢 ∈ 𝑉, we write 𝐵𝑛 (𝑢) = 𝐵𝐺 𝑛 (𝑢) to denote both the set {𝑣 ∈ 𝑉 : dist(𝑢, 𝑣) ≤ 𝑛} and the subgraph of 𝐺 that this set of vertices induces. Hopefully it will 2 always be clear from context which of these definitions is intended. We also adopt the convention that 𝐵𝑛 := 𝐵𝑛 (𝑜). Local and global similarity It helps to think of the geometry of a given infinite transitive graph 𝐺 in two parts: the small-scale, local structure and the large-scale, global structure. Indeed, an important principle in the study of percolation on transitive graphs is that certain key features such as the behaviour of the model around the critical point, or the value of the critical point itself, should be entirely determined by one part or the other. Rather than attempt to intrinsically define what the “local” and “global” structure are of a given infinite transitive graph 𝐺, let us instead define what it means for a given pair of graphs to have similar global or local structures. The local metric on the space of transitive graphs is given by, for each pair 𝐺, 𝐻 of transitive graphs, distloc (𝐺, 𝐻) := 2− sup{𝑛≥1:𝐵𝑛 𝐵𝑛 } , 𝐺 𝐻 𝐻 where 𝐵𝐺 𝑛  𝐵𝑛 means that the corresponding rooted subgraphs (rooted at 𝑜) are isomorphic. The topology this induces on the space of all transitive graphs is called thelocal (aka Benjamini Schramm) topology. For example, the sequence of tori (Z/𝑛Z) 2 : 𝑛 ≥ 1 and the sequence of cylinders (Z × (Z/𝑛Z) : 𝑛 ≥ 1) both converge locally to the square lattice Z2 . On the other hand, quasi-isometry provides a way to measure the global similarity of graphs. Given metric spaces (𝑉1 , 𝑑1 ) and (𝑉2 , 𝑑2 ), and given constants 𝐶, 𝐷 > 0, we say that 𝑑1 is (𝐶, 𝐷)-quasiisometric to 𝑑2 if there exists a function 𝜙 : 𝑉1 → 𝑉2 such that for all 𝑢, 𝑣 ∈ 𝑉1 , 1 𝑑1 (𝑢, 𝑣) − 𝐷 ≤ 𝑑2 (𝜙(𝑢), 𝜙(𝑣)) ≤ 𝐶𝑑1 (𝑢, 𝑣) + 𝐷, 𝐶 and for every vertex 𝑥 ∈ 𝑉2 , there exists some 𝑢 ∈ 𝑉1 with 𝑑2 (𝑥, 𝜙(𝑢)) ≤ 𝐷. In other words, 𝜙 roughly preserves distances and is roughly surjective. We simply say that that 𝑑1 and 𝑑2 are quasi-isometric if there exists 𝐶, 𝐷 > 0 such that 𝑑1 is (𝐶, 𝐷)-quasi-isometric to 𝑑2 , and it can be easily verified that this defines an equivalence relation. We naturally extend all of these definitions to graphs by identifying each graph with its graph metric. For example, the square lattice (graph) Z2 is quasi-isometric to the plane R2 (with its usual metric), and the cylinder (Z × (Z/100Z)) is quasi-isometric to the line Z. When working with infinite transitive graphs, we will not be so interested in the constants of quasiisometries, and we simply regard two quasi-isometric infinite transitive graphs as having “the same” global structure. However, this notion is not useful when working with finite transitive graphs, since 3 all finite graphs are quasi-isometric to each other. Instead, the more relevant way to measure the global similarity of two non-trivial compact metric spaces (𝑉1 , 𝑑1 ) and (𝑉2 , 𝑑2 ) (such as the graph metrics of two finite graphs) is to first normalise, i.e. consider the metrics 𝑑˜1 := diam1 𝑑1 𝑑1 on 𝑉1 and 𝑑˜2 := diam1 𝑑2 𝑑2 on 𝑑2 (where diam denotes the diameter of a metric space) then take the GromovHausdorff distance distGH ( 𝑑˜1 , 𝑑˜2 ). Recall that this Gromov-Hausdorff distance1 is defined to be the infimum 𝜀 > 0 such that there exists a third metric space (𝑉3 , 𝑑3 ) and isometric embeddings 𝜙1 : (𝑉1 , 𝑑1 ) → (𝑉3 , 𝑑3 ) and 𝜙2 : (𝑉2 , 𝑑2 ) → (𝑉3 , 𝑑3 ) such that the Hausdorff distance with respect to 𝑑3 between the images 𝜙1 (𝑉1 ) and 𝜙2 (𝑉3 ) in 𝑉3 is at most 𝜀. For example, the diameter-rescaled  2 sequence of tori 2𝜋 Z/𝑛Z : 𝑛 ≥ 1 (where, given a graph 𝐺 and a constant 𝑐 > 0, we write 𝑐𝐺 𝑛 for the graph metric on 𝐺 where all distances are scaled by 𝑐) Gromov-Hausdorff converges to the torus 𝑆 1 × 𝑆 1 with the 𝐿 1 metric. Growth rates Let G be the set of all infinite transitive graphs. An important way to begin classifying the elements of G is according to their growth rate, i.e. the asymptotic behaviour of the function Gr : N → N sending Gr : 𝑛 ↦→ |𝐵𝑛 |, where |𝐵𝑛 | denotes the number of vertices in 𝐵𝑛 . In an ideal world, we would have a classification of all possible global geometries of infinite transitive graphs, similar in spirit to the classification of all finite simple groups. For example, we might wish for a family of easy to describe infinite transitive graphs such that every infinite transitive graph is quasi-isometric to one in this list. Unfortunately, this is not the case, and no one expects that such a classification would ever be possible. However, spectacularly, there is a kind of classification if we restrict to those 𝐺 ∈ G with polynomial growth, i.e. those for which there there exist 𝐶, 𝐷 < ∞ such that Gr(𝑛) ≤ 𝐶𝑛 𝐷 for all 𝑛. Indeed, thanks to deep, classical work of Gromov and Trofimov, every infinite transitive graph of polynomial growth is quasi-isometric to the Cayley graph of a special kind of group called a nilpotent group. While we will not define what it means to be a nilpotent group here, let us simply say that it is a generalisation of being abelian, so for example, the group Z𝑑 is always nilpotent. Thanks to this so-called structure theory for infinite transitive graphs of polynomial growth, certain important results about probability on general infinite transitive graphs proceed by a structure vs expansion dichotomy: either the graph has polynomial growth, in which case we can apply this detailed structure theory, or the graph has super-polynomial growth, in which case this fast growth 1 Up to doubling 𝜀, the same notion can be expressed in terms of quasi-isometries with constants, although this latter definition is less commonly used. 4 can itself be exploited directly. At the other extreme from polynomial growth, we say that an infinite transitive graph 𝐺 has exponential growth if there exist 𝑐, 𝛿 > 0 such that Gr(𝑛) ≥ 𝑐𝑒 𝛿𝑛 for all 𝑛. For example, this includes the 𝑑-regular tree for every 𝑑 ≥ 3. The hypothesis of exponential growth can itself be helpful when studying percolation. For example, the analogue of the “notorious open problem” about critical percolation on Z3 was proved for all graphs of exponential growth. There also exist mysterious transitive graphs that have neither polynomial growth nor exponential growth, and are therefore said to have intermediate growth (e.g. the “Grigorchuk group”). Expansion One way that having a fast growth rate (“expansion” in the “structure vs expansion” dichotomy) can be exploited is as follows: for transitive graphs, a fast growth rate implies good isoperimetric properties; good isoperimetric properties imply that a random walk on the graph has good escape properties; and these escape properties can be used to prove geometric facts about a graph that serve percolation arguments. This is a common theme running through the heart of both our work on non-triviality and locality (discussed in later sections), for example. Let us be more precise about what we mean by “good isoperimetric and escape properties”. Let 𝐺 = (𝑉, 𝐸) be an infinite graph with bounded vertex degrees. Given a set of vertices 𝑆 ⊆ 𝑉, let 𝜕𝑆 be the edge boundary of 𝑆, i.e. the set of all edges having one endpoint in 𝑆 and the other in 𝑉\𝑆. Given 𝑑 ≥ 1, we say that 𝐺 satisfies a 𝑑-dimensional isoperimetric inequality if there exists 𝑐 > 0 such that for every finite set of vertices 𝑆, 𝑑−1 |𝜕𝑆| ≥ 𝑐 |𝑆| 𝑑 . Correspondingly, we define the isoperimetric dimension of 𝐺 to be Dim 𝐺 := sup {𝑑 ≥ 1 : 𝐺 satisfies a 𝑑-dimensional isoperimetric inequality} . For example, Dim Z𝑑 = 𝑑 for all 𝑑 ≥ 1. Some classical results about random walks on general graphs are that if Dim 𝐺 > 2, then simple random walk is transient (i.e. has a positive probability to never return to its starting point.), and if Dim 𝐺 > 4, then the paths of two independent simple random walks have a positive probability to never intersect. An important consequence of the aforementioned structure theory for polynomial growth is that an infinite transitive graph 𝐺 has polynomial growth if and only if Dim 𝐺 < ∞. In particular, if a graph does not have polynomial growth, then Dim 𝐺 = ∞ and hence, for example, simple random walk on 𝐺 is transient. 5 Another notion of good isoperimetry, which is stronger than having Dim 𝐺 = ∞ and which also plays an important role in percolation, is nonamenability. We say that 𝐺 is nonamenable if there exists 𝑐 > 0 such that for every finite set of vertices 𝑆, |𝜕𝑆| ≥ 𝑐 |𝑆| . This is the infinite-graph analogue of being an expander graph. A graph that is not nonamenable is said to be amenable. A fundamental result in the study of percolation on general graphs is that if 𝐺 is amenable, then P 𝑝 (𝜔 contains at least two infinite clusters) = 0 for all 𝑝 ∈ [0, 1], and a major open conjecture is that the converse is true too. 1.3 Non-triviality The first basic question when embarking on a general study of percolation on arbitrary infinite transitive graphs is to understand for which graphs percolation undergoes a non-trivial phase transition, meaning that the critical point 𝑝 𝑐 is strictly between 0 and 1. It is easy to show (exercise!) that 𝑝 𝑐 > 0, and indeed this holds for every infinite (not necessarily transitive) graph whose vertex degrees are bounded above uniformly. So the real question is to understand when 𝑝 𝑐 < 1. This question about whether percolation on a given graph undergoes a non-trivial phase transition is actually equivalent to the analogous questions for many other statistical mechanics models on the same graph, most notably for the Ising model of magnetism. General graphs Before restricting to transitive graphs, let us start by considering a more general setup. Let 𝐺 be any infinite (not necessarily transitive) graph2 . Does “𝑝 𝑐 (𝐺) < 1” have a geometric counterpart? A celebrated argument from 1930s physics gives a geometric condition P that is sufficient (but a priori not necessary) for 𝑝 𝑐 < 1, defined in terms of cutsets. Given a vertex 𝑣, we say that a set of edges Π is a cutset from 𝑣 to ∞ if 𝑣 belongs to a finite component of the graph (𝑉, 𝐸\Π). We say that a cutset Π from 𝑣 to ∞ is minimal if Π does not contain a proper subset Π′ that is also a minimal cutset from 𝑣 to ∞. Consider the exponential growth rate for the number of minimal cutsets of size 𝑛: 𝜅(𝐺) := sup sup |{Π : Π is a minimal cutset from 𝑣 to ∞ of size 𝑛}| 1/𝑛 , 𝑛≥1 𝑣∈𝑉 2 As always, we assume that 𝐺 is connected, undirected, etc. 6 and consider the following uniform analogue of 𝑝 𝑐 (𝐺): 𝑝 ∗𝑐 (𝐺) := inf{𝑝 ∈ [0, 1] : inf P 𝑝 (𝑣 is in an infinite component) > 0}. 𝑣∈𝑉 Note that in general, 𝑝 ∗𝑐 (𝐺) ≥ 𝑝 𝑐 (𝐺), and if 𝐺 happens to be transitive, then 𝑝 ∗𝑐 (𝐺) = 𝑝 𝑐 (𝐺). Now the so-called Peierls argument establishes the following: Theorem 1.3.1 (Peierls, 1936). For every infinite graph, if 𝜅 < ∞ then 𝑝 ∗𝑐 < 1. With Severo and Tassion, we recently established that the converse holds too: Theorem 1.3.2 (Easo, Severo, Tassion). For every infinite graph, 𝜅 < ∞ if (and only if) 𝑝 ∗𝑐 < 1. Transitive graphs It is easy to see that the line Z satisfies 𝑝 𝑐 = 1, and indeed, so does every infinite transitive graph that is quasi-isometric to Z. In their original work introducing percolation on general transitive graphs, Benjamini and Schramm conjectured that conversely, these are in fact the only infinite transitive graphs with 𝑝 𝑐 = 1. Conjecture 1.3.3 (Benjamini and Schramm 1996). Every infinite transitive graph that is not quasiisometric to Z satisfies 𝑝 𝑐 < 1. Babson and Benjamini then made the following (a priori) stronger conjecture, which is actually completely deterministic. Conjecture 1.3.4 (Babson and Benjamini3 1999). Every infinite transitive graph that is not quasiisometric to Z satisfies 𝜅 < ∞. Babson and Benjamini proved their own conjecture in the special case of Cayley graphs of finitely presented groups, and Timar later showed that the property “𝜅 < ∞” is invariant under quasiisometries. By the structure theory of polynomial growth, these results imply that for every infinite transitive graph 𝐺 satisfying Dim 𝐺 < ∞, if 𝐺 is not quasi-isometric to Z, then 𝜅 < ∞ and (hence) 𝑝 𝑐 < 1. In particular, to prove either of the above conjectures, it suffices to work with graphs satisfying Dim 𝐺 = ∞. In a breakthrough work, Duminil-Copin, Goswami, Raoufi, Severo, and Yadin resolved the 𝑝 𝑐 < 1 conjecture of Benjamini and Schramm. More precisely, they proved the following theorem, and 3 To be historically accurate, they made this conjecture in the case of Cayley graphs, and Benjamini later asked about arbitrary transitive graphs more generally. 7 by the previous paragraph, this was sufficient to resolve the conjecture. Their proof involved an intricate multi-scale interpolation scheme comparing probabilities of certain events for percolation to those of a percolation-like model arising from the Gaussian free field. Theorem 1.3.5 (Duminil-Copin, Goswami, Raoufi, Severo, and Yadin). Every infinite graph with Dim 𝐺 > 4 satisfies 𝑝 ∗𝑐 < 1. Our proof of Theorem 1.3.2 can be tweaked to also prove the following theorem. This yields a much simpler and shorter proof of Theorem 1.3.5 and resolves the 𝜅 < ∞ conjecture of Babson and Benjamini. It remains open to extend this result to Dim 𝐺 > 1. Theorem 1.3.6 (Easo, Severo, Tassion). Every infinite graph with Dim 𝐺 > 2 satisfies 𝜅 < ∞. 1.4 Locality As explained in the previous section, 𝑝 𝑐 is strictly between 0 and 1 for every infinite transitive graph that is not quasi-isometric to Z. A natural follow-up question is: what is the value of 𝑝 𝑐 exactly? Unfortunately, there is no reason in general to expect a simple formula for 𝑝 𝑐 , even in natural examples like the three-dimensional lattice Z3 . (There are a few spectacular exceptions to this rule.) A weaker question is: which aspect of the geometry of 𝐺 is responsible for determining the value of 𝑝 𝑐 ? Schramm’s locality conjecture asserted that when 𝑝 𝑐 is strictly between 0 and 1, the exact value of 𝑝 𝑐 should be entirely determined by the local geometry of 𝐺. More precisely, for any 𝜀 > 0, there should exist 𝑛 ≥ 1 such that for every pair of infinite transitive graphs 𝐺 and 𝐻, 𝐻 neither of which is quasi-isometric to Z, if 𝐵𝐺 𝑛 is isomorphic to 𝐵𝑛 , then | 𝑝 𝑐 (𝐺) − 𝑝 𝑐 (𝐻)| ≤ 𝜀. Equivalently, letting H denote the set of all infinite transitive graphs that are not quasi-isometric to Z, endowed with the local topology, the function 𝑝 𝑐 : H → (0, 1) should be continuous. Thanks to fundamental work of Grimmett and Marstrand (which preceded Schramm’s conjecture about general transitive graphs), this was know in the special case of Euclidean lattices. In particular, it
follows from their work that for every 𝑑 ≥ 2, the sequence of graphs Z𝑑 × (Z/𝑛Z) : 𝑛 ≥ 1 , which converges locally to Z𝑑+1 , satisfies     𝑝 𝑐 Z𝑑 × (Z/𝑛Z) → 𝑝 𝑐 Z𝑑+1 as 𝑛 → ∞. Various authors verified special cases of Schramm’s locality conjecture (and some variants to others models). Most notably, by exploiting the structure theory discussed earlier, Contreras, Martineau, and Tassion proved Schramm’s locality conjecture for all infinite transitive graphs of polynomial 8 growth. Building on this work and new ideas, Hutchcroft and I were able to resolve Schramm’s locality conjecture in its entirety. Theorem 1.4.1 (Easo and Hutchcroft). The function 𝑝 𝑐 : H → (0, 1) is continuous. This theorem, which in this formulation appears quite abstract, can in fact be reduced to a more concrete statement about propagating connnection bounds. Very roughly, it suffices to prove that the statement P (𝜀, 𝑛, 𝑝) that “every pair of vertices at distance ≤ 𝑛 apart are connected under P 𝑝 with probability at least 𝜀” satisfies a general implication of the form P (𝜀, 𝑛, 𝑝) P (𝜀′, 𝑛′, 𝑝 + 𝛿), =⇒ where 𝛿 > 0, 𝜀′ < 𝜀, 𝑛′ > 𝑛, and which is quantitatively strong, i.e. 𝛿 is small, 𝑛′ is large, and 𝜀′ is not too small. (The actual implication we end up proving is quite a bit more intricate.) To prove such an implication, we split into various cases depending on the way that the graph looks around the current scale 𝑛, exploiting a kind of “structure vs expansion” dichotomy as mentioned earlier. In one case, we apply a refinement of the structure theory for transitive graphs of polynomial growth in order to implement the methods of Contreras, Martineau, and Tassion. (In fact, we had to engage with this structure theory more substantially than did previous works on percolation, and in particular, Hutchcroft and I ended up writing a companion paper about groups of polynomial growth.) In another case, we exploit the hypothesis that |𝐵𝑛 | is large in order to implement a new percolation argument that works more efficiently when we have “more vertices packed in a smaller space”. In a third case, we use the trajectories of random walks to probabilistically prove deterministic facts about the geometry of 𝐺 around scale 𝑛. These deterministic facts then play the role that structure theory did in the first case, again allowing us to implement the methods of Contreras, Martineau, and Tassion. The two key difficulties were that (1) different transitive graphs can have very different geometric properties, e.g. a regular tree and a Euclidean lattice, and therefore require different arguments, and (2) a single transitive graph can have very different geometric properties from one scale to the next, e.g. a graph with (eventually) polynomial growth may look like a tree up to a large finite scale, so we need to thriftily exploit geometric hypotheses that can only be assumed to hold at a single scale. 1.5 Finite graphs The story of percolation theory on infinite transitive graphs is only one half of the story of this model. In 1960, roughly when Broadbent and Hammersley introduced so-called percolation 9 theory on Euclidean lattices, Erdős and Rényi introduced percolation on the complete graph 𝐾𝑛 with 𝑛 vertices. This is the well-known Erdős-Rényi model (or simply random graph) model in combinatorics. In this setting, one studies the threshold for the emergence of a giant (as opposed to infinite) cluster under percolation. The fundamental result is that percolation on 𝐾𝑛 undergoes a phase transition around 𝑝 = 𝑛1 in the sense that for every fixed 𝜀 > 0, there exists 𝛿 > 0 such that 𝑛 (the largest cluster contains ≥ 𝛿𝑛 vertices) = 1, lim P𝐾1+𝜀 𝑛→∞ 𝑛 whereas for every 𝛿 > 0, lim P𝐾1−𝑛𝜀 (the largest cluster contains ≥ 𝛿𝑛 vertices) = 0. 𝑛→∞ 𝑛 We say that the sequence (1/𝑛) 𝑛≥1 is the percolation threshold for the sequence of complete graphs. (While we call this the threshold, note that a percolation threshold is only ever unique up to multiplication by 1 + 𝑜(1).) There has since been a tremendous amount of work on this model and on percolation on the sequence of hypercubes ({0, 1}𝑛 )𝑛≥1 . Note that hypercubes and complete graphs are both examples of transitive graphs4 . We have been working to develop a theory of percolation on general finite transitive graphs that forms a bridge between these canonical finite graph models and the rich theory of percolation on infinite transitive graphs. As in the Erdős-Rényi model, in this theory, we study the phase transition for the emergence of a giant cluster. Roughly speaking, we can think of the theory of percolation on infinite transitive graphs as the theory of percolation on microscopic (i.e. 𝑂 (1)) scales in bounded-degree finite transitive graphs. In this sense, the finite graph theory generalises the infinite graph theory. (A limitation of this maxim is that not every infinite transitive graph can be locally approximated by finite transitive graphs.) In particular, certain basic questions in the finite graph theory have no natural analogues in the infinite graph theory. For example, the uniqueness/non-uniqueness of giant clusters is not directly related to the uniqueness/non-uniqueness of infinite clusters, which is instead related to the microscopic metric distortion of giant clusters. Hutchcroft and I established the basic features of the supercritical phase of percolation on finite transitive graphs. Together, we showed that the giant cluster is almost surely unique (except in the trivial cases), resolving a conjecture of Benjamini from 2001. Our argument has already been applied by others to spherical Gaussian ensembles, and with Hutchcroft, we are extending 4 There has also been a great deal of work on families of finite graphs that are not necessarily transitive, e.g. arbitrary dense graphs. 10 our arguments to establish results about the Potts model that are new even for tori. In a sequel to this work on uniqueness, with Hutchcroft we established that the density of this unique giant concentrates around some limit and fully characterised those graphs for which this limiting density is mean-field (meaning like on a tree). Our argument in this sequel relies on a new application of sharp-threshold theory, applied (unusually) to study events that obviously do not have sharp thresholds. This analysis of the supercritical phase did not address the basic question of whether, (in the language of Bollobás, Borgs, Chayes, Riordan for example), percolation on finite transitive graphs has a phase transition, meaning that a giant cluster emerges suddenly at some threshold. By combining the supercritical analysis with Vanneuville’s new proof of an old result about infinite graphs, we proved that such a phase transition does always occur (except in the trivial cases). Note that this result (and those about the supercritical phase in the previous paragraph) hold for all finite transitive graphs, with possibly diverging vertex degrees. In particular, this result recovers the classical fact that percolation on large complete graphs or hypercubes undergoes a phase transition but via very soft and general arguments. The theory of percolation on finite transitive graphs is intimately linked with the theory on infinite transitive graphs if we restrict our study to families of finite transitive graphs with uniformly bounded vertex degrees. Indeed, with respect to the local topology, every infinite set G of finite transitive graphs with bounded degrees is relatively compact, and every graph in the boundary of G is infinite. Our very recent work combined several ideas from the works discussed above about finite graphs and about Schramm’s locality conjecture to establish that in some sense the sudden emergence of a giant cluster for percolation on bounded-degree finite transitive graphs is “the same” phase transition as the usual emergence of an infinite cluster for percolation on infinite transitive graphs. More precisely, we considered the following pair of questions, which are actually equivalent: 1. Does percolation on a large bounded-degree finite transitive graph 𝐺 have a sharp phase transition? This means that in the subcritical phase of percolation, the largest cluster is with high probability not just sublinear but logarithmic in the total number of vertices in 𝐺. 2. If a finite transitive graph 𝐺 and an infinite transitive graph 𝐻 are close in the local sense, does the critical point for the emergence of a giant cluster in 𝐺 approximately coincide with the critical point for the emergence of an infinite cluster in 𝐻? 11 Unfortunately, the answer to both of these questions in general is no. For example, take the
∞ sequence Z𝑛 × Z 𝑓 (𝑛) 𝑛=1 for any 𝑓 : N → N growing fast. This sequence always converges locally to Z2 , where the critical point for the emergence of an infinite cluster is 𝑝 𝑐 = 12 . On the other hand, provided that 𝑓 grows sufficiently fast, the threshold for the emergence of a giant cluster in Z𝑛 × Z 𝑓 (𝑛) will be as in the sequence of cycles, around 𝑝 𝑐 = 1. Moreover, for percolation of any fixed parameter 𝑝 ∈ ( 12 , 1) on Z𝑛 × Z 𝑓 (𝑛) , the order of the largest cluster will then typically be much larger than logarithmic but much smaller than linear in the total number of vertices. The problem is that these graphs are long and thin, coarsely resembling long cycles. In particular, after suitably rescaling, their graph metrics (rapidly) converge in the Gromov-Hausdorff metric to the unit circle. We proved that this is in fact the only possible obstacle. This theorem provides a direct way to translate results and conjectures about percolation on general (non-one-dimensional) infinite transitive graphs into statements about percolation on general (non-one-dimensional) finite transitive graphs. 12 PAPERS INCLUDED The first paper explores the question: when does percolation on an infinite graph 𝐺 undergo a non-trivial phase transition? 1. Counting minimal cutsets and 𝑝 𝑐 < 1 Joint with Severo and Tassion The next two papers resolve Schramm’s locality conjecture that the value of the critical point for percolation on a given infinite transitive graph 𝐺 is typically entirely determined by the local (i.e. microscopic) geometry of 𝐺. The first paper gives the proof itself, and the second paper establishes a required ingredient about groups of polynomial growth. 2. The critical percolation probability is local Joint with Hutchcroft 3. Uniform finite presentation for groups of polynomial growth Joint with Hutchcroft (Published in Discrete Analysis: https://doi.org/10.19086/da.127778) The next two papers establish the basic features of the giant clusters that form in the supercritical phase of percolation on finite transitive graphs. 4. Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component Joint with Hutchcroft (Published in Duke Mathematical Journal: https://doi.org/10.1215/00127094-2023-0066) 5. Supercritical percolation on finite transitive graphs II: Concentration, locality, and equicontinuity of the giant’s density Joint with Hutchcroft The next paper establishes that percolation on a finite transitive graph typically undergoes a phase transition in the sense that a giant cluster emerges suddenly around a single critical point. 6. Existence of a percolation threshold on finite transitive graphs (Published in International Mathematics Research Notices: https://doi.org/10.1093/imrn/rnad222) The next paper establishes that this phase transition for percolation on a bounded-degree finite transitive graph 𝐺 is typically sharp in the sense that the subcritical clusters are logarithmically small. Equivalently, if 𝐺 approximates an infinite transitive graph 𝐻, then the critical point for the emergence of a giant cluster in 𝐺 approximately coincides with the critical point for the emergence of an infinite cluster in 𝐻. 7. Sharpness and locality for percolation on finite transitive graphs 14 Chapter 2 15 COUNTING MINIMAL CUTSETS AND 𝑝 𝑐 < 1 Joint work with Franco Severo and Vincent Tassion Abstract We prove two results concerning percolation on general graphs. • We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies 𝑝 𝑐 < 1, then the number of minimal cutsets of size 𝑛 separating a given vertex from infinity is bounded above exponentially in 𝑛. This resolves a conjecture of Babson and Benjamini from 1999. • We prove that 𝑝 𝑐 < 1 for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that 𝑝 𝑐 < 1 for every transitive graph of superlinear growth. 2.1 Main results Let 𝐺 = (𝑉, 𝐸) be an infinite, connected, locally finite graph. A set of edges 𝐹 ⊂ 𝐸 is called a cutset from a vertex 𝑣 to ∞ if 𝑣 belongs to a finite connected component of (𝑉, 𝐸 \ 𝐹). A cutset is called minimal if no proper subset of it is a cutset. Let Q𝑛 (𝑣) be the set of minimal cutsets from 𝑣 to ∞ of cardinality 𝑛 and consider the quantity 𝑞 𝑛 := sup |Q𝑛 (𝑣)|. (2.1.1) 𝑣∈𝑉 Here |∅| := 0. We emphasize that 𝑞 𝑛 = ∞ is possible, for example for 𝐺 = Z and 𝑛 = 2. In this paper, we are interested in cases where the number of cutsets 𝑞 𝑛 grows at most exponentially with 𝑛, and we define 𝜅(𝐺) := sup 𝑞 1/𝑛 (2.1.2) 𝑛 . 𝑛≥1 Let P 𝑝 denote (Bernoulli bond) percolation of parameter 𝑝 ∈ [0, 1] on 𝐺, where each edge is open with probability 𝑝 independently of the other edges. Consider the percolation probabilities 𝜃 𝑣 ( 𝑝) := P 𝑝 (𝑣 ↔ ∞), where 𝑣 ↔ ∞ denotes the event that 𝑣 belongs to an infinite open connected component. We define the critical parameter for uniform percolation as 𝑝 ∗𝑐 (𝐺) := inf{𝑝 ∈ [0, 1] : 𝜃 ∗ ( 𝑝) > 0}, (2.1.3) where 𝜃 ∗ ( 𝑝) := inf 𝑣∈𝑉 𝜃 𝑣 ( 𝑝). By the classical Peierls argument [Pei36a] if 𝜅(𝐺) < ∞, then percolation on 𝐺 has a uniformly percolating phase in the sense that 𝑝 ∗𝑐 (𝐺) < 1. Our first theorem establishes the converse. For every infinite, connected, locally finite graph 𝐺 we have 𝑝 ∗𝑐 (𝐺) < 1 ⇐⇒ 𝜅(𝐺) < ∞. (2.1.4) Currently, the geometric condition 𝜅(𝐺) < ∞ is not well understood. Our second result gives a sufficient condition based on the simple random walk. Given a vertex 𝑣, let P𝑣 be the law of a ∞ on 𝐺 starting at 𝑣. We say that 𝐺 is uniformly transient if simple random walk (𝑋𝑡 )𝑡=0 inf [𝑑𝑣 · P𝑣 (∀𝑡 ≥ 1 : 𝑋𝑡 ≠ 𝑣)] > 0, 𝑣∈𝑉 (2.1.5) where 𝑑𝑣 denotes the degree of 𝑣. Let 𝐺 be an infinite, connected, locally finite graph. If 𝐺 is uniformly transient, then 𝜅(𝐺) < ∞. 2.2 Consequences and comments In this section, all graphs are assumed to be infinite, connected, and locally finite. Given a set of vertices 𝑆 in a graph 𝐺 = (𝑉, 𝐸), we define the boundary 𝜕𝑆 to be the set of all edges {𝑢, 𝑣} ∈ 𝐸 Í such that 𝑢 ∈ 𝑆 but 𝑣 ∉ 𝑆, and we define the weight |𝑆| 𝐺 := 𝑢∈𝑆 𝑑𝑢 . The isoperimetric dimension of 𝐺 is given by         |𝜕𝑆| Dim(𝐺) := sup 𝑑 ≥ 1 : inf > 0 . 𝑑−1   𝑆⊆𝑉 𝑑   |𝑆| 0<|𝑆|<∞   𝐺 1. We remark that the uniform critical parameter 𝑝 ∗𝑐 (𝐺) slightly differs from the most classical (non-uniform) one given by 𝑝 𝑐 (𝐺) := inf{𝑝 ∈ [0, 1] : 𝜃 ( 𝑝) > 0}, where 𝜃 ( 𝑝) := sup𝑣∈𝑉 𝜃 𝑣 ( 𝑝). However, these notions often coincide, such as for (quasi-)transitive graphs. See the introduction of [Dum+20a] for a survey of the rich history of the “𝑝 𝑐 < 1” question and its place in statistical mechanics. Let us just recall that all of the results about percolation here can be translated into analogous statements about many other models, most notably the Ising model. 2. Duminil-Copin, Goswami, Raoufi, Severo, and Yadin proved that every quasi-transitive graph of superlinear growth satisfies 𝑝 𝑐 < 1 [Dum+20a]. This had previously been a long-standing conjecture of Benjamini and Schramm [BS96a]. In fact, the authors of [Dum+20a] established 16 that 𝑝 ∗𝑐 < 1 for every (not necessarily transitive) bounded degree graph 𝐺 satisfying Dim(𝐺) > 4, and this was known to imply the conjecture about transitive graphs by the classical works of Gromov [Gro81a] and Trofimov [Tro84a]. 3. Theorem 2.1 establishes that 𝑝 ∗𝑐 < 1 for every graph 𝐺 satisfying Dim(𝐺) > 2, since such graphs are uniformly transient (see e.g. [LP16a, Theorem 6.41]). We therefore obtain stronger results than [Dum+20a], through a completely new proof. Theorem 2.1 fully realises the idea at the heart of [Dum+20a] to exploit the transience of a simple random walk to prove 𝑝 𝑐 < 1. In particular, we resolve [Dum+20a, Problem 1.4]. 4. Our proofs of Theorems 1 and 2 can also be run on finite graphs to establish the analogous results about giant clusters. (See [HT21e] for background.) In this setting, to define 𝑞 𝑛 , one should instead count the number of minimal cutsets of cardinality 𝑛 from a vertex 𝑣 to another vertex 𝑢 (and take the supremum over all choices for distinct 𝑢 and 𝑣). The corresponding notion of uniform transience for a given family of finite graphs is that there exists a constant 𝐶 < ∞ such that every graph 𝐺 = (𝑉, 𝐸) in the family satisfies max R 𝐺 (𝑢, 𝑣) ≤ 𝐶, 𝑢,𝑣∈𝑉 where R 𝐺 (𝑢, 𝑣) denotes the effective resistance from 𝑢 to 𝑣 in the graph 𝐺. 5. Babson and Benjamini conjectured that 𝜅 < ∞ for every transitive1 graph of superlinear growth [BB99a]. Notice that this purely geometric conjecture is a priori stronger than the above 𝑝 𝑐 < 1 conjecture of Benjamini and Schramm. Babson and Benjamini verified their conjecture in the special case of Cayley graphs of finitely presented groups by establishing that minimal cutsets in such graphs are coarsely connected. By [Tim07; Gro81a; Tro84a] (see also [CMT24, Lemma 2.1]), this extends to all transitive graphs satisfying Dim(𝐺) < ∞. Given these results, it suffices to show that 𝜅 < ∞ for every transitive graph satisfying Dim(𝐺) = ∞. Theorem 2.1 therefore resolves the 𝜅 < ∞ conjecture of Babson and Benjamini. (Alternatively, taking the results of [Dum+20a] for granted, this conjecture follows from Theorem 2.1.) 6. We establish the existence of a universal constant 𝜀 > 0 such that every transitive graph 𝐺 satisfies 𝑝 𝑐 = 1 or 𝑝 𝑐 ≤ 1 − 𝜀. When 𝐺 is recurrent, this follows from the proof of [HT21e, 1 In fact, Babson and Benjamini originally made this conjecture in the case of Cayley graphs, and Benjamini later extended this conjecture to allow arbitrary transitive graphs. 17 Theorem 1.7], and when 𝐺 is transient, this follows from our proof of Theorem 2.1 because there exists a universal constant 𝑐 > 0 such that a simple random walk in any transient transitive graph has probability at least 𝑐 never to return to where it started [TT20a, Corollary 1.3]. Previous works had established this result if 𝜀 is allowed to depend on the degree of vertices in 𝐺 [HT21e, Theorem 1.7], or if we instead consider site percolation on a Cayley graph [PS23a; Lyo+23a]. By the proof of Theorem 2.1, we also obtain a universal constant 𝐾 < ∞ such that 𝜅 < 𝐾 for every transitive graph of superlinear growth. 7. Much work has been motivated by a desire to find a sharp geometric criterion for a graph 𝐺 to satisfy 𝑝 𝑐 < 1. Indeed, a well-known open conjecture of Benjamini and Schramm is that every (not necessarily transitive) graph 𝐺 with Dim(𝐺) > 1 satisfies 𝑝 𝑐 < 1 [BS96a]. We were very surprised to find that the geometric criterion 𝜅 < ∞ (which is arguably simpler and more natural than the isoperimetric criterion) is not just sharp but exact. Nevertheless, in light of Theorem 2.1 and this conjecture of Benjamini and Schramm, we encourage the reader to investigate the following: Every graph 𝐺 with Dim(𝐺) > 1 satisfies 𝜅 < ∞. The Peierls argument can be used to deduce results that are (a priori) much stronger than 𝑝 𝑐 < 1. To explore these, it helps to consider the isoperimetric profile 𝜓 of a graph 𝐺 = (𝑉, 𝐸), given by 𝜓(𝑛) := inf |𝜕𝑆| . 𝑆⊆𝑉 𝑛≤|𝑆| 𝐺 <∞ 8. Every graph 𝐺 = (𝑉, 𝐸) satisfying 𝜅 < ∞ admits a strongly percolating phase in the sense that for all 𝑝 ∈ (1 − 1/𝜅, 1], there is a constant 𝑐 > 0 such that P 𝑝 (𝑆 ↮ ∞) ≤ 𝑒 −𝑐𝜓(|𝑆|) for every finite set 𝑆 ⊆ 𝑉; P 𝑝 (𝑛 ≤ |𝐶𝑣 | < ∞) ≤ 𝑒 −𝑐𝜓(𝑛) for every 𝑛 ≥ 1 and 𝑣 ∈ 𝑉 . (2.2.1) Thus our work resolves [Dum+20a, Problem 1.6] and implies that percolation on every transitive graph of superlinear growth has a strongly percolating phase. It remains an important open problem to establish that on these graphs, such bounds hold for all 𝑝 ∈ ( 𝑝 𝑐 , 1]. Indeed, this is the “upper bound” half of [HH21b, Conjecture 5.1]. 9. Conversely, our proof of Theorem 2.1 (more precisely, Proposition 2.5.1) can be used to show that for every transitive graph 𝐺 = (𝑉, 𝐸) and for every 𝑝 > 𝑝 𝑐 , there is a constant 𝑐 > 0 such 18 that P 𝑝 (𝑛 ≤ |𝐶𝑣 | < ∞) ≥ 𝑒 −𝑐𝜓(𝑛) for every 𝑛 ≥ 1 and 𝑣 ∈ 𝑉 . This establishes the “lower bound” half of [HH21b, Conjecture 5.1]. 10. A major motivation for studying anchored isoperimetric inequalities for graphs and manifolds is the belief that — unlike (uniform) isoperimetric inequalities — anchored inequalities should typically be robust under small perturbations of the space [BLS99, Section 6]. We obtain the following concrete statement to this effect by combining Theorem 2.1 with an argument of Pete [Pet08, Theorem 4.1]: for every graph 𝐺 satisfying 𝑝 ∗𝑐 (𝐺) < 1, there exists 𝜀 > 0 such that if 𝐺 satisfies a 𝑑-dimensional anchored isoperimetric inequality for any 𝑑 ≥ 1 (or 𝑓 anchored isoperimetric inequality for any function 𝑓 ) then so does every infinite cluster formed by percolation of parameter 1 − 𝜀. 11. By combining the previous item with Theorem 2 and results of Thomassen [Tho92] and Pemantle and Peres [PP96], we deduce that for every graph 𝐺 = (𝑉, 𝐸) with Dim(𝐺) > 2, and for every probability measure 𝜇 on (0, ∞), the random weighted network (𝑉, 𝐶) with 𝐶 = (𝐶 (𝑒) : 𝑒 ∈ 𝐸) ∼ 𝜇 ⊗𝐸 is almost surely transient. (This was previously known if Dim(𝐺) > 4 [Hut23a].) 12. A standard analysis of Karger’s algorithm from computer science establishes that every finite
graph 𝐺 = (𝑉, 𝐸) with exactly 𝑛 vertices contains at most 𝑛2 minimum cuts, i.e. sets of edges 𝐹 such that (𝑉, 𝐸\𝐹) is disconnected but there is no set of edges 𝐹 ′ with |𝐹 ′ | < |𝐹 | such that (𝑉, 𝐸\𝐹 ′) is also disconnected. In the same spirit, in the present paper, we design randomized algorithms to instead count minimal cutsets. Acknowledgements: We are very grateful to Benny Sudakov for telling us about Karger’s algorithm from computer science. This seed is what prompted us to investigate probabilistic approaches to bounding the number of minimal cutsets, ultimately leading to the present work. We thank Itai Benjamini for bringing the 𝜅(𝐺) < ∞ question to our attention in the first place. PE is grateful for the hospitality provided by ETH Zurich during this project. This project was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 851565). FS was supported by the ERC grant Vortex (No 101043450). 19 2.3 Background and notation In this section, we fix 𝐺 = (𝑉, 𝐸) a locally finite, connected graph. Paths and connectivity Let 𝑆 ⊂ 𝑉, 𝑢, 𝑣 ∈ 𝑆. A path from 𝑢 to 𝑣 in 𝑆 is a finite sequence 𝛾 = (𝛾0 , 𝛾1 , . . . , 𝛾ℓ ) of distinct vertices of 𝑆 such that 𝛾0 = 𝑢, 𝛾ℓ = 𝑣 and {𝛾𝑖−1 , 𝛾𝑖 } ∈ 𝐸 for every 𝑖 ∈ {1, . . . , ℓ}. When such a path exists, we say that 𝑢 is connected to 𝑣 in 𝑆. By extension, a set 𝐴 is said to be connected to a set 𝐵 in 𝑆 if there exists a vertex of 𝐴 that is connected to a vertex of 𝐵 in 𝑆. A path from 𝑢 to ∞ in 𝑆 is an infinite sequence of distinct vertices 𝛾0 , 𝛾1 , . . . in 𝑆 such that 𝛾0 = 𝑢 and {𝛾𝑖−1 , 𝛾𝑖 } ∈ 𝐸 for every 𝑖 ∈ {1, 2, . . .}. When such a path exists, we say that 𝑢 is connected to ∞ in 𝑆. Exposed boundary Let 𝑆 ⊂ 𝑉 be a finite set. The exposed boundary of 𝑆 is the set 𝜕∞ 𝑆 of all the edges {𝑢, 𝑣} such that 𝑢 ∈ 𝑆 and 𝑣 is connected to ∞ in 𝑉 \ 𝑆. Notice that the exposed boundary is a subset of the standard boundary defined at the beginning of Section 2.2: for every finite set 𝑆 ⊂ 𝑉, we have 𝜕∞ 𝑆 ⊂ 𝜕𝑆. Percolation configurations An element 𝜔 ∈ {0, 1} 𝐸 is called a percolation configuration. Given such a configuration, an edge 𝑒 ∈ 𝐸 is said to be open if 𝜔(𝑒) = 1 and closed if 𝜔(𝑒) = 0. By extension, a path is said to be open if all its edges are open. The cluster of a vertex 𝑢 ∈ 𝑉 is the connected component of 𝑢 in the graph (𝑉, {𝑒 ∈ 𝐸 : 𝜔(𝑒) = 1}). Percolation events A measurable subset 𝐴 ⊂ {0, 1} 𝐸 is called a percolation event. Given 𝑆 ⊂ 𝑉 and 𝑢, 𝑣 ∈ 𝑆, we 𝑆 denote by 𝑢 ← → 𝑣 the event that there exists an open path from 𝑢 to 𝑣 in 𝑆, and simply write 𝑢 ↔ 𝑣 when 𝑆 = 𝑉. Finally, 𝑢 ↔ ∞ denotes the event that there exists an open path from 𝑢 to ∞ in 𝑉. Percolation measures A percolation measure on 𝐺 is a probability measure on the product space {0, 1} 𝐸 . For 𝑝 ∈ [0, 1], we denote by P 𝑝 the standard Bernoulli percolation measure, under which each edge is open with probability 𝑝 independently of the other edges. Positive association A percolation event E is called increasing if for all percolation configurations 𝜔, 𝜉 satisfying 𝜔 ≤ 𝜉 for the standard product (partial) ordering, we have 𝜔 ∈ E =⇒ 𝜉 ∈ E. Typical examples of 20 𝑆 increasing events are the connection events (such as 𝑢 ← → 𝑣) introduced above. A percolation measure P is said to be positively associated if P[E ∩ F ] ≥ P[E]P[F ] (2.3.1) for all increasing events E, F . This property is often referred to as the FKG inequality. We will use that Bernoulli percolation P 𝑝 is positively associated (for every fixed 𝑝 ∈ [0, 1]) as established by Harris [Har60]. 2.4 Exposed boundaries and cutsets In this section, we fix 𝐺 = (𝑉, 𝐸) an infinite, connected, locally finite graph. In our paper, we will use that minimal cutsets can be obtained by considering the exposed boundary of finite connected sets. In this section, we recall some well-known facts relating the two notions. The first elementary result is that the exposed boundary of a finite connected set is a minimal cutset. Lemma 2.4.1. Let 𝑆 ⊂ 𝑉 be a finite connected set. For every 𝑢 ∈ 𝑆, 𝜕∞ 𝑆 is a minimal cutset from 𝑢 to ∞. Proof. Any path from 𝑢 to ∞ in 𝑉 must traverse an edge in 𝜕∞ 𝑆 (consider the last edge traversed by this path intersecting 𝑆). Therefore, 𝜕∞ 𝑆 is a cutset from 𝑢 to ∞. To prove that it is minimal, consider an edge 𝑒 ∈ 𝜕∞ 𝑆. Since 𝑆 is connected, there exists a path from 𝑢 to an endpoint of 𝑒 in 𝑆 and by definition of the exposed boundary, there must exist a path from the other endpoint of 𝑒 to ∞ in 𝑉 \ 𝑆. The concatenation of these two paths with 𝑒 connects 𝑢 to ∞ without using any edges of 𝜕∞ 𝑆 other than 𝑒. Hence 𝜕∞ 𝑆 \ {𝑒} is not a cutset from 𝑢 to ∞. □ The second elementary result identifies the exposed boundary under some simple conditions. Lemma 2.4.2. Let 𝑢 ∈ 𝑉, let Π be a minimal cutset from 𝑢 to ∞. Let 𝐴 be the connected component of 𝑢 in (𝑉, 𝐸 \ Π) and 𝐵 = {𝑒 ∩ 𝐴, 𝑒 ∈ Π} be the set of inner vertices of Π. For every set 𝑆 of vertices, we have (𝐵 ⊂ 𝑆 ⊂ 𝐴) =⇒ (𝜕∞ 𝑆 = Π). (2.4.1) Proof. Since 𝐴 is a maximal connected set in (𝑉, 𝐸 \ Π), all the edges at the boundary of 𝐴 belong to Π, and therefore 𝜕∞ 𝐴 ⊂ 𝜕 𝐴 ⊂ Π. By Lemma 2.4.1, 𝜕∞ 𝐴 is a cutset from 𝑢 to ∞, hence, by the minimality of Π, the two inclusions above must be equalities: 𝜕∞ 𝐴 = 𝜕 𝐴 = Π. 21 (2.4.2) Now, let 𝑆 be a set satisfying 𝐵 ⊂ 𝑆 ⊂ 𝐴. Let 𝑒 ∈ Π. Since 𝑒 ∈ 𝜕∞ 𝐴, one endpoint of 𝑒 must belong to 𝐵 and the other endpoint is connected to ∞ in 𝑉 \ 𝐴. Therefore, by hypothesis, one endpoint of 𝑒 belongs to 𝑆 and the other endpoint is connected to ∞ in 𝑉 \ 𝑆. This proves the inclusion Π ⊂ 𝜕∞ 𝑆. (2.4.3) Let 𝑒 ∈ 𝜕∞ 𝑆. Let 𝑢 be the endpoint of 𝑒 in 𝑆, and let 𝑣 be the endpoint of 𝑒 connected to ∞ in 𝑉\𝑆. Then, by hypothesis, 𝑢 ∈ 𝐴 and 𝑣 is connected to ∞ in 𝑉\𝐵. Since Π = 𝜕 𝐴, every edge in Π intersects 𝐴 and hence intersects 𝐵. Therefore, there must exist an infinite path starting at 𝑣 in the subgraph (𝑉, 𝐸\Π). In particular, 𝑣 ∉ 𝐴, and hence 𝑒 ∈ 𝜕 𝐴 = Π. This proves that the inclusion above must be an equality. □ 2.5 Full connectivity via positive association In this section, we consider the following problem: Let 𝐵 be a finite set in a graph, and P be a percolation measure. What is the probability that all the vertices of 𝐵 are all connected to each other? Or, in other words, what is the probability that all the vertices of 𝐵 lie in the same cluster? We prove that this probability is at least exponential in the size of 𝐵 when the measure is positively associated, and the probability for a point to be connected to 𝐵 is uniformly lower bounded. This result, formally stated below, will allow us to construct random sets with a prescribed boundary. Proposition 2.5.1. Let 𝐺 = (𝑉, 𝐸) be a finite, connected graph. Let P be a positively associated percolation measure on 𝐺. Let 𝐵 ⊂ 𝑉, let 𝜃, 𝑝 ∈ (0, 1] and suppose that P(𝑢 ↔ 𝐵) ≥ 𝜃 for every 𝑢 ∈ 𝑉, and P(𝑒 is open) ≥ 𝑝 for every 𝑒 ∈ 𝐸. Then for every 𝑜 ∈ 𝑉, ! Ù P {𝑜 ↔ 𝑏} ≥ 𝑐 |𝐵| , 𝑏∈𝐵  where 𝑐 := 𝑝𝜃 2  3/𝜃 . Proof. Say that a finite sequence of vertices 𝑥 1 , . . . , 𝑥 𝑘 is chained if 𝑥 1 = 𝑜 and for all 𝑖 ∈ {2, . . . , 𝑘 }, 𝑝𝜃 𝜃 ≤ P (𝑥𝑖 ↔ {𝑥 1 , . . . , 𝑥𝑖−1 }) ≤ . 2 2 (P1) Since there exists at least one chained sequence (take 𝑘 = 1) and 𝑉 is finite, there must exist a chained sequence 𝑥 1 , . . . , 𝑥 𝑘 that is maximal in the sense that for every vertex 𝑥 𝑘+1 , the sequence 𝑥 1 , . . . , 𝑥 𝑘+1 is not chained. Fix a maximal chained sequence 𝑥 1 , . . . , 𝑥 𝑘 , and let 𝑋 := {𝑥 1 , . . . , 𝑥 𝑘 }. 22 We claim that, in addition to (P1), this sequence satisfies the following two properties, where 𝑛 := |𝐵|: ∀𝑢 ∈ 𝑉 𝑘≤ P(𝑢 ↔ 𝑋) ≥ 𝜃 , 2 (P2) 2𝑛 . 𝜃 (P3) To prove (P2), consider the set of vertices 𝑊 ⊂ 𝑉 that are connected to 𝑋 with probability at least 𝜃/2 and suppose for contradiction that 𝑊 ≠ 𝑉. Since 𝑊 is non empty (because 𝑋 ⊂ 𝑉) and 𝐺 is connected, we can consider an edge {𝑢, 𝑣} such that 𝑢 ∈ 𝑊 and 𝑣 ∉ 𝑊. By positive association, 𝜃/2 > P(𝑣 ↔ 𝑋) ≥ P({𝑢, 𝑣} is open) · P(𝑢 ↔ 𝑋) ≥ 𝑝𝜃 . 2 In particular, 𝑥 1 , . . . , 𝑥 𝑘 , 𝑣 is a chained sequence, contradicting the maximality of 𝑥1 , . . . , 𝑥 𝑘 . We now prove (P3). To this aim, for each 𝑖 ∈ {1, . . . , 𝑘 }, let 𝑁𝑖 denote the number of clusters that intersect both {𝑥1 , . . . , 𝑥𝑖 } and 𝐵. For every 𝑖 ∈ {2, . . . , 𝑘 }, the increment 𝑁𝑖 − 𝑁𝑖−1 is equal to 1 if 𝑥𝑖 is connected to 𝐵 but not to the previous points {𝑥 1 , . . . , 𝑥𝑖−1 }, and it is equal to 0 otherwise. Therefore, for every 𝑖 ∈ {2, . . . , 𝑘 }, we have the deterministic inequality 𝑁𝑖 − 𝑁𝑖−1 ≥ 1𝑥𝑖 ↔𝐵 − 1𝑥𝑖 ↔{𝑥1 ,...,𝑥𝑖−1 } . (2.5.1) Taking the expectation, using our hypothesis and (P1), for every 𝑖 ∈ {2, . . . , 𝑘 }, we get E(𝑁𝑖 ) − E(𝑁𝑖−1 ) ≥ P(𝑥𝑖 ↔ 𝐵) − P(𝑥𝑖 ↔ {𝑥 1 , . . . , 𝑥𝑖−1 }) ≥ 𝜃/2. | {z } | {z } ≥𝜃 (2.5.2) ≤𝜃/2 Summing over 𝑖 ∈ {2, . . . , 𝑘 } and using E(𝑁1 ) = P(𝑥 1 ↔ 𝐵) ≥ 𝜃 ≥ 𝜃/2, we get E[𝑁 𝑘 ] ≥ 2𝜃 𝑘. Since 𝑁 𝑘 is deterministically bounded above by |𝐵| = 𝑛, this concludes the proof of (P3). We now explain how the three properties above of the chained sequence imply the desired lower bound in the proposition. First, we estimate the event that all the vertices of 𝑋 are connected to 𝑜: By (P1), (P3) and positive association, we have !   𝑘−1   2𝑛𝜃 𝑘 Ù Ö 𝑝𝜃 𝑝𝜃 P {𝑜 ↔ 𝑢} ≥ P (𝑥𝑖 ↔ {𝑥 1 , . . . , 𝑥𝑖−1 }) ≥ ≥ . (2.5.3) 2 2 𝑢∈𝑋 𝑖=2 Second, we estimate the event that all the vertices of 𝐵 are connected to 𝑋: By (P2) and positive association, we have !   𝑛 Ù 𝜃 . P {𝑏 ↔ 𝑋 } ≥ 2 𝑏∈𝐵 23 If all the vertices of 𝑋 are connected to 𝑜 and all the vertices of 𝐵 are connected to 𝑋, then all the vertices of 𝐵 are connected to 𝑜. Hence, by the two displayed equations above and positive association, we obtain !   2𝑛   ! ! 𝑛 Ù Ù Ù 𝑝𝜃 𝜃 𝜃 {𝑏 ↔ 𝑋 } ≥ {𝑜 ↔ 𝑏} ≥ P {𝑜 ↔ 𝑢} · P · P ≥ 𝑐𝑛 , 2 2 𝑢∈𝑋 𝑏∈𝐵 𝑏∈𝐵  where 𝑐 := 2.6 𝑝𝜃 2  3/𝜃 . □ Proof of Theorem 2.1 Let 𝐺 = (𝑉, 𝐸) be an infinite, connected, locally finite graph. In this section, we prove Theorem 2.1, in the following form. ∃𝑝 < 1 ∃𝜃 > 0 ∀𝑢 ∈ 𝑉 P 𝑝 (𝑢 ↔ ∞) ≥ 𝜃
⇐⇒ ∃𝐾 < ∞ ∀𝑢 ∈ 𝑉 ∀𝑛 ≥ 1
|Q𝑛 (𝑢)| ≤ 𝐾 𝑛 . (2.6.1) The implication ⇐ is well-known, and follows from the Peierls argument [Ben13a, Theorem 4.11], which we now recall for completeness. Let 𝑢 ∈ 𝑉. If the cluster of 𝑢 is finite, then by Lemma 2.4.1, its exposed boundary is a finite minimal cutset from 𝑢 to ∞, and all its edges are closed. Hence, by the union bound, for every 𝑝 ∈ [0, 1] we have ∑︁ P 𝑝 (|𝐶𝑢 | < ∞) ≤ 𝑞 𝑛 (1 − 𝑝) 𝑛 . (2.6.2) 𝑛≥1 If 𝑞 𝑛 ≤ 𝐾 𝑛 for some constant 𝐾 < ∞, then the right hand side above converges to 0 as 𝑝 tends to 1. Since the bound is uniform in 𝑢, there exists 𝑝 < 1 such that ∀𝑢 ∈ 𝑉 P 𝑝 (𝑢 ↔ ∞) ≥ 1/2. (2.6.3) We now prove the implication ⇒. Fix 𝜃, 𝑝 ∈ (0, 1) such that P 𝑝 (𝑢 ↔ ∞) ≥ 𝜃 for every 𝑢 ∈ 𝑉. Fix 𝑜 ∈ 𝑉 and 𝑛 ≥ 1. Writing 𝐶 for the cluster of 𝑜, we show that for every minimal cutset Π from 𝑜 to ∞ with |Π| = 𝑛, P 𝑝 (𝜕∞𝐶 = Π) ≥ 1/𝐾 𝑛 , (2.6.4) where 𝐾 = 𝐾 ( 𝑝, 𝜃) ∈ (0, ∞) is a finite constant depending on 𝑝 and 𝜃 only (in particular it does not depend on the chosen vertex 𝑜). This concludes the proof since 1≥ ∑︁ P 𝑝 (𝜕∞𝐶 = Π) Π∈Q𝑛 (𝑜) 24 (2.6.4) ≥ |Q𝑛 (𝑜)|/𝐾 𝑛 . (2.6.5) Let us now prove the lower bound (2.6.4). As in Lemma 2.4.2, let 𝐴 be the connected component of 𝑜 in (𝑉, 𝐸 \ Π) and 𝐵 the set of inner vertices of Π. Since any infinite open path from a vertex 𝑢 ∈ 𝐴 must intersect 𝐵 before exiting 𝐴, the hypothesis P 𝑝 (𝑢 ↔ ∞) ≥ 𝜃 implies ∀𝑢 ∈ 𝐴 𝐴 P 𝑝 (𝑢 ← → 𝐵) ≥ 𝜃. (2.6.6) Let E be the event that every vertex in 𝐵 is connected to 𝑜 by an open path in 𝐴. By Proposition 2.5.1 applied to the finite subgraph of 𝐺 induced by 𝐴, we have P 𝑝 (E) ≥ 𝑐 𝑛 , where 𝑐 = ( 𝑝𝜃/2) 3/𝜃 > 0. Let F be the event that all the edges of Π are closed. By independence, we have P 𝑝 (E ∩ F ) = P 𝑝 (E)P 𝑝 (F ) ≥ 𝑐 𝑛 (1 − 𝑝) 𝑛 . (2.6.7) If the event E ∩ F occurs, then the cluster 𝐶 of 𝑜 satisfies 𝐵 ⊂ 𝐶 ⊂ 𝐴. Hence, by Lemma 2.4.2 we must have 𝜕∞𝐶 = Π. This concludes that P 𝑝 (𝜕∞𝐶 = Π) ≥ P 𝑝 (E ∩ F ) ≥ 𝑐 𝑛 (1 − 𝑝) 𝑛 , (2.6.8) 1 1 = ( 𝑝𝜃/2) 3/𝜃 . which establishes the desired lower bound (2.6.4) with 𝐾 = 𝑐(1−𝑝) (1−𝑝) 2.7 A covering lemma for Markov chains In this section, we give conditions under which a killed Markov chain survives long enough to visit every state and then return to its initial state2 . We will apply this in the next section to prove Theorem 2.1. Here [𝑛] denotes the set {1, . . . , 𝑛}. Lemma 2.7.1. Let 𝑛 ≥ 1. Let 𝑃 = ( 𝑝𝑖, 𝑗 )𝑖, 𝑗 ∈[𝑛] be a symmetric matrix of non-negative entries such Í that3 𝑗 ∈[𝑛] 𝑝(𝑖, 𝑗) ≤ 1 for all 𝑖 ∈ [𝑛]. Let Γ be the set of all sequences 𝛾 = (𝛾0 , 𝛾1 , . . . , 𝛾 𝑘 ) in [𝑛] (for any 𝑘 ≥ 1) with 𝛾0 = 1 such that the unique element 𝑖 ∈ [𝑘] satisfying both 𝛾𝑖 = 1 and {𝛾0 , 𝛾1 , . . . , 𝛾𝑖 } = [𝑛] is 𝑖 = 𝑘. For every such sequence 𝛾, define 𝑝(𝛾) := 𝑘 Ö 𝑝 (𝛾𝑖−1 , 𝛾𝑖 ) . 𝑖=1 For each 𝜀 > 0, if every non-empty proper subset 𝐼 of [𝑛] satisfies ∑︁ ∑︁ 𝑝(𝑖, 𝑗) ≥ 𝜀, (2.7.1) 𝑖∈𝐼 𝑗 ∈[𝑛]\𝐼 2 In fact, we lower bound the probability that this occurs in ≤ 2𝑛 − 2 steps (which is optimal), where 𝑛 is the number of states. Contrast this with [BGM13; DK21], both called Linear cover time is exponential unlikely; we give conditions under which linear cover time is exponentially likely. 3 We can think of 𝑃 as the transition matrix of a Markov chain which is killed at 𝑖 with probability 1 − Í 𝑝(𝑖, 𝑗). 𝑗 25 2 𝜀 then 𝛿 := 16𝑒 2 satisfies ∑︁ 𝑝(𝛾) ≥ 𝛿 𝑛 . 𝛾∈Γ Proof. Let 𝑒 1 , . . . , 𝑒 2𝑛−2 ∈ [𝑛] 2 ⊔ {∅} be an iid sequence of random variables such that for all 𝑢, 𝑣 ∈ [𝑛], 𝑝(𝑢, 𝑣) P (𝑒 1 = (𝑢, 𝑣)) = . 𝑛 Such random variables exist because these probabilities sum to at most 1. Let 𝐻 be the undirected multigraph with vertex set [𝑛] and edges 𝑒 1 , . . . , 𝑒 2𝑛−2 . Even though [𝑛] 2 consists of ordered pairs, we think of each 𝑒𝑖 ∈ [𝑛] 2 as encoding an undirected edge, loops allowed. (When 𝑒𝑖 = ∅, we simply do not include an edge.) Consider the iid spanning subgraphs 𝐻1 and 𝐻2 of 𝐻 that contain only the edges 𝑒 1 , . . . , 𝑒 𝑛−1 and 𝑒 𝑛 , . . . , 𝑒 2𝑛−2 , respectively. We will lower bound the probability that each of these graphs is connected. Consider any 𝑘 ∈ [𝑛 − 1]. Suppose that we are given all of the connected components 𝐶1 , . . . , 𝐶𝑟 of the spanning subgraph of 𝐻 that contains only the edges 𝑒 1 , . . . , 𝑒 𝑘−1 . If 𝑟 ≥ 2, then the conditional probability that 𝑒 𝑘 connects two of these components is 𝑟 ∑︁ ∑︁ ∑︁ 𝑧=1 𝑖∈𝐶𝑧 𝑗 ∈[𝑛]\𝐶 𝑧 𝑝(𝑖, 𝑗) (2.7.1) 𝑟𝜀 . ≥ 𝑛 𝑛 𝑛 Therefore by induction on 𝑘, and by using the elementary bound 𝑛𝑛! ≤ 𝑒 𝑛 in the third inequality, 𝑛 Ö 𝑟𝜀 𝑛! · 𝜀 𝑛 𝜀𝑛 P (𝐻1 is connected) ≥ ≥ ≥ 𝑛. 𝑛 𝑛𝑛 𝑒 𝑟=2 (2.7.2) Let 𝛾 = (𝛾0 , 𝛾1 , . . . , 𝛾 𝑘 ) be a sequence in Γ. Say that 𝛾 is present if there exists an injection 𝜎 : [𝑘] → [2𝑛 − 2] such that for every 𝑖 ∈ [𝑘], we have 𝑒 𝜎(𝑖) = (𝛾𝑖−1 , 𝛾𝑖 ) or (𝛾𝑖 , 𝛾𝑖−1 ). Assume that 𝑘 ≤ 2𝑛 − 2, and note that 𝛾 cannot be present otherwise. There are at most (2𝑛 − 2) 𝑘 choices of 𝜎, and given 𝜎, for each 𝑖, the probability that 𝑒 𝜎(𝑖) = (𝛾𝑖−1 , 𝛾𝑖 ) is the same as the probability that 𝑒 𝜎(𝑖) = (𝛾𝑖 , 𝛾𝑖−1 ), both given by 𝑛1 𝑝(𝛾𝑖−1 , 𝛾𝑖 ) = 𝑛1 𝑝(𝛾𝑖 , 𝛾𝑖−1 ). So by a union bound, P (𝛾 is present) ≤ (2𝑛 − 2) 𝑘 𝑘 Ö 2 𝑛 𝑖=1 𝑝 (𝛾𝑖−1 , 𝛾𝑖 ) ≤ 4 𝑘 𝑝(𝛾) ≤ 42𝑛 𝑝(𝛾). (2.7.3) On the other hand, when 𝐻1 is connected and 𝐻2 is connected, then some 𝛾 ∈ Γ must be present in 𝐻 because every multigraph that contains two edge-disjoint spanning trees must also contain a 26 spanning subgraph that is connected and Eulerian [Cat92]4 . Thanks to (2.7.2), this occurs with probability at least 𝜀 2𝑛 /𝑒 2𝑛 . So by a union bound, ∑︁ 𝜀 2𝑛 ∑︁ 2𝑛 𝑝(𝛾). ≤ P(𝛾 is present) ≤ 4 𝑒 2𝑛 𝛾∈Γ 𝛾∈Γ The conclusion follows by rearranging. 2.8 (2.7.4) □ Proof of Theorem 2.1 Let 𝐺 = (𝑉, 𝐸) be an infinite, connected, locally finite graph such that for some constant 𝜀 > 0, for ∞ on 𝐺 started at 𝑣 satisfies every vertex 𝑣 ∈ 𝑉, the simple random walk (𝑋𝑡 )𝑡=0 𝑑𝑣 P𝑣 (∀𝑡 ≥ 1 : 𝑋𝑡 ≠ 𝑣) ≥ 𝜀. Let 𝐺 ′ = (𝑉 ′, 𝐸 ′) be the graph5 obtained from 𝐺 by replacing each edge by a path of length 2. View 𝑉 as a subset of 𝑉 ′, and let 𝑚 : 𝐸 → 𝑉 ′ map each edge to its midpoint. Let P′𝑢 be the law of simple random walk in 𝐺 ′ started from a given vertex 𝑢, and let 𝜏 := sup{𝑡 ≥ 0 : 𝑋𝑡 = 𝑋0 }. We claim that for all 𝑧 ∈ 𝑉 ′, 2𝜀 𝑑 𝑧 P′𝑧 (𝜏 = 0) ≥ 𝜀1 := . (2.8.1) 4+𝜀 This is trivial when 𝑧 ∈ 𝑉, even with 𝜀1 = 𝜀/2, because simple random walk on 𝐺 ′ induces lazy simple random walk on 𝐺. Otherwise, when 𝑧 = 𝑚({𝑢, 𝑣}) for some {𝑢, 𝑣} ∈ 𝐸, this follows from the corresponding bounds for 𝑢 and 𝑣 by rearranging the following elementary calculation, where ℓ𝑥 := |{𝑡 ≥ 0 : 𝑋𝑡 = 𝑥}|: ∑︁ ∑︁ 1 ′ 𝑛 ′ = P𝑧 (𝜏 > 0) = E𝑧 [ℓ𝑧 ] = P′𝑧 (𝑋𝑡 = 𝑧) ′ P𝑧 (𝜏 = 0) 𝑛≥0 𝑡≥0  ∑︁  1 1 ′ ′ =1+ P𝑧 (𝑋𝑡−1 = 𝑢) · + P𝑧 (𝑋𝑡−1 = 𝑣) · 𝑑𝑢 𝑑𝑣 𝑡≥1 E′𝑧 [ℓ𝑢 ] E′𝑧 [ℓ𝑣 ] + 𝑑𝑢 𝑑𝑣 ′ ′ E [ℓ𝑢 ] E𝑣 [ℓ𝑣 ] 1 1 ≤ 1+ 𝑢 + =1+ + . ′ ′ 𝑑𝑢 𝑑𝑣 𝑑𝑢 P𝑢 (𝜏 = 0) 𝑑𝑣 P𝑣 (𝜏 = 0) =1+ 4 This general fact can be proved directly as follows: Let 𝐸 be the set of edges in the multigraph, and let 𝑇 and 𝑇 1 2 be the two trees. Let 𝑜1 , . . . , 𝑜 𝑘 be the vertices that have odd degree in 𝑇1 . Since the sum of the degrees of all of the vertices in a given graph is always even, we can write 𝑘 = 2𝑙 for some non-negative integer 𝑙. For each 𝑖 ∈ [𝑙], pick a path 𝑃𝑖 in 𝑇2 from 𝑜2𝑖−1 to 𝑜2𝑖 . Then, viewing 𝑇1 and each 𝑃𝑖 as elements of the Z/2Z-vector space {0, 1} 𝐸 , the required subgraph is given by 𝑇1 + 𝑃1 + · · · + 𝑃𝑙 . 5 This construction is a technicality that is only necessary if 𝐺 has unbounded vertex degrees. 27 Let 𝐶 := {𝑋𝑡 : 0 ≤ 𝑡 ≤ 𝜏} and 𝜕 := {𝑒 ∩ 𝐶 : 𝑒 ∈ 𝜕∞𝐶}. Fix 𝑜 ∈ 𝑉, and pick a neighbour 𝑜′ ∈ 𝑚(𝐸) of 𝑜 in 𝐺 ′. Fix a finite minimal cutset Π from 𝑜 to ∞ in 𝐺, and set 𝑛 := |Π|. We will show that for some finite constant 𝐾 = 𝐾 (𝜀) ∈ (0, ∞) depending only on 𝜀, P′𝑜′ (𝜕 = 𝑚(Π)) ≥ 1/𝐾 𝑛 . (2.8.2) This implies that 𝜅(𝐺) < ∞ because for all 𝑜 ∈ 𝑉 and 𝑛 ≥ 1, 1≥ ∑︁ (2.8.2) P′𝑜′ (𝜕 = 𝑚(Π)) ≥ |Q𝑛 (𝑜)| /𝐾 𝑛 . Π∈Q𝑛 (𝑜) Let 𝐴 be the connected component of 𝑜 in (𝑉, 𝐸\Π), let 𝑈 := 𝑚(Π) ∪ {𝑜′ }, and let 𝐼 := 𝐴 ∪ 𝑚({𝑒 ∈ 𝐸 : 𝑒 ⊂ 𝐴}). For all 𝑢, 𝑣 ∈ 𝑈 ∪ 𝐼, let 𝑝(𝑢, 𝑣) := P′𝑢 (∃𝑡 ≥ 1 : 𝑋1 , . . . , 𝑋𝑡−1 ∈ 𝐼\{𝑢} and 𝑋𝑡 = 𝑣) . Í Extend this to sets of vertices by 𝑝(𝐿, 𝑅) := 𝑢∈𝐿;𝑣∈𝑅 𝑝(𝑢, 𝑣), and similarly, 𝑝(𝑢, 𝐿) := 𝑝({𝑢}, 𝐿) and 𝑝(𝐿, 𝑢) := 𝑝(𝐿, {𝑢}). We would like to apply Lemma 2.7.1 to the matrix 𝑃 := ( 𝑝(𝑢, 𝑣))𝑢,𝑣∈𝑈 . By time-reversing trajectories, we have 𝑝(𝑢, 𝑣) = 𝑝(𝑣, 𝑢) whenever 𝑑𝑢 = 𝑑𝑣 , which is for example the case when 𝑢, 𝑣 ∈ 𝑈. So 𝑃 is symmetric, and clearly the entries of 𝑃 are non-negative and sum to at most 1 along each row. We claim that for every non-trivial partition 𝑈 = 𝐿 ⊔ 𝑅, 𝑝(𝐿, 𝑅) ≥ 𝜀2 := 𝜀12 /64. (2.8.3) Indeed, for each 𝑥 ∈ 𝑈 ∪ 𝐼, consider the function (the unit voltage) 𝐹 (𝑥) := P′𝑥 (∃𝑡 ≥ 0 : 𝑋0 , . . . , 𝑋𝑡−1 ∉ 𝐿 and 𝑋𝑡 ∈ 𝑅) . Given 𝑢 ∈ 𝐿, there exists6 𝑥 ∈ 𝐴 such that {𝑢, 𝑥} ∈ 𝐸 ′, and if 𝐹 (𝑥) ≥ 1/2, then we are done because 𝑝(𝑢, 𝑅) ≥ P′𝑢 (𝑋1 = 𝑥) · 𝐹 (𝑥) ≥ 1/2 · 1/2 ≥ 𝜀2 . In particular, we may assume that there exists 𝑥 ∈ 𝐴 with 𝐹 (𝑥) < 1/2. By a similar argument, we may assume that there exists 𝑦 ∈ 𝐴 with 𝐹 (𝑦) > 1/2. Since 𝐴 is connected in 𝐺, we can therefore find {𝑥 ′, 𝑦′ } ∈ 𝐸 satisfying 𝐹 (𝑥 ′) ≤ 1/2 ≤ 𝐹 (𝑦′). Let 𝑧 := 𝑚({𝑥 ′, 𝑦′ }), which has degree 2. Note that 𝐹 (𝑧) ≥ P′𝑧 (𝑋1 = 𝑦′) · 𝐹 (𝑦′) ≥ 1/2 · 1/2, 6 If 𝑢 = 𝑜 ′ , take 𝑥 := 𝑜. If 𝑢 = 𝑚(𝑒) where 𝑒 ∈ Π, take 𝑥 where {𝑥} = 𝑒 ∩ 𝐴, which exists by Lemma 2.4.2. 28 and by a union bound, 𝐹 (𝑧) ≤ ∞ ∑︁ P′𝑧 (𝜏 > 0) 𝑛 𝑝(𝑧, 𝑅) = 𝑛=0 𝑝(𝑧, 𝑅) (2.8.1) 𝑝(𝑧, 𝑅) . ≤ P′𝑧 (𝜏 = 0) 𝜀1 /2 So by rearranging, 𝑝(𝑧, 𝑅) ≥ 𝜀1 /8. By a similar argument (i.e. by replacing 𝐹 by 1 − 𝐹, which switches the roles of 𝐿 and 𝑅, and by recalling that 𝑝(𝐿, 𝑧) = 𝑝(𝑧, 𝐿)), we deduce that 𝑝(𝐿, 𝑧) ≥ 𝜀1 /8. Now (2.8.3) follows because 𝑝(𝐿, 𝑅) ≥ 𝑝(𝐿, 𝑧) 𝑝(𝑧, 𝑅). Therefore by Lemma 2.7.1, the event E that the random walk visits every vertex in 𝑈 then returns to 𝑜′ before exiting 𝑈 ∪ 𝐼 satisfies P′𝑜′ (E) ≥ 𝜀3|𝑈| ≥ 𝜀3𝑛+1 (2.8.4) for some constant 𝜀3 > 0 depending only on 𝜀2 . So by Lemma 2.4.2 and the strong Markov property, P′𝑜′ (𝜕 = 𝑚 (Π)) ≥ P′𝑜′ (E) · P′𝑜′ (𝜏 = 0) (2.8.1),(2.8.4) ≥ 𝜀3𝑛+1 · 𝜀1 /2. (2.8.5) By expanding the definitions of 𝜀1 , 𝜀2 , 𝜀3 we deduce that (2.8.2) holds with 𝐾 := 220 /𝜀 5 . 2.9 Alternative proof of Theorem 2.1 using the Gaussian free field Here we sketch an alternative, slightly less elementary proof of Theorem 2.1 along the lines of the proof of Theorem 2.1. Let 𝐺 be an infinite, connected, locally finite graph that is uniformly ˜ 𝐸) ˜ obtained by replacing each edge by a path of length transient. Consider the graph 𝐺˜ = (𝑉, 3. Similarly to the proof in Section 2.8, one can prove that 𝐺˜ is also uniformly transient. Let ˜ 𝜑 ∈ R𝑉 with law P be the (centered) Gaussian free field (GFF) on 𝐺˜ – see e.g. [BP24] for the required background and definitions. Uniform transience implies that there exists 𝜀 > 0 such that ˜ Var(𝜑(𝑥)) ≤ 1/𝜀 for every 𝑥 ∈ 𝑉. Fix 𝑜 ∈ 𝑉 and let 𝐶˜ be the cluster of 𝑜 in the percolation model induced by the excursion set {𝜑 ≥ 0} := {𝑥 ∈ 𝑉˜ : 𝜑(𝑥) ≥ 0}. Given every edge 𝑒 of 𝐺, we associate the corresponding ˜ with both endpoints of degree 2. For a subset Π of edges in 𝐺, we denote by Π̃ mid-edge 𝑒˜ in 𝐺, ˜ We claim that there exists 𝑐 = 𝑐(𝜀) > 0, depending only on the associated set of mid-edges in 𝐺. 𝜀, such that for every Π ∈ Q𝑛 (𝑜), P(𝜕∞𝐶˜ = Π̃) ≥ 𝑐 𝑛 . (2.9.1) Similarly to the previous sections, Theorem 2.1 follows readily from (2.9.1). We now proceed to prove (2.9.1). Enumerate Π̃ by 𝑒˜𝑖 = {𝑥𝑖 , 𝑦𝑖 }, 1 ≤ 𝑖 ≤ 𝑛, where 𝑥𝑖 and 𝑦𝑖 are the inner and outer endpoints, respectively. We first observe that, for some constant 𝑐 1 = 𝑐 1 (𝜀) > 0, P(𝜑(𝑦𝑖 ) ∈ [−2, −1] and 𝜑(𝑥𝑖 ) ∈ [1, 2] ∀ 0 ≤ 𝑖 ≤ 𝑛) ≥ 𝑐 𝑛1 . 29 (2.9.2) Indeed, this follows by successively demanding the desired event at each vertex. Here we use the Markov property of the GFF (see [BP24]) and the fact that the conditional variance of the next vertex given the previous ones is between 1/2 (since they have degree 2) and 1/𝜀, while the conditional mean remains bounded between −2 and 2. ˜ 𝐸˜ \ Π̃). Notice that Let F be the event in (2.9.2) and 𝐴 be the component of 𝑜 in (𝑉, ! ! 𝑛 𝑛 Ù Ù {𝜑≥0}∩𝐴 {𝜑≥0}∩𝐴 P(𝜕∞𝐶˜ = Π̃) ≥ P(F )P {0 ←−−−−−→ 𝑥𝑖 } F ≥ 𝑐 𝑛 P {0 ←−−−−−→ 𝑥𝑖 } F . 1 𝑖=1 𝑖=1 By the Markov property, conditionally on F , the process {𝜑 ≥ 0} ∩ 𝐴 stochastically dominates {𝜑 𝐴 ≥ −1}, where 𝜑 𝐴 is the centered GFF on 𝐴 (i.e. associated to the random walk on 𝐴 killed when reaching 𝜕 𝐴 = {𝑥 1 , . . . , 𝑥 𝑛 }). Therefore, it is enough to prove that, for some constant 𝑐 2 = 𝑐 2 (𝜀) > 0, ! 𝑛 Ù {𝜑 𝐴 ≥−1} P {𝑜 ←−−−−−→ 𝑥𝑖 } ≥ 𝑐 𝑛2 . (2.9.3) 𝑖=1 Indeed, since the GFF is positively associated (see [BP24]), the desired inequality (2.9.3) follows readily from Proposition 2.5.1 and the following inequality {𝜑 𝐴 ≥−1} P(𝑢 ←−−−−−→ 𝜕 𝐴) ≥ E(sgn(𝜑 𝐴 (𝑢) + 1)) ≥ 𝑐 3 , (2.9.4) for some constant 𝑐 3 = 𝑐 3 (𝜀) > 0. The latter follows easily from the Markov property of the GFF. Indeed, let S be the union of all clusters of {𝜑 𝐴 ≥ −1} intersecting 𝜕 𝐴 and note that its closure S (i.e. the union of 𝑆 with its neighbours) is a stopping set. Clearly, one has sgn(𝜑 𝐴 (𝑢) + 1) = 1 {𝜑 𝐴 ≥−1} almost surely on the event G := {𝑢 ←−−−−−→ 𝜕 𝐴} = {𝑢 ∈ S}. On the complementary event G 𝑐 and conditionally on the field on S, the Markov property implies that we have a GFF on 𝐴 \ S with boundary conditions < −1. In particular, sgn(𝜑 𝐴 (𝑢) + 1) has a negative conditional expectation on G 𝑐 . These observations readily imply the first inequality of (2.9.4). The second inequality follows from the fact that the variance of 𝜑 𝐴 (𝑢) is at most 1/𝜀. 30 Chapter 3 31 THE CRITICAL PERCOLATION PROBABILITY IS LOCAL Joint work with Tom Hutchcroft Abstract We prove Schramm’s locality conjecture for Bernoulli bond percolation on transitive graphs: If (𝐺 𝑛 )𝑛≥1 is a sequence of infinite vertex-transitive graphs converging locally to a vertex-transitive graph 𝐺 and 𝑝 𝑐 (𝐺 𝑛 ) ≠ 1 for every 𝑛 ≥ 1 then lim𝑛→∞ 𝑝 𝑐 (𝐺 𝑛 ) = 𝑝 𝑐 (𝐺). Equivalently, the critical probability 𝑝 𝑐 defines a continuous function on the space G ∗ of infinite vertex-transitive graphs that are not one-dimensional. As a corollary of the proof, we obtain a new proof that 𝑝 𝑐 (𝐺) < 1 for every infinite vertex-transitive graph that is not one-dimensional. 3.1 Introduction In Bernoulli bond percolation, the edges of a connected, locally finite graph 𝐺 are chosen to be either retained (open) or deleted (closed) independently at random, with probability 𝑝 ∈ [0, 1] of retention. The law of the resulting random subgraph is denoted P 𝑝 = P𝐺𝑝 . Percolation theory is concerned primarily with the geometry of the connected components of this random subgraph, which are known as clusters. Much of the interest in the model arises from the fact that it undergoes a phase transition: If we define the critical probability  𝑝 𝑐 (𝐺) = inf 𝑝 ∈ [0, 1] : there exists an infinite cluster P 𝑝 -almost surely then “most” interesting graphs have 𝑝 𝑐 (𝐺) strictly between 0 and 1, so that there is a non-trivial phase where infinite clusters do not exist followed by non-trivial phase where they do exist. We will be primarily interested in percolation on (vertex-)transitive graphs, i.e., graphs for which any vertex can be mapped to any other vertex by an automorphism of the graph. We allow our graphs to contain loops and multiple edges, and make the implicit assumption throughout the paper that all transitive graphs are connected and locally finite (i.e. have finite vertex degrees). Many interesting features of percolation on an infinite transitive graph at and near the critical point are expected to be universal, meaning that they depend only on the graph’s large-scale geometry and not its microscopic structure. For example, the critical exponents governing the power-law behaviour of various interesting quantities at and near criticality are believed to depend only on the volume-growth dimension of the graph, and should therefore take the same values on e.g. the square and triangular lattice. In contrast, Schramm conjectured around 2008 [BNP11a, Conjecture 1.2] that the value of the critical probability 𝑝 𝑐 (𝐺) should be entirely determined by the local (microscopic) geometry of the graph, subject to the global constraint that 𝑝 𝑐 (𝐺) < 1. More precisely, he conjectured that if 𝐺 𝑛 is a sequence of infinite transitive graphs converging to an infinite transitive graph 𝐺 in the local topology (defined below) and lim sup𝑛→∞ 𝑝 𝑐 (𝐺 𝑛 ) < 1 then 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺) as 𝑛 → ∞. The assumption that lim sup𝑛→∞ 𝑝 𝑐 (𝐺) < 1 is needed to rule out degenerate one-dimensional examples such as the cylinder Z × (Z/𝑛Z) (which converges to the square grid Z2 but which has 𝑝 𝑐 (Z × (Z/𝑛Z)) = 1 ↛ 𝑝 𝑐 (Z2 ) = 1/2), and is now known to be equivalent to the graphs 𝐺 𝑛 having superlinear volume growth for all sufficiently large 𝑛 [DGRSY20; HT21a]. The fact that 𝑝 𝑐 is lower semi-continuous follows straightforwardly from standard facts about percolation on transitive graphs as observed in [Pet, §14.2] and [DT16a, p.4] and does not require the assumption that the graphs are not one-dimensional; the difficult part of the conjecture is to prove upper semi-continuity. The locality conjecture has inspired a great deal of subsequent work, including both partial progress on the original conjecture [BNP11a; MT17; Hut20a; HH21a; CMT23a], which we review in detail below, and analogous results in other settings including self-avoiding walk [GL17; GL18], the random cluster model [DT19], finite random graphs [BNP11a; KLS20; Sar21a; Hof21; RS22a; ABS22; BZ23; ABS23], and geometric random graphs [HM22; LLMS23]. In this paper we give a complete proof of Schramm’s locality conjecture. Theorem 3.1.1. Let G ∗ be the set of all infinite transitive graphs that are not one-dimensional, endowed with the local topology. Then the function 𝑝 𝑐 : G ∗ → (0, 1) is continuous. Here, the local topology (a.k.a. the Benjamini-Schramm topology) on the space of transitive ∞ converges to 𝐺 if and only if, for each 𝑟 ≥ 1, graphs, denoted by G, is defined so that (𝐺 𝑛 )𝑛=1 the balls of radius 𝑟 in 𝐺 𝑛 and 𝐺 are isomorphic as rooted graphs for all sufficiently large 𝑛. We say that an infinite transitive graph is one-dimensional if it has linear volume growth (i.e., if its balls 𝐵𝑛 satisfy |𝐵𝑛 | = 𝑂 (𝑛) as 𝑛 → ∞); it follows from (a simple special case of) the structure theory of transitive graphs of polynomial growth that an infinite transitive graph is one-dimensional if and only if it is rough-isometric to Z [TY16], while the results of [DGRSY20] imply that an infinite transitive graph has 𝑝 𝑐 < 1 if and only if it is not one-dimensional. (In fact the proof of Theorem 8.1.1 also yields a new proof of this theorem as we discuss in detail in Section 3.3.) 32 Remark 3.1.1. In our forthcoming paper [EH23+a], we prove related results on the locality of the density of the infinite cluster, implying in particular that the percolation probability 𝜃 ( 𝑝, 𝐺) = P𝐺𝑝 (𝑜 ↔ ∞) is a continuous function of (𝐺, 𝑝) in the supercritical set {(𝐺, 𝑝) : 𝐺 ∈ G ∗ , 𝑝 > 𝑝 𝑐 (𝐺)}. (Theorem 8.1.1 implies that this set is open.) An alternative proof of this result using the methods developed in the present paper is sketched in Section 3.7. Previous work In this section we overview previous work on locality, the 𝑝 𝑐 < 1 problem, and the structure theory of transitive graphs of polynomial growth. Our proof will employ many ideas and methods from these earlier papers, including most notably the works [Hut20a; Hut20e; CMT22] and the structure theory developed in [BGT12a; TT21a]. Euclidean lattices. Well before the formulation of Schramm’s conjecture, the first significant work on locality was carried out in the seminal work of Grimmett and Marstrand [GM90a], who proved that the critical probability for percolation on a “slab” Z𝑑−𝑘 × {0, . . . , 𝑛} 𝑘 converges to 𝑝 𝑐 (Z𝑑 ) as 𝑛 → ∞ provided that 𝑑 − 𝑘 ≥ 2. (In this context Z𝑑 and Z𝑑−𝑘 × {0, . . . , 𝑛} 𝑘 refer to the Cayley graphs of these groups with their standard generating sets.) This theorem and the methods developed to prove it are of central importance to the study of supercritical percolation in three and more dimensions. (Moreover, one of the motivations for the work of Grimmett and Marstrand was to get closer to proving that the percolation phase transition on Z𝑑 is continuous, and indeed similar methods were used in [BGN91] to prove continuity for half-spaces.) Although it is not strictly an instance of Schramm’s conjecture since slabs are not transitive, the Grimmett–Marstrand theorem trivially implies that the analogous statement holds for “slabs with periodic boundary conditions” (i.e., Z𝑑−𝑘 × (Z/𝑛Z) 𝑘 with its standard generating set), which are transitive. The proof of the Grimmett–Marstrand theorem relies heavily on renormalization techniques exploiting the full symmetries of Z𝑑 and scale-invariance of Euclidean space R𝑑 , and does not readily generalize to other transitive graphs. Let us also mention that a quantitative version of the Grimmett–Marstand theorem was proven in the more recent work of Duminil-Copin, Kozma, and Tassion [DKT21] which was very influential in both our work and [CMT22]. Remark 3.1.2. A further classical Euclidean result in the spirit of the locality conjecture was established by Kesten [Kes90], who proved that 𝑝 𝑐 (Z𝑑 ) ∼ 1/(2𝑑 − 1) as 𝑑 → ∞. See [ABS04a] for a simple proof and [Sla06] for more refined results. While not strictly an instance of the locality conjecture since Z𝑑 does not converge in the local topology, the intuitive reason for this result to hold is that Z𝑑 is “locally tree-like” when 𝑑 → ∞ in the sense that small cycles have a negligible effect on the behaviour of the percolation model, so that one can define a kind of “local limit” of Z𝑑 33 as 𝑑 → ∞ (in a different technical sense to the one we consider here) in terms of Aldous’s Poisson weighted infinite tree (PWIT) [AS04]. Progress on locality. Previous works on the locality conjecture can be divided into two strains, with completely different set of methods and domains of application associated to them: The first strain concerns graphs that satisfy various strong, “infinite-dimensional” expansion conditions, while the second concerns “finite-dimensional” graphs (i.e., graphs of polynomial volume growth) where one can hope to develop appropriate generalizations of Grimmett–Marstrand theory. The first strain splits further into two cases according to whether or not the graphs in question are unimodular, a technical condition1 that holds for most familiar examples of transitive graphs including every Cayley graph and every amenable transitive graph [SW88]. Although nonunimodular graphs are often considered to be “pathological” compared to their unimodular cousins, it turns out that nonunimodularity is actually a very helpful assumption: in [Hut20e] the second author carried out a very detailed analysis of critical percolation on nonunimodular transitive graphs, which he then used to prove the nonunimodular case of locality in [Hut20a]. Moreover, it was proven in [Hut20a, Corollary 5.5] that the set of nonunimodular transitive graphs is both closed and open in G, so that to prove Theorem 8.1.1 it now suffices to consider the case that all graphs in question are unimodular. Let us now discuss previous results for unimodular graphs in the “infinite-dimensional” setting. The first result in this direction was due to Benjamini, Nachmias, and Peres [BNP11a], who proved the conjecture for nonamenable graph sequences satisfying a certain high girth condition (e.g., uniformly nonamenable graph sequences of divergent girth; unimodularity is not required). More recently, the second author [Hut20a] proved the conjecture for graph sequences of uniform exponential growth (meaning that the balls of radius 𝑟 in the graphs 𝐺 𝑛 all have volume lowerbounded by 𝑒 𝑐𝑟 for some constant 𝑐 independent of 𝑛 and 𝑟), and Hermon and the second author [HH21a] proved the conjecture for sequences of graphs satisfying a certain uniform stretchedexponential heat kernel upper bound, a class that includes certain examples of intermediate volume growth (i.e., volume growth that is superpolynomial but subexponential; note however that the spectral condition of [HH21a] is not implied by any growth condition). The works [Hut20a; Hut20e; HH21a] all establish locality for the families of graphs they consider by proving quantitative tail estimates on critical percolation clusters that hold uniformly for all graphs in the family, yielding 1 Here is the definition: A transitive graph 𝐺 = (𝑉, 𝐸) is unimodular if it satisfies the mass-transport principle, Í Í meaning that 𝑥 𝐹 (𝑜, 𝑥) = 𝑥 𝐹 (𝑥, 𝑜) for every 𝐹 : 𝑉 2 → [0, ∞] that is diagonally invariant in the sense that 𝐹 (𝑥, 𝑦) = 𝐹 (𝛾𝑥, 𝛾𝑦) for every automorphism 𝛾 of 𝐺. We will not directly engage with unimodularity in this paper, but it will appear as a hypothesis in many of our intermediate results since it is needed to apply the two-ghost inequality of [Hut20a]. 34 much more than just locality. (In particular, they also imply that the graphs in question have continuous percolation phase transitions.) Remark 3.1.3. Although the techniques developed in [Hut20a; HH21a] have yet to be made to work for nearest-neighbour percolation models in finite dimension, versions of these arguments have been used in [Hut21a] to analyze certain long-range percolation models in finite-dimensional spaces. The results of [Hut21a] can be used to prove versions of the locality conjecture for certain large families of long-range percolation models on unimodular transitive graphs (under the assumption that the long-range edge kernel has a sufficiently heavy tail uniformly throughout the sequence). Further results on locality for long-range percolation can be found in [MS96; Ber02]. Polynomial growth and structure theory. We now discuss the second strain of results, concerning graphs of polynomial volume growth. Let us first briefly review the structure theory of transitive graphs of polynomial growth, which plays an important role in these developments. Recall that G is the space of all infinite transitive graphs and that 𝐺 ∈ G is said to have polynomial growth if for some positive reals 𝐶 and 𝑑, the number of vertices contained in a ball of radius 𝑛, denoted Gr(𝑛) := |𝐵𝑛 (𝑜)|, satisfies Gr(𝑛) ≤ 𝐶𝑛 𝑑 for all 𝑛 ≥ 1. The geometry of such graphs is highly constrained: it is a consequence of Gromov’s theorem [Gro81b] and Trofimov’s theorem [Tro84b] that every 𝐺 ∈ G with polynomial growth is necessarily quasi-isometric to the Cayley graph of a nilpotent group. In particular, for every such graph 𝐺, there is a positive real 𝐶 and a unique positive integer 𝑑 such that 𝐶 −1 𝑛 𝑑 ≤ Gr(𝑛) ≤ 𝐶𝑛 𝑑 for all 𝑛 ≥ 1. The integer 𝑑 is called the (volume growth) dimension of 𝐺; it coincides with the isoperimetric dimension and spectral dimension of 𝐺 by a theorem of Coulhon and Saloff-Coste [CS93]. These results are often used in the study of probability on transitive graphs as part of a “structure vs. expansion dichotomy”, wherein each graph either satisfies a high-dimensional isoperimetric inequality (which is often a helpful assumption) or else is quasi-isometric to a nilpotent group of bounded step and rank (which is useful because these graphs are highly explicit and well-behaved); a detailed overview of the structure theory of transitive graphs of polynomial growth and its applications to probability is given in the introduction to [EH23d]. More recently, finitary versions of these results have been established, first for groups in the landmark work of Breuillard, Green, and Tao [BGT12a], then for transitive graphs by Tessera and Tointon [TT21a]. These results imply, for instance, that for each constant 𝐾 < ∞ there exists 𝑁 < ∞ such that if we observe that Gr(3𝑛) ≤ 𝐾 Gr(𝑛) for some 𝑛 ≥ 𝑁, then 𝐺 is (1, 𝐶𝑛)-quasi isometric the Cayley graph of a virtually nilpotent group where the constant 𝐶 along with the rank, step, and index of the nilpotent subgroup are all bounded above by some function of 𝐾 (see Section 3.5 for 35 further details). This finitary structure theory is extremely useful in applications to problems such as locality in which one wishes to argue in a way that is uniform over some family of graphs. For example, it follows from [TT21a, Corollary 1.5] that for each 𝑑 ≥ 1 the set of transitive graphs of polynomial growth with dimension at most 𝑑 is an open subset of G, and moreover that if 𝐺 𝑛 → 𝐺 with 𝐺 of polynomial growth of dimension 𝑑 then there exists 𝑛0 < ∞ and a constant 𝐶 such that the ball of radius 𝑟 in 𝐺 𝑛 has volume at most 𝐶𝑟 𝑑 for every 𝑛 ≥ 𝑛0 , with constants independent of 𝑛. As such, to prove locality in the case that the limit has polynomial growth, it suffices to consider the case that all graphs in the sequence satisfy a uniform polynomial upper bound on their growth as well as various other forms of strong uniform control on their geometry. Besides the original work of Grimmett and Marstrand, the first result on locality for graphs of polynomial growth was due to Martineau and Tassion [MT17], who proved that locality holds for Cayley graphs of abelian groups. Their proof employs a variation on the Grimmett–Marstrand argument, overcoming significant technical difficulties arising due to the loss of rotational and reflection symmetry. This result was greatly extended in the recent work of Contreras, Martineau, and Tassion, who developed a version of Grimmett–Marstrand theory for transitive graphs of polynomial growth in [CMT22] and used this theory together with the finitary structure theory discussed above to deduce the polynomial growth case of the locality conjecture in [CMT23a]. As with the aforementioned works in the infinite-dimensional setting, the works [CMT22; CMT23a] establish not just locality but also many further strong quantitative results about percolation on the classes of graphs they consider. However, while [Hut20a; HH21a] established quantitative estimates on finite clusters in critical percolation, [CMT22; CMT23a] instead establish strong results about the geometry of the infinite cluster in supercritical percolation. This reflects a fundamental distinction between the two approaches, with the analysis of the polynomial growth case involving estimates on uniqueness of annuli crossings etc. that are simply not true for “big” graphs like the 3-regular tree. What challenges remain? Given the previous results discussed above, it appears that there are two main cases of the locality conjecture left to consider: arbitrary sequences of superpolynomial growth transitive graphs converging to a superpolynomial growth graph, and the “diagonal” case in which a sequence of polynomial-growth graphs converges to a graph of superpolynomial growth. Moreover, the second case might be split further according to whether the graphs in the sequence have bounded or divergent dimension. (As we will soon explain, our proof will in fact work through a different and less obvious kind of case analysis.) In the first case, a key difficulty is that the geometry of transitive graphs of superpolynomial growth can be highly arbitrary, with 36 the space of all such graphs being an ineffably complex object in some senses: the challenge is precisely that we need an argument that is robust enough to work for all possible transitive graphs, for which there is nothing like a general classification. Moreover, the best known uniform lower bound on the growth of groups of superpolynomial growth, due to Shalom and Tao [ST10b], is 𝑐 of the form 𝑛 (log log 𝑛) for a small constant 𝑐 > 0. This growth lower bound (which has not yet been proven for transitive graphs that are not Cayley) is vastly weaker2 than the assumptions used in the works [BNP11a; Hut20a; HH21a] discussed above. In the diagonal case, one must contend with these same difficulties again together with the total incompatibility of the methods that have thus far been used to handle the high-growth and polynomial growth cases. As such, while the main difficulties in the locality conjecture arise from “unknown enemies” hiding deep within the unknowable expanse of the space of all transitive graphs, there are also explicit examples that seem difficult to handle within existing frameworks. (One such example is the standard Cayley graph of the free step-𝑠 nilpotent group on two generators, which converges to a 4-regular tree as 𝑠 → ∞ but has finite dimension for each finite 𝑠.) Parallels with 𝑝 𝑐 < 1. Before we begin to describe our proof of the locality conjecture, let us first discuss how its history closely parallels the (older) history of the 𝑝 𝑐 < 1 problem. It follows from the classical work of Peierls [Pei36b] that 𝑝 𝑐 (Z𝑑 ) < 1 for every 𝑑 ≥ 2. In their highly influential work [BS96b], Benjamini and Schramm conjectured that 𝑝 𝑐 < 1 for every transitive graph that is not one-dimensional. Benjamini and Schramm also proved in the same paper that 𝑝 𝑐 < 1 for every (not necessarily transitive) nonamenable graph, while earlier results of Lyons [Lyo95] implied that exponential growth suffices in the transitive case. On the other hand, it is a simple consequence of the structure theory that every transitive graph of polynomial volume growth that is not onedimensional contains a subgraph that is quasi-isometric to Z2 , which easily implies that every such graph has 𝑝 𝑐 < 1 (see e.g. [HT21a, Section 3.4] for details). As such, for many years the problem remained open only for groups of intermediate growth. Even in this case the problem was solved for most “known” examples of graphs of intermediate growth, such as the Grigorchuk group [MP01; RY17], with the main remaining difficulty coming from “unknown enemies” as discussed above. See the introduction of [DGRSY20] for a detailed account of this partial progress including several further references. The 𝑝 𝑐 < 1 problem was eventually solved in full generality by Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [DGRSY20]. More precisely, they established that 𝑝 𝑐 < 1 for any (not necessarily 2 While a well-known conjecture of Grigorchuk [Gri14] states that every superpolynomial growth transitive graph has growth at least exp[𝑐𝑛1/2 ], this conjecture is completely open, somewhat controversial, and in any case would still require a significant advance on the methods of [Hut20a; HH21a] to be applicable towards the locality conjecture. 37 transitive) bounded degree graph satisfying a (4 + 𝜀)-dimensional isoperimetric inequality, with the structure theory and classical results above handling all remaining transitive graphs. Their proof uses a comparison between percolation and the Gaussian free field which works only for values of 𝑝 very close to 1, making their methods unsuitable for the locality problem. A finitary version of the results of [DGRSY20] was developed by the second author and Tointon in [HT21a], who proved in particular that sequences of finite transitive graphs have a non-trivial phase in which a giant cluster exists provided that they are “not one dimensional” in an appropriate quantitative sense. This finitary approach also allowed them to prove a uniform version of the main result of [DGRSY20], stating that for each 𝑑 ≥ 1 there exists 𝜀 > 0 such that every infinite transitive graph that has degree at most 𝑑 and is not one-dimensional has 𝑝 𝑐 < 1 − 𝜀. (For Cayley graphs it is now known that 𝜀 can be taken independently of the degree [PS23b]; see Section 3.7 for some related conjectures.) Can we do something similar? Continuing to follow the path set by this previous work on the 𝑝 𝑐 < 1 problem, one might hope to prove locality via a similar dichotomy, finding some method that handles all graphs that are “high-dimensional” in some sense, then using the structure theory to separately analyze the remaining “low-dimensional” examples. One technical problem with this approach, which was already a major hurdle in [HT21a], is that one is forced to consider sequences of graphs that may look high-dimensional up to some divergently large scale then switch to looking low-dimensional, meaning that one must find a way to “patch together” the outputs of the two different case analyses at the crossover scale. A more fundamental problem, however, is that to date there have simply been no viable approaches to prove locality under the assumption that the graphs are high dimensional. As we will see, our proof will instead follow a more subtle approach in which we first dichotomize into two much less obvious cases according to whether or not the graph has quasi-polynomial growth on the relevant scale. (Here, a function is said to have quasi-polynomial growth if it is bounded by a function of the form exp[(log 𝑛) 𝑂 (1) ].) In the low growth case, we then employ a second, subordinate dichotomization according to whether the rate of growth on the relevant scale is low-dimensional or high-dimensional; it is in this second dichotomy that we can make use of the structure theory. A key technical difficulty when arguing this way is that (as far as we know with the current structure theory), the same graph might oscillate between the quasi-polynomial and super-quasi-polynomial regimes infinitely many times as we go up the scales, so that any “case analysis” we do via this dichotomy must be able to handle this oscillation. Moreover, the delicate nature of our proof leads us to engage with the structure theory literature in a deeper way than had 38 previously been necessary in applications to probability. Indeed, our proof relies in part on our structure-theoretic companion paper [EH23d] in which we prove a “uniform finite presentation” theorem for groups of polynomial volume growth which we use to make some of the arguments from [CMT22] finitary. Besides this, we must also contend with the fact that the assumption of quasi-polynomial growth is highly non-standard, so that we must spend a significant amount of the paper studying the deterministic geometry of transitive graphs of quasi-polynomial growth (at some scale) with a view to eventually generalizing the methods of [CMT22] from polynomial to quasi-polynomial growth. Interestingly, our proof of locality yields as an immediate corollary a new proof that 𝑝 𝑐 < 1 for transitive graphs that are not one-dimensional, recovering the main result of [DGRSY20]. This proof works directly with Bernoulli percolation and does not rely on the comparison to the GFF in any way. (On the other hand it is also much more complicated than the original proof!) About the proof In this section we give an overview of our proof. Let us first establish some relevant notation that will be used throughout the paper. Recall that G denotes the space of all (vertex-)transitive graphs (which we always take to be connected and locally finite) and that G ∗ ⊆ G is the space of infinite transitive graphs that are not one-dimensional. We also write U ⊆ G for the space of all unimodular transitive graphs and write U ∗ = U ∩ G ∗ . Given 𝑑 ∈ N, we write G𝑑 , G𝑑∗ , and U𝑑∗ for the subsets of these spaces in which every graph has degree 𝑑. Given sets of vertices 𝐴 and 𝐵 in a graph, we write {𝐴 ↔ 𝐵} for the event that there is a path from 𝐴 to 𝐵 in the given configuration. We also use the notation 𝐴 ↔ ∞ to mean that there is an infinite cluster that intersects 𝐴. When 𝐴 = {𝑢} and 𝐵 = {𝑣} are singletons, we may simply write 𝑢 ↔ 𝑣 and 𝑢 ↔ ∞ instead. For each transitive graph 𝐺 we will write 𝑜 for an arbitrarily chosen root vertex of 𝐺 which we will refer to as the origin. We write 𝐵𝑛 for the graph-distance ball of radius 𝑛 around 𝑜 in 𝐺 and write 𝑆𝑛 for the set of vertices at distance exactly 𝑛 from 𝑜. Uniform estimates and finite-size criteria. Recall that percolation is said to undergo a continuous phase transition on an infinite graph 𝐺 if the function 𝜃 𝐺 : 𝑝 ↦→ P𝐺𝑝 (𝑜 ↔ ∞) is continuous (it is a theorem of Schonmann [Sch99] that 𝜃 𝐺 is always continuous at every point other than 𝑝 𝑐 ). One of the best-known conjectures in the study of percolation on general infinite transitive graphs is that percolation should undergo a continuous phase transition on every 𝐺 ∈ G ∗ [BS96b, Conjecture 4]; this is famously still open when 𝐺 is the three-dimensional cubic lattice. Although it might not be obvious from their statements, the locality conjecture and the continuity conjecture are very closely related. To see why, let us give two equivalent formulations of Theorem 8.1.1, which will inform 39 the remainder of our analysis. (For notational convenience we define P 𝑝 := P1 when 𝑝 > 1.) Reformulation of the locality conjecture 1. For each 𝑑 ∈ N and 𝜀, 𝛿 > 0, there exists 𝑟 ∈ N such that P𝐺𝑝 (𝑜 ↔ 𝑆𝑟 ) ≥ 𝛿 =⇒ P𝐺𝑝+𝜀 (𝑜 ↔ ∞) > 0 (3.1.1) for every 𝐺 ∈ G𝑑∗ and 𝑝 ∈ [0, 1]. Reformulation of the locality conjecture 2. For each 𝑑 ∈ N and 𝜀 > 0, there exists a function ℎ𝜀 = ℎ 𝑑,𝜀 : N → (0, ∞) with ℎ𝜀 (𝑟) → 0 as 𝑟 → ∞ such that P𝐺𝑝𝑐 −𝜀 (𝑜 ↔ 𝑆𝑟 ) ≤ ℎ𝜀 (𝑟) (3.1.2) for every 𝐺 ∈ G𝑑∗ and 𝑟 ≥ 1. These two reformulations are trivially equivalent to one another (the difference amounts to considering either ℎ𝜀 (𝑛) or its inverse), but we find the two different viewpoints on the same statement to be illuminating: the first is formulated in terms of a “finite-size criterion for not being very subcritical” while the second is formulated in terms of “uniform estimates on subcritical percolation”. In either formulation, the 𝜀 = 0 analogue of the same statement would imply both the locality conjecture and a strengthened, “uniform in 𝐺” version of the conjecture concerning the continuity of the phase transition; this is precisely what the arguments of [Hut20a; HH21a] establish for the classes of graphs they consider. (In fact this uniform version of continuity is implied by locality together with the non-uniform version of continuity by a simple compactness argument.) This suggests a close connection between the two problems, while the freedom to use “sprinkling” (i.e., to increase 𝑝 by small amounts in an appropriate manner) may make locality significantly more tractable. Let us now briefly explain why Reformulation 1 is equivalent to the locality conjecture (or more accurately to the upper semi-continuity of 𝑝 𝑐 , which is the difficult part of locality). In one direction, suppose that 𝐺 𝑛 → 𝐺 and that 𝑝 > 𝑝 𝑐 (𝐺). Since 𝑝 > 𝑝 𝑐 (𝐺), the connection probability P𝐺𝑝 (𝑜 ↔ 𝑆𝑟 ) does not decay as 𝑟 → ∞. Since for each fixed 𝑟 we also have that P𝐺𝑝 𝑛 (𝑜 ↔ 𝑆𝑟 ) = P𝐺𝑝 (𝑜 ↔ 𝑆𝑟 ) for every sufficiently large 𝑛 by the definition of local convergence, we may apply Reformulation 1 with 𝜀 = P𝐺𝑝 (𝑜 ↔ ∞) > 0 and 𝛿 an arbitrary positive number to deduce that 𝑝 𝑐 (𝐺 𝑛 ) ≤ 𝑝 + 𝛿 for all sufficiently large 𝑛. This implies the desired upper semi-continuity since 𝑝 > 𝑝 𝑐 (𝐺) and 𝛿 > 0 were arbitrary. In the other direction, suppose that Reformulation 1 is false, so that there exists 𝑑 ∈ N, 𝜀 > 0, and 𝛿 > 0 such that for each 𝑟 ≥ 1 there exists 𝐺 𝑟 ∈ G𝑑∗ and 𝑝𝑟 ∈ [0, 1] with 𝑝 𝑐 (𝐺 𝑟 ) ≥ 𝑝𝑟 + 𝜀 and P𝐺𝑝 𝑟 (𝑜 ↔ 𝑆𝑟 ) ≥ 𝛿. Since G𝑑∗ is compact, there exists a subsequence along which 𝐺 𝑟 converges locally to some transitive graph 𝐺 ∈ G𝑑∗ and 𝑝𝑟 converges to some 𝑝 ∈ [0, 1], 40 which must satisfy P𝐺𝑝 (𝑜 ↔ 𝑟) ≥ 𝛿 for every 𝑟 ≥ 1 and hence that 𝑝 𝑐 (𝐺) ≤ 𝑝. Since the lim inf of 𝑝 𝑐 (𝐺 𝑟 ) along this subsequence is at least 𝑝 + 𝜀 we deduce that 𝑝 𝑐 is not continuous on G ∗ , completing the proof of the equivalence. Remark 3.1.4. The lower semi-continuity of 𝑝 𝑐 follows by a very similar argument to the deduction of upper semi-continuity from Reformulation 1 that we just gave, but where the relevant finite-size criteria or uniform bounds follow easily from standard facts about the sharpness of the phase transition: The argument of [DT16a] is formulated in terms of finite-size criteria for subcriticality, while that of [Pet] is formulated in terms of uniform bounds on supercritical percolation. Remark 3.1.5. Although the perspective on the locality problem we have just discussed is highly influential on our approach, we will in fact follow a slightly different approach in order to circumvent some technical problems related to estimating the “burn-in”, i.e., the amount of sprinkling needed to perform the base case of our multi-scale induction scheme. As such, we do not obtain an explicit function ℎ𝜀 as in Reformulation 2 in this paper. The additional steps required to make our argument completely quantitative and obtain an explicit bound on the function ℎ𝜀 will be carried out in a forthcoming companion paper [EH23+b]. A non-trivial reformulation. Let us now begin to go into more detail about our methods of proof. As discussed above, the results of [Hut20a; Hut20e] completely resolve the nonunimodular case of the conjecture, and since the space U ∗ is both closed and open in G ∗ we may restrict from now on to the case that all graphs are unimodular. In this case, the sharpness of the phase transition [Men86; AB87a; DT16a] together with the methods of [Hut20a] allow us to further reformulate Theorem 8.1.1 as follows. Reformulation of the locality conjecture 3. For all 𝑑 ∈ N and all 𝜀, 𝛿 > 0, there exists 𝑛 ∈ N such that 1 log P 𝑝+𝛿 (𝑜 ↔ 𝑆 𝑚 ) = 0 min P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜀 =⇒ lim 𝑚→∞ 𝑚 𝑢,𝑣∈𝐵 𝑛 for every 𝐺 ∈ U𝑑∗ and 𝑝 ∈ [0, 1]. The fact that we can replace the statement that 𝜃 ( 𝑝 + 𝛿) > 0 appearing on the right hand side of Reformulation 1 with the statement that the radius has a subexponential tail appearing on the right hand side of Reformulation 3 follows directly from the sharpness of the phase transition [Men86; AB87a; DT16a], which implies that the radius has an exponential tail whenever 𝑝 < 𝑝 𝑐 . (This does not require unimodularity.) The fact that we can replace the lower bound on the tail of the radius appearing on the left hand side of Reformulation 1 with the lower bound on point-to-point 41 connection probabilities appearing on the left hand side of Reformulation 3 is a much less obvious3 fact and is the main content of [Hut20a]. (Indeed, in the exponential growth setting studied in [Hut20a], a uniform upper bound on critical point-to-point connection probabilities was already established in [Hut16].) The argument needed to see that Reformulation 3 implies Theorem 8.1.1 is explained in more detail in Sections 3.2 and 3.3. Sprinkled renormalization of the two-point function. We now explain our unconditional proof4 of Reformulation 3, which occupies the bulk of the paper. At a very high level, we will use a “sprinkled multi-scale induction argument”, in which we start with estimates concerning percolation on some scale and deduce similar estimates at a much larger scale after increasing 𝑝 by some appropriately small amount; if we can do this efficiently enough, so that the total sprinkling is small when we start at a large scale, we can carry the induction up to infinitely many scales and (hopefully) prove that the resulting slightly larger parameter is supercritical (or at least that it is not subcritical). Since our actual induction hypothesis is rather complicated, let us first illustrate how such an argument might work in principle. Let 𝑑 ≥ 1 be fixed and suppose that we were able to prove an implication of the form min P𝐺𝑝 (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛) min P𝐺𝑝+𝜀(𝑛) (𝑢 ↔ 𝑣) ≥ 𝜂(𝑛) =⇒ 𝑢,𝑣∈𝐵𝑛 𝑢,𝑣∈𝐵 𝜙 (𝑛) (3.1.3) held for all 𝐺 ∈ U𝑑∗ , 𝑝 ∈ [0, 1], and 𝑛 ≥ 1, where 𝜙 : N → N is strictly increasing and 𝜀, 𝛿, 𝜂 : N → (0, 1] are decreasing. We claim that such an implication would suffice to prove locality provided that sufficiently strong quantitative relationships hold between the various functions that appear. For example, the argument would work provided that 𝜂(𝑛) ≥ 𝛿(𝜙(𝑛)), that 𝛿(𝑛) is Í 𝑘 𝑘 subexponentially small as a function of 𝑛, and that ∞ 𝑘=0 𝜀(𝜙 (𝑛)) < ∞, where 𝜙 denotes the 𝑘fold convolution of 𝜙. (Consider for example 𝜙(𝑛) = (𝑛 + 1) 2 , 𝛿(𝑛) = (log 𝑛) −1 , 𝜂(𝑛) = (2 log 𝑛) −1 , and 𝜀(𝑛) = (log log 𝑛) −2 .) To see this, note that if we define the sequence (𝑛 𝑘 ) 𝑘 ≥1 by 𝑛0 = 1 and 𝑛 𝑘+1 = 𝜙(𝑛 𝑘 ) for each 𝑘 ≥ 1 then, under this assumption, the implication (3.1.3) implies that min P𝐺𝑝 (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛 𝑘 ) min P𝐺𝑝+𝜀(𝑛 𝑘 ) (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛 𝑘+1 ) =⇒ 𝑢,𝑣∈𝐵 𝑛𝑘 𝑢,𝑣∈𝐵 𝑛𝑘+1 3 Indeed, unlike the tail of the radius, it is possible that point-to-point connection probabilities continue to decay in the supercritical regime, as is the case on the 3-regular tree. This breaks the argument that Reformulation 1 implies Theorem 8.1.1 that we gave above. 4 As discussed in Remark 3.1.5, we do not quite proceed via Reformulation 3 in order to circumvent certain technical obstacles. Despite these caveats, we still think of our argument to be best understood as “morally” going via Reformulation 3, which in any case is implied by Theorem 8.1.1. 42 and hence by induction that min P𝐺𝑝 (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛 𝑘 ) 𝑢,𝑣∈𝐵𝑛𝑘 for some 𝑘 ≥ 1 =⇒ min P𝐺𝑝+Í𝑖−1 𝜀(𝑛 ) (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛 𝑘+1 ) 𝑢,𝑣∈𝐵 𝑛𝑖 𝑗=𝑘 for every 𝑖 ≥ 𝑘. 𝑗 Since the conclusion on the right hand side implies that connection probability are subexponential Í 𝐺 in the distance at 𝑝 + ∞ 𝑗=𝑘 𝜀(𝑛 𝑗 ), it would follow from (3.1.3) that if min𝑢,𝑣∈𝐵𝑛𝑘 P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝛿(𝑛 𝑘 ) Í∞ Í∞ for some 𝑘 then 𝑝 𝑐 ≤ 𝑝 + 𝑗=𝑘 𝜀(𝑛 𝑗 ). The assumption that 𝑗=0 𝜀(𝑛 𝑗 ) ensures that the tail sum appearing here is small, verifying Reformulation 3. Following this approach, we are led to the problem of how to extend point-to-point connection lower bounds from one scale to a much larger scale after sprinkling by a small amount. More specifically, we want to do this as efficiently as possible, with the hope of obtaining an inductive statement that is sufficiently strong to imply locality. Snowballing. In Section 3.4, we develop a new method based on Talagrand’s theory of sharp thresholds [Tal94] and “cluster repulsion” inequalities inspired by the work of Aizenman-KestenNewman [AKN87b] that allows us to prove a bootstrapping implication of the form (3.1.3), where the function 𝜙 depends on the volume of the ball of radius 𝑛. While the basic idea of using Talagrand together with Aizenman-Kesten-Newman is already present in [DKT21; CMT22] (both in the polynomial growth case), we find a new way of both implementing and applying5 this argument using ghost fields that works directly in infinite volume and uses the two-ghost inequality of [Hut20a] rather than the classical Aizenman-Kesten-Newman inequality. We call this the snowballing method. While the methods of [DKT21; CMT22] needed upper bounds on the growth to work, our method actually becomes more efficient as the growth gets larger; if the growth is large enough, the bootstrapping implication we obtain (which is of the form (3.1.3)) is strong enough to prove locality by the argument outlined above. Optimizing the snowballing argument as much as we could (see Remark 3.4.5), we found that this method could prove locality for graph 𝐶 sequences satisfying a uniform growth lower bound of the form 𝑛𝑐(log log 𝑛) for some universal constant 𝐶 and any 𝑐 > 0. For Cayley graphs, the hypothesis needed for this argument to work is of course frustratingly close to the Shalom-Tao bound Gr(𝑛) ≥ exp(𝑐 log 𝑛(log log 𝑛) 𝑐 ) [ST10b], where 𝑐 is a small universal 5 In particular, the argument we use to efficiently extend two-point estimates to a higher scale given the outputs of this sharp threshold argument is also novel. 43 constant, which holds for every Cayley graph of superpolynomial growth. Thus, even a modest improvement to Shalom-Tao or the snowballing argument would allow us to prove locality for all sequences of superpolynomial growth via this method. Since we were not able to improve either argument to the required extent (and in any case must also deal with the “diagonal” case of locality), we will instead attack the locality conjecture in a “pincer movement”, where we push the polynomial growth methods of [CMT22] to handle all growths that are too slow to be treated by the snowballing argument. In fact we will push these methods to handle all graphs of quasi-polynomial growth Gr(𝑛) ≤ exp((log 𝑛) 𝐶 ), so that there is a considerable overlap in the growth regimes handled by the two methods. As mentioned above, a key technical difficulty is that (as far as we know) the same graph may oscillate between the two growth regimes infinitely often, so that the two methods must be harmonized in some way to allow for this. Chaining via orange peeling. A well-known general approach to efficiently extend connection lower bounds from one scale to another is by a method we will call chaining. Consider the event E 𝑅 that 𝑆 𝑅 ↔ 𝑆10𝑅 and the event U 𝑅 that there is at most one cluster intersecting both 𝑆2𝑅 and 𝑆5𝑅 . Suppose we knew that for some large 𝑅 and small 𝜀 > 0, P 𝑝 (E ) ≥ 1 − 𝜀 and P 𝑝 (U ) ≥ 1 − 𝜀. (3.1.4) Then, by Harris’ inequality and a union bound, we could deduce that any pair of vertices 𝑢 and 𝑣 at distance 𝑘 𝑅 satisfy P 𝑝 (𝑢 ↔ 𝑣) ≥ ′ min ′ 𝑢 ,𝑣 ∈𝐵10𝑅 P 𝑝 (𝑢′ ↔ 𝑣 ′) 2 · [1 − 𝑘𝜀] − 𝑘𝜀. (3.1.5) If we also had a way to use (3.1.5) to deduce a version of (3.1.4) at scale 𝑘 𝑅 in place of 𝑅 (possibly after a small increase of 𝑝), and this argument is sufficiently efficient quantitatively, we might be able to formulate an inductive argument yielding both connection lower bounds and uniqueness of annuli crossings at all scales. (Of course such an argument cannot work on e.g. the 3-regular tree.) This is roughly what is done in [CMT22] (although they consider supercritical percolation, so that crossing probabilities do not decay a priori), who establish that U 𝑅 holds with high probability when 𝐺 has polynomial growth and 𝑅 → ∞. Their method, which (following [Gri99]) they refer to as orange peeling6 and is inspired by the earlier work [BT17], relies on knowing a two-point lower bound within annuli of the form 𝐵 𝑅+Δ \𝐵 𝑅 for all large 𝑅 and some appropriate Δ(𝑅) ≪ 𝑅. This information needs to be proven using information at lower scales, so that the argument is a kind of multi-scale induction or coarse-grained renormalization. The idea is to argue that as two 6 We do not completely understand the metaphor; perhaps onion peeling would be more appropriate? 44 clusters cross the thick annulus 𝐵5𝑅 \𝐵2𝑅 , they are very likely to become connected to each other because they have to cross many disjoint annuli of the form 𝐵 𝑅′ +Δ \𝐵 𝑅′ , and in each such annulus they have a good probability to become connected to each other after sprinkling. If there are enough opportunities for any two crossing clusters to merge then all crossing clusters will have merged by the time they reach the outer sphere. The full argument of [CMT22] is highly technical and sophisticated, with the discussion in this paragraph presenting only a simplified cartoon version of some selected parts of their argument. Working with quasi-polynomial growth. To run the something like the above “orange peeling” method, it is helpful to know that annuli 𝐵 𝑅2 \𝐵 𝑅1 are in some sense well-connected. For example, in the setting of [CMT22], the authors used the fact that for any pair of vertices 𝑢, 𝑣 in the exposed sphere 𝑆 ∞ 𝑅 , which is a certain subset of the usual sphere 𝑆 𝑅 , there is a path from 𝑢 to 𝑣 that stays within 𝐵 𝑅+Δ \𝐵 𝑅−Δ . To prove this they used the structure theory of graphs of polynomial growth (i.e. the fact that such graphs are finitely presented and are one-ended when they are not one-dimensional). As such, this method does not easily generalize to other graphs. (Indeed, any reasonable connectivity-of-annuli statement cannot hold in complete generality — annuli in regular trees are as poorly connected as sets at a given distance can possibly be.) We will prove that a weak connectivity property of annuli does hold if we assume that a quasi-polynomial growth upper bound Gr(𝑅) ≤ exp((log 𝑅) 𝐶 ) holds around the relevant scales. This statement, which we call the polylog-plentiful tubes condition, says that for any two sets of vertices 𝐴 and 𝐵 crossing a thick annulus 𝐵3𝑅 \𝐵 𝑅 , we can find many (i.e., at least (log 𝑅) Ω(1) ) paths from 𝐴 to 𝐵 that are not too long (i.e., have length at most 𝑅(log 𝑅) 𝑂 (1) ) and that are well-separated from each other (i.e., any two paths in the set have distance at least (log 𝑅) Ω(1) ). The proof of this polylog-plentiful tubes condition will exploit a structure vs. randomness dichotomy, where we use the structure theory of [EH23d] to handle the low-dimensional case and handle the high-dimensional case by building the required disjoint tubes using coupled families of random walks. Once this polylog-plentiful tubes condition is verified, we then show it can be used to push the methods of [CMT22] to handle graphs of quasi-polynomial growth. (This requires significant technical changes throughout their entire argument, with the proofs of some intermediate steps being completely different.) We might hope that the low- and high-growth arguments both imply a common statement similar to (3.1.3). Unfortunately our induction statement is more involved than this. The issue is that our lowgrowth argument works with point-to-point connections within finite sets such as balls and tubes, while the high-growth argument works directly in infinite volume, and our induction hypothesis must be able to handle oscillation between these two cases. Our actual induction statement, which 45 we explain in detail in Section 3.3, therefore has two parts: a lower bound on the full-space (infinite volume) two-point function, with no restriction on the geometry of the graph at the relevant scale, together with a lower-bound on connection probabilities inside tubes that holds only when the thickness of the tube happens to belong to a quasi-polynomial growth scale. (The length of these tubes are permitted to be quasi-polynomial in their thickness, so that this estimate tells us not just about quasi-polynomial growth scales but also many subsequent larger scales.) Thus, we are able to formulate a multi-scale inductive implication that holds in all growth regimes and is strong enough to imply Theorem 8.1.1. Organization and overview We now outline the structure of the rest of the paper. Section 2: In this section we review both the classical Aizenman-Kesten-Newman inequality [AKN87b] and its consequences for the “uniqueness zone” as derived in [CMT22] as well as the two-ghost inequality of [Hut20a], which is a kind of infinite-volume version of Aizenman-KestenNewman analyzing volumes of clusters rather than their diameters. We also state a useful lemma of [Hut20a] that applies this inequality and that leads to Reformulation 3 above. Section 3: In this section we formulate the multi-scale induction framework used to prove locality, stating the induction step as Proposition 3.3.1. We then explain how this technical proposition both implies Theorem 8.1.1 and leads to a new proof of 𝑝 𝑐 < 1 for transitive graphs that are not one-dimensional as originally proven in [DGRSY20]. Section 4: In this section we develop a new method of deducing two-point function lower bounds at a large scale from lower bounds at a smaller scale, which we call snowballing. This is based in part on an idea already present in [DKT21; CMT22], in which one uses Aizenman-Kesten-Newman bounds as an input to Talagrand’s sharp threshold theorem [Tal94]. Unlike those works, we use the two-ghost formulation of Aizenman-Kesten-Newman from [Hut20a] to work directly with volumes of clusters and prove bounds that hold in arbitrary unimodular transitive graphs; this causes the inequalities we derive to become vastly more efficient as the growth of the graph gets larger. The proof of the main snowballing proposition also involves a novel ghost-based chaining argument to convert this sharp threshold statement into a statement about extending point-to-point connection probabilities. All the arguments in this section work more efficiently as the growth gets larger, but still have content in the polynomial growth case. In Section 2.2, we explain how the snowballing method implies part of the main multi-scale induction step and also easily implies the full locality conjecture for graphs satisfying a mild superpolynomial growth lower bound. (Here the growth 46 assumption is needed only to ensure that the total amount of sprinkling is small when we start at a large scale and induct up to infinity; no further geometric properties of graphs of high growth are used.) The tools we develop in Section 2.1 also play an indispensable role in our low-growth arguments in Section 3.6. Section 5: In this section we establish deterministic geometric features of transitive graphs of quasi-polynomial growth (at some scale) that will be used to analyze percolation on these graphs in Section 3.6. The main question addressed is as follows: how can one harness low — but not necessarily polynomial — volume growth to establish that annuli are well-connected? Here, the “well-connectedness” of annuli is made precise via what we call the polylog-plentiful tubes condition. The proof that the polylog-plentiful tubes condition holds for graphs of quasi-polynomial growth uses a “structure vs. expansion” dichotomy according to whether the rate of growth on the relevant scale is low or high dimensional; in the low-dimensional case we employ the structure theory of [BGT12a; TT21a; EH23d] while in the high-dimensional case we construct large families of disjoint tubes using certain coupled random walks; the quasi-polynomial growth assumption is used to ensure that we can couple two walks started at distance 𝑛 to coalesce within distance 𝑛(log 𝑛) 𝑂 (1) . (The low dimensional case of the analysis is the only place where we directly use the fact that our graphs are not one-dimensional, where it is needed to ensure that “exposed spheres” are well-connected in a certain sense.) Section 6: Using the polylog-plentiful tubes condition, we run a chaining argument for graphs of quasi-polynomial growth (on some scale) that is inspired by [CMT22]. Many changes to their argument are required to deal with this new geometric setting, and our arguments in this section also make use of the snowballing method developed in Section 3.4. We then use the outputs of this analysis to complete the proof of the induction step and hence of Theorem 8.1.1. Section 7: In this section we first briefly sketch how the methods of Section 3.4 can be used to prove locality of the density of the infinite cluster in the supercritical phase; a significant generalization of this result is proven in full detail (via a different method) in our forthcoming paper [EH23+a]. We finish by discussing some open problems in Section 3.7. 3.2 Cluster repulsion and the a priori uniqueness zone In this section we review various bounds on the probability that two large clusters either meet at a single edge or both intersect a small ball, together with some important consequences of these inequalities. These inequalities originate inexplicitly in the work of Aizenman, Kesten, and Newman [AKN87b] and were brought to the wider attention of the community in the work of Cerf 47 [Cer15]. These inequalities come in two flavours: finite-volume estimates that work with the radii of clusters, and infinite-volume estimates that work with the volume of clusters. The first kind of inequality, which is closer to the original vision of [AKN87b; Cer15], works best under a lowgrowth assumption and plays an important role in Contreras, Martineau, and Tassion’s analysis of percolation on transitive graphs of polynomial growth [CMT22; CMT23a], while the second kind of inequality, introduced in [Hut20a] and known as two-ghost inequalities, become more useful the faster the graph grows. We will also review the argument of [Hut20a] which allows us to deduce locality of 𝑝 𝑐 from uniform bounds on the two-point function using the two-ghost inequality. Aizenman-Kesten-Newman and the a priori uniqueness zone We now review the finite-volume Aizenman-Kesten-Newman inequality as presented in [Cer15; CMT22]. Finite-volume two-arm estimates. Given 𝑚, 𝑛 ∈ (0, ∞) with 𝑚 ≤ 𝑛, we define Piv[𝑚, 𝑛] to be the event that there exist two distinct clusters in 𝜔 ∩ 𝐵𝑛 that each intersect both spheres 𝑆𝑛 and 𝑆 𝑚 . (That is, Piv[𝑚, 𝑛] is the event that there is more than one cluster crossing the annulus from 𝑚 to 𝑛.) The following lemma is a minor variation on [CMT22, Proposition 4.1]. Proposition 3.2.1. For each 0 < 𝜀 < 1/2, 0 < 𝜂 < 1, and 𝑑 ≥ 1 there exists a constant 𝐶 = 𝐶 (𝜀, 𝜂, 𝑑) such that if 𝐺 is a connected, transitive graph of vertex degree 𝑑 then  log Gr(𝑛) P 𝑝 (Piv[1, 𝑛]) ≤ 𝐶 𝑛  12 −𝜀 , for every 𝑝 ∈ [𝜂, 1] and 𝑛 ≥ 1. To deduce this proposition from the proof of [CMT22, Proposition 4.1], we will require the following elementary fact about the growth of balls. Here 𝜕𝐵𝑚 (𝑜) denotes the set of edges that have one endpoint in 𝐵𝑚 (𝑜) and the other endpoint in the complement of 𝐵𝑚 (𝑜). Lemma 3.2.2. For each 𝑑 ≥ 1 there exists a constant 𝐶𝑑 such that the following holds. Let 𝐺 be a graph of maximum vertex degree at most 𝑑, and let 𝑜 be a vertex of 𝐺. For each integer 𝑛 ≥ 1 there exists an integer 𝑛 ≤ 𝑚 ≤ 2𝑛 − 1 such that |𝜕𝐵𝑚 (𝑜)| 𝐶𝑑 ≤ log |𝐵2𝑛 (𝑜)| . |𝐵𝑚 (𝑜)| 𝑛 Proof of Lemma 3.2.2. Write Gr(𝑚) := |𝐵𝑚 (𝑜)| for every 𝑚 ∈ N. Since 𝐺 has vertex degrees bounded above by 𝑑, we have that |𝜕𝐵𝑚 (𝑜)| ≤ 𝑑 (Gr(𝑚 + 1) − Gr(𝑚)) and hence that 48 |𝜕𝐵𝑚 (𝑜)| /Gr(𝑚) ≤ 𝑑 (Gr(𝑚 + 1)/Gr(𝑚) − 1). It follows that  2𝑛−1 2𝑛−1 ∑︁  Gr(𝑚 + 1) ∑︁ |𝜕𝐵𝑚 (𝑜)| ≤𝑑 −1 . |𝐵𝑚 (𝑜)| Gr(𝑚) 𝑚=𝑛 𝑚=𝑛 Now, since we also have that Gr(𝑚+1) Gr(𝑚) ≤ 𝑑, there exists a constant 𝐶 = 𝐶𝑑 such that Gr(𝑚 + 1) Gr(𝑚 + 1) − 1 ≤ 𝐶𝑑 log Gr(𝑚) Gr(𝑚) for every 𝑚 ≥ 0. As such, it follows that 2𝑛−1 ∑︁ 2𝑛−1 ∑︁ |𝜕𝐵𝑚 (𝑜)| Gr(𝑚 + 1) Gr(2𝑛) ≤ 𝑑𝐶𝑑 log = 𝑑𝐶𝑑 log |𝐵𝑚 (𝑜)| Gr(𝑚) Gr(𝑛) 𝑚=𝑛 𝑚=𝑛 which is easily seen to imply the claim. □ Proof of Proposition 3.2.1. This follows by exactly the same proof as [CMT22, Proposition 4.1] except that we use our Lemma 3.2.2 instead of their Lemma 4.2 (which is the same estimate specialized to the polynomial growth setting). □ The a priori uniqueness zone. We now discuss how two-arm bounds at a single edge can be used to deduce bounds on the probability of having multiple clusters crossing an annulus. The following lemma, essentially due to Cerf [Cer15], lets us apply Proposition 3.2.1 to bound the probability that there are two distinct crossings of an annulus. Lemma 3.2.3 ([CMT22, Lemma 6.2]). Let 𝐺 be a connected transitive graph. Then

P 𝑝 Piv[𝑟, 𝑛] ≤ P 𝑝 Piv[1, 𝑛/2] · |𝑆𝑟 | 2 · Gr(𝑚) 𝐵𝑚 min𝑎,𝑏∈𝑆𝑟 P 𝑝 𝑎 ←−→ 𝑏 (3.2.1)
for every 𝑟, 𝑚, 𝑛 ∈ (1, ∞) with 𝑟 ≤ 𝑚 ≤ 𝑛/2 and every 𝑝 ∈ (0, 1). Remark 3.2.1. The authors of [CMT22] stated this lemma in their context of infinite graphs with polynomial growth. However, their proof works exactly the same for arbitrary connected transitive graphs. (The statement given in [CMT22] has 𝐵2𝑚 in place of 𝐵𝑚 — which is slightly stronger — but this appears to be a typo.) Since any geodesics from 𝑜 to two vertices 𝑎, 𝑏 ∈ 𝐵𝑚 are both open with probability at least 𝑝 2𝑚 and the growth satisfies the trivial upper bound Gr(𝑚) ≤ 𝑑 𝑚+1 , the quantity multiplying the probability
P 𝑝 Piv[1, 𝑛/2] on the right hand side of (3.2.1) is at most exponential in 𝑚 when 𝑝 is bounded away from 0. Proposition 3.2.1 and Lemma 3.2.3 therefore have the following immediate corollary. 49 Corollary 3.2.4 (Trivial a priori uniqueness zone). Let 𝐺 be a connected, unimodular transitive graph with vertex degree 𝑑 and let 𝜀, 𝜂 ∈ (0, 1). There exist positive constants 𝑐 = 𝑐(𝑑, 𝜂, 𝜀), 𝐶 = 𝐶 (𝑑, 𝜂, 𝜀), and 𝑛0 = 𝑛0 (𝑑, 𝜂, 𝜀) such that  log Gr(𝑛) P 𝑝 Piv[𝑐 log 𝑛, 𝑛] ≤ 𝐶 𝑛
 1/2−𝜀 for every 𝑛 ≥ 𝑛0 and 𝑝 ∈ [𝜂, 1]. In other words, if 𝐺 has subexponential growth then the probability of having two distinct crossings of the annulus from 𝑐 log 𝑛 to 𝑛 is always small for an appropriately small constant 𝑐. (The restriction that 𝑝 is small could be removed by noting that it is very unlikely for there to be any crossings of the annulus when 𝑝 ≤ 1/𝑑.) The two-ghost inequality We now recall the two-ghost inequality of [Hut20a], a form of the Aizenman-Kesten-Newman bound that holds for any unimodular transitive graph, without any growth assumptions. (This version of the bound does not imply uniqueness of the infinite cluster since it requires at least one of the clusters to be finite.) Let 𝐺 = (𝑉, 𝐸) be a graph. For each edge 𝑒 of 𝐺 and 𝑛 ≥ 1 we define S𝑒,𝑛 to be the event that 𝑒 is closed and that the endpoints of 𝑒 are in distinct clusters, each of which has volume at least 𝑛 and at least one of which is finite. Theorem 3.2.5 (Two-ghost inequality). Let 𝐺 be a unimodular transitive graph of degree 𝑑. There exists a constant 𝐶𝑑 such that  1/2  1− 𝑝 (3.2.2) P 𝑝 (S𝑒,𝑛 ) ≤ 𝐶𝑑 𝑝𝑛 for every 𝑒 ∈ 𝐸 (𝐺), 𝑝 ∈ (0, 1] and 𝑛 ≥ 1. Proof. This is an immediate consequence of [Hut20a, Corollary 1.7]: The statement given there concerns the edge volume rather than the vertex volume, but this only makes the statement stronger. □ Theorem 3.2.5 has the following useful consequence concerning the probability that two large distinct clusters come close to one another; this lets us convert bounds on the two-point function into bounds on the tail of the volume and underlies the fact that Reformulation 3 implies the unimodular case of Theorem 8.1.1. 50 Lemma 3.2.6. Let 𝐺 be an infinite, connected, unimodular transitive graph with vertex degree 𝑑. There exists a constant 𝐶𝑑 such that P 𝑝 (|𝐾𝑢 | ≥ 𝑛 and |𝐾𝑣 | ≥ 𝑛 but 𝑢 ↮ 𝑣) ≤ 𝐶𝑑 · 𝑑 (𝑢, 𝑣) 𝑝 −𝑑 (𝑢,𝑣)−1 𝑛−1/2 , for every 𝑢, 𝑣 ∈ 𝑉 (𝐺), 𝑛 ≥ 1, and 𝑝 < 𝑝 𝑐 (𝐺). Proof. This follows from Theorem 3.2.5 by the same argument used to prove equation (4.2) of [Hut20a]. (Again, the only difference is that we are using vertex volumes rather than edge volumes.) Í𝑑 (𝑢,𝑣) −𝑖+1 𝑝 . □ The quantity 𝑑 (𝑢, 𝑣) 𝑝 −𝑑 (𝑢,𝑣)−1 arises as a simple upper bound on ((1 − 𝑝)/𝑝) 1/2 𝑖=1 3.3 The multi-scale induction step In this section we state our key technical proposition, Proposition 3.3.1, which encapsulates the multi-scale induction used to prove Theorem 8.1.1. We introduce relevant definitions in Section 3.3, give the statement of the proposition in Section 3.3, and explain how it implies our main theorem in Section 3.3. In Section 3.3 we also explain how Proposition 3.3.1 yields a new proof of the fact that 𝑝 𝑐 < 1 for all infinite, connected, transitive graphs that are not one-dimensional. Definitions In this section we establish the notation necessary to state the main multi-scale induction proposition in Section 3.3. Natural coordinates for sprinkling. We define the sprinkling function Spr : (0, 1) × R → (0, 1) by 𝜆 𝜆 Spr( 𝑝; 𝜆) = 1 − (1 − 𝑝) 𝑒 so that (1 − Spr( 𝑝; 𝜆)) = (1 − 𝑝) 𝑒 . The sprinkling functions (Spr( · ; 𝜆))𝜆∈R form a semigroup in the sense that Spr( 𝑝; 𝜆 + 𝜇) = Spr(Spr( 𝑝; 𝜆); 𝜇) for every 𝜆, 𝜇 ∈ R and 𝑝 ∈ (0, 1). For each 0 < 𝑝, 𝑞 < 1 we define   log(1 − max{𝑝, 𝑞}) , so that max{𝑝, 𝑞} = Spr(min{𝑝, 𝑞}; 𝛿( 𝑝, 𝑞)). 𝛿( 𝑝, 𝑞) = log log(1 − min{𝑝, 𝑞}) Note that if 𝑝 ≥ 1/𝑑, as we will assume throughout most of the paper, then for each 𝐷 < ∞ there exists a positive constant 𝑐 = 𝑐(𝑑, 𝐷) such that  𝛿 2𝑑 − 1 𝑒 𝛿 −1 (1 − Spr( 𝑝; 𝛿)) = (1 − 𝑝) (1 − 𝑝) ≤ (1 − 𝑝) ≤ (1 − 𝑐𝛿)(1 − 𝑝) (3.3.1) 2𝑑 51 for every 0 ≤ 𝛿 ≤ 𝐷, so that Bernoulli bond percolation with parameter Spr( 𝑝; 𝛿) stochastically dominates the independent union of a Bernoulli-𝑝 bond percolation configuration and a Bernoulli(𝑐𝛿) bond percolation configuration. (Note however that 1 − Spr( 𝑝; 𝛿) is much smaller than (1 − 𝑐𝛿) (1 − 𝑝) when 𝑝 is close to 1.) The corridor function. Given a path 𝛾, we write len(𝛾) for its length. Given a path 𝛾 and some Ðlen(𝛾) 𝑟 ∈ (0, ∞), we define 𝐵𝑟 (𝛾) := 𝑖=0 𝐵𝑟 (𝛾𝑖 ), and call a set of this form a tube. We refer to len(𝛾) and 𝑟 as the length and thickness of the tube respectively. (Note that these parameters depend on the choice of representation of the tube 𝐵𝑟 (𝛾), and are not determined by the tube as a set of vertices.) Following [CMT22], we define the corridor function by   𝐵 𝑛 (𝛾) 𝜅 𝑝 (𝑚, 𝑛) := inf P 𝑝 𝛾0 ←−−→ 𝛾len(𝛾) 𝛾:len(𝛾)≤𝑚 for each 𝑝 ∈ (0, 1) and 𝑛, 𝑚 ≥ 1, so that 𝜅 𝑝 (𝑚, 𝑛) measures the difficulty of connecting points within tubes of thickness 𝑛 and length at most 𝑚. We may also take 𝑛 = ∞ in the definition of the corridor function, where the restriction for connections to lie in a tube disappears and we have simply that 𝜅 𝑝 (𝑚, ∞) = 𝜅 𝑝 (𝑚) = inf{P 𝑝 (𝑥 ↔ 𝑦) : 𝑑 (𝑥, 𝑦) ≤ 𝑚}. Note that the corridor function 𝜅 𝑝 (𝑚, 𝑛) is increasing in 𝑝 and 𝑛 and decreasing in 𝑚. Low growth scales. Given a transitive graph 𝐺 and a parameter 𝐷 > 0, we define the set of low growth scales to be n o L (𝐺, 𝐷) = 𝑛 ≥ 1 : log Gr(𝑚) ≤ (log 𝑚) 𝐷 for all 𝑚 ∈ [𝑛1/3 , 𝑛] , so that n o n o 𝑛 ≥ 1 : log Gr(𝑛) ≤ 3−𝐷 (log 𝑛) 𝐷 ⊆ L (𝐺, 𝐷) ⊆ 𝑛 ≥ 1 : log Gr(𝑛) ≤ (log 𝑛) 𝐷 . (We will sometimes call these quasi-polynomial growth scales since the function exp[(log 𝑥) 𝐶 ] is sometimes known as a quasi-polynomial.) For the purposes of the proof of the main theorem, we will apply this definition only with the (somewhat arbitrary) choice of constant 𝐷 = 20. The burn-in sprinkle. The first step of our induction will have a different form to the others, which necessitates a possibly larger amount of sprinkling. We now introduce notation describing this initial amount of sprinkling. Given a transitive graph 𝐺, 𝑝 ∈ (0, 1), and 𝑚 ≥ 1 define n o 1 𝑏(𝑚) = 𝑏(𝑚, 𝑝) = max 𝑏 ∈ N : 1 ≤ 𝑏 ≤ 𝑚 1/3 and P 𝑝 (Piv[4𝑏, 𝑚 1/3 ]) ≤ (log 𝑚) −1 , 8 52 setting 𝑏(𝑚) = 0 if the set being maximized over is empty, and define the burn-in to be Burn(𝑛, 𝑝) = Burn(𝐺, 𝑛, 𝑝) ( )  1/4 h i log log 𝑚 = max : 𝑚 ∈ L (𝐺, 20) ∩ (log 𝑛) 1/2 , 𝑛 , min{log 𝑚, log Gr(𝑏(𝑚))} setting Burn(𝑛, 𝑝) = 0 if the set being maximized over is empty and setting Burn(𝑛, 𝑝) = ∞ if there exists 𝑚 belonging to the set for which 𝑏(𝑚) ≤ 1. (If 𝑏(𝑚) > 1 then log Gr(𝑏(𝑚)) and log 𝑚 are both positive.) Note that Burn(𝐺, 𝑛, 𝑝) is determined by the ball of radius 𝑛 in 𝐺, so that two graphs whose balls of this radius are isomorphic have the same value of Burn(𝑛, 𝑝) for each 𝑝 ∈ (0, 1). Statement of the induction step We are now ready to state our main multi-scale induction proposition. We recall that, when applied to logical propositions, the symbol “∨” means “or” while the symbol “∧” means “and”. The condition 𝑛0 ≥ 16 appearing in the proposition ensures that log log 𝑛0 ≥ 1. Proposition 3.3.1 (The main multi-scale induction step). For each 𝑑 ∈ N there exist constants 𝐾 = 𝐾 (𝑑) and 𝑁 = 𝑁 (𝑑) ≥ 16 such that the following holds. Let 𝐺 be an infinite, connected, unimodular transitive graph with vertex degree 𝑑 that is not one-dimensional, let 𝑝 0 ∈ (0, 1), and let 𝑛0 ≥ 16. Let 𝑛−1 = (log 𝑛0 ) 1/2 , let 𝛿0 = 1 + 𝐾 · Burn(𝑛0 , 𝑝 0 ), (log log 𝑛0 ) 1/2 define sequences (𝑛𝑖 )𝑖≥1 and (𝛿𝑖 )𝑖≥1 recursively by   𝑛𝑖 := exp((log 𝑛𝑖−1 ) 9 ) = exp◦3 log◦3 (𝑛0 ) + 𝑖 log 9 and 𝛿𝑖 := (log log 𝑛𝑖 ) −1/2 = 3−𝑖 (log log 𝑛0 ) −1/2 , and let ( 𝑝𝑖 )𝑖≥1 be an increasing sequence of probabilities satisfying 𝑝𝑖+1 ≥ Spr( 𝑝𝑖 ; 𝛿𝑖 ) for each 𝑖 ≥ 0. For each 𝑖 ≥ 0 define the statement !   Full-Space(𝑖) = P 𝑝𝑖 (𝑢 ↔ 𝑣) ≥ exp −(log log 𝑛𝑖 ) 1/2 for all 𝑢, 𝑣 ∈ 𝐵𝑛𝑖 and for each 𝑖 ≥ 1 define the statement ! Corridor(𝑖) = 𝜅 𝑝𝑖 (𝑒 [log 𝑚] 10  , 𝑚) ≥ exp −(log log 𝑛𝑖 ) 53  1/2 for every 𝑚 ∈ L (𝐺, 20) ∩ [𝑛𝑖−2 , 𝑛𝑖−1 ] . If 𝑛0 ≥ 𝑁, 𝑝 0 ≥ 1/𝑑, and 𝛿0 ≤ 1 then the implications h i Full-Space(0) =⇒ Corridor(1) ∨ ( 𝑝 1 ≥ 𝑝 𝑐 ) , h Full-Space(𝑖) ∧ 𝑖 Û i Corridor(𝑘) =⇒ h =⇒ h (C0 ) i Corridor(𝑖 + 1) ∨ ( 𝑝𝑖+1 ≥ 𝑝 𝑐 ) , and (C) 𝑘=1 h Full-Space( 𝑗) ∧ Corridor( 𝑗 + 1) i Full-Space( 𝑗 + 1) ∨ ( 𝑝 𝑗+1 ≥ 𝑝 𝑐 ) i (F) hold for every 𝑖 ≥ 1 and 𝑗 ≥ 0. Remark 3.3.1. Note that Corridor(1) has a significantly different form than Corridor(𝑖) for 𝑖 ≥ 2 since 𝑛−1 is much smaller than the natural extrapolation of the sequence (𝑛𝑖 )𝑖≥0 to 𝑖 = −1. Remark 3.3.2. The condition 𝑝 ≥ 1/𝑑 appearing in the hypotheses of Proposition 3.3.1 is in fact redundant: An elementary path counting argument yields that P 𝑝 (𝑢 ↔ 𝑣) ≤ P 𝑝 (there is a simple open path of length at least 𝑑 (𝑢, 𝑣) starting from 𝑢) 1 𝑑 · · ( 𝑝(𝑑 − 1)) 𝑑 (𝑢,𝑣) ≤ 𝑑 − 1 1 − 𝑝(𝑑 − 1) for every 𝑝 < 1/(𝑑 − 1) and 𝑢, 𝑣 ∈ 𝑉 (𝐺), so that if 𝑛0 is sufficiently large and Full-Space(0) holds then 𝑝 ≥ 1/𝑑. We include this redundant assumption anyway to clarify the structure of the proof. Remark 3.3.3. In the statement Corridor(𝑖), the tubes that arise in the relevant corridor function 10 𝜅 𝑝𝑖 (𝑒 [log 𝑚] , 𝑚) have thickness given by the low-growth scale 𝑚, but can have length equal to 10 the much larger value 𝑒 [log 𝑚] . As such, the statement Corridor(𝑖) gives us strong control of percolation not just at low-growth scales but at a large range of scales above each low-growth scale. 1/3 10 9 In particular, provided 𝑛𝑖 is sufficiently large that 𝑒 [log(𝑛𝑖 )] ≥ 𝑒 (log 𝑛𝑖 ) = 𝑛𝑖+1 , the implication (F) holds trivially whenever 𝑛𝑖 ∈ L (𝐺, 20). Remark 3.3.4. The “∨( 𝑝𝑖+1 ≥ 𝑝 𝑐 )” that appears on the right hand side of the implications (C0 ), (C), and (F) can be removed if one assumes that 𝐺 is amenable, or if one works with "wired" connections as discussed in Section 3.7. This would be useful if one wished to use (a modification of) our methods to study the geometry of the infinite cluster in graphs of low growth, extending the results of [CMT22; Hut23b] to this setting. Most of the paper is dedicated to proving Proposition 3.3.1: The implication (F) is proven in Section 3.4 while the implications (C0 ) and (C) are proven in Sections 3.5 and 3.6. 54 Deduction of the main theorem from the induction step In this section we deduce our main theorem, Theorem 8.1.1, from Proposition 3.3.1. We write 𝑝 ∞ = lim𝑖→∞ 𝑝𝑖 for the limit of the parameters ( 𝑝𝑖 )𝑖≥0 defined recursively by 𝑝𝑖+1 = Spr( 𝑝𝑖 ; 𝛿𝑖 ), so that ∞  ∑︁    𝑝 ∞ = Spr 𝑝 0 ; 𝛿𝑖 ≤ Spr 𝑝 0 ; 2(log log 𝑛0 ) −1/2 + 𝐾 · Burn(𝑛0 , 𝑝 0 ) . 𝑖=0 We will apply Proposition 3.3.1 via the following corollary, which is a consequence of Proposition 3.3.1 together with the sharpness of the phase transition. Corollary 3.3.2. Let 𝑑 ≥ 1 and let 𝐾 = 𝐾 (𝑑) and 𝑁 = 𝑁 (𝑑) be the constants from Proposition 3.3.1. Let 𝐺 be an infinite, connected, unimodular transitive graph with vertex degree 𝑑 that is not onedimensional. Then the implication h P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑒 −(log log 𝑛) 1/2 i  for every 𝑢, 𝑣 ∈ 𝐵𝑛 ∧ [𝛿0 ≤ 1] h  i =⇒ 𝑝 𝑐 (𝐺) ≤ Spr 𝑝; 2(log log 𝑛) −1/2 + 𝐾 · Burn(𝑛, 𝑝) , holds for every 𝑛 ≥ 𝑁 and 𝑝 ≥ 1/𝑑. Proof of Corollary 3.3.2 given Proposition 3.3.1. It suffices to prove that 𝑝 𝑐 ≤ 𝑝 ∞ whenever 𝑛 ≥ 𝑁 1/2 and 𝑝 ≥ 1/𝑑 are such that P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑒 −(log log 𝑛) for every 𝑢, 𝑣 ∈ 𝐵𝑛 and 𝛿0 ≤ 1. Fix one such 𝑛 and 𝑝. Set 𝑛0 = 𝑛 and define (𝑛𝑖 )𝑖≥0 as in Proposition 3.3.1. Since Full-Space(0) holds by assumption, it follows from Proposition 3.3.1 that either 𝑝𝑖 ≥ 𝑝 𝑐 for some 𝑖 ≥ 1 or Full-Space(𝑖) and Corridor(𝑖) hold for every 𝑖 ≥ 1. In the former case we may trivially conclude that 𝑝 𝑐 ≤ 𝑝 ∞ , while in the latter case we have that P 𝑝∞ (𝑢 ↔ 𝑣) ≥ 𝑒 −(log log 𝑛𝑖 ) 1/2 for every 𝑖 ≥ 0 and every 𝑢, 𝑣 ∈ 𝐵𝑛𝑖 . (3.3.2) On the other hand, it follows from the sharpness of the phase transition [Men86; AB87a] that for each 𝑝 < 𝑝 𝑐 there exists a positive constant 𝑐 𝑝 such that P 𝑝 (𝑢 ↔ 𝑣) ≤ 𝑒 −𝑐 𝑝 𝑑 (𝑢,𝑣) for every 𝑢, 𝑣 ∈ 𝑉 (𝐺). This is incompatible with the (very) subexponential lower bound (3.3.2), so that 𝑝 ∞ ≥ 𝑝 𝑐 in this case also. □ We now apply Corollary 3.3.2 to prove Theorem 8.1.1. The proof we give here will rely on the results of both [Hut20a; Hut20e] (to deal with the nonunimodular case) and [CMT22; CMT23a] (to deal with the case that the limit has polynomial growth). We remark that the quantitative proof of the theorem given in [EH23+b] yields a completely self-contained and “uniform in the 55 graph” deduction of locality from Proposition 3.3.1 in the unimodular case that does not rely on the results7 of [CMT22; CMT23a]. (Doing this requires non-trivial bounds on the burn-in which can be avoided in the case that the limit has superpolynomial growth as we will see below.) Proof of Theorem 8.1.1 given Proposition 3.3.1. Let (𝐺 𝑚 )𝑚≥1 be a sequence of infinite, connected, transitive graphs converging locally to some infinite transitive graph 𝐺, and suppose that the graphs 𝐺 𝑚 all have superlinear growth. We want to prove that 𝑝 𝑐 (𝐺 𝑚 ) → 𝑝 𝑐 (𝐺). If 𝐺 is nonunimodular the claim follows from the results of [Hut20a; Hut20e] (specifically [Hut20a, Theorem 5.6]), while if 𝐺 has polynomial growth the result follows from the main result of [CMT23a]. Thus, we may assume that 𝐺 is unimodular and has superpolynomial growth. Since the set of nonunimodular graphs is both closed and open in G [Hut20a, Corollary 5.5], we may assume that the graphs (𝐺 𝑛 )𝑛≥1 are all unimodular. We may also assume that the graphs 𝐺 𝑛 and 𝐺 all have the same vertex degree 𝑑. Let 𝑁1 = 𝑁1 (𝑑) and 𝐾 = 𝐾 (𝑑) be the constants from Proposition 3.3.1. Suppose for contradiction that 𝑝 𝑐 (𝐺 𝑚 ) does not converge to 𝑝 𝑐 (𝐺). Since 𝑝 𝑐 is lower semicontinuous ([Pet, §14.2] and [DT16a, p.4]), we have that lim inf 𝑚→∞ 𝑝 𝑐 (𝐺 𝑚 ) ≥ 𝑝 𝑐 (𝐺). Thus, by taking a subsequence, we may assume that inf 𝑚≥1 𝑝 𝑐 (𝐺 𝑚 ) = 𝑝 ∗ > 𝑝 𝑐 (𝐺). Let 𝑝 0 = ( 𝑝 𝑐 (𝐺) + 𝑝 ∗ )/2 so that 𝑝 𝑐 (𝐺) < 𝑝 0 < 𝑝 ∗ . For each 𝑛 ≥ 1, let 𝑚(𝑛) be minimal such that the balls of radius 𝑛 are isomorphic in 𝐺 𝑚 and 𝐺 for all 𝑚 ≥ 𝑚(𝑛). Let 𝑐 > 0 be the constant from Corollary 3.2.4. Since 𝐺 has superpolynomial growth, we have by [Gro81b; Tro84b] that log log 𝑛 = 0, 𝑛→∞ log Gr(𝑐 log 𝑛; 𝐺) lim where Gr(𝑚; 𝐺) denotes the volume of the ball of radius 𝑚 in 𝐺, and it follows from Corollary 3.2.4 that lim sup sup Burn(𝐺 𝑚 , 𝑛, 𝑝) = 0. (3.3.3) 𝑛→∞ 𝑝∈[1/𝑑,1] 𝑚≥𝑚(𝑛) In particular, there exists 𝑁2 ≥ 𝑁1 (depending on the superpolynomial graph 𝐺 and the sequence (𝐺 𝑚 )) such that if 𝑛0 ≥ 𝑁2 then (log log 𝑛0 ) −1/2 + 𝐾 · Burn(𝐺 𝑚 , 𝑛0 , 𝑝 0 ) ≤ 1 and Spr( 𝑝 0 ; 2(log log 𝑛0 ) −1/2 + 𝐾 · Burn(𝐺 𝑚 , 𝑛0 , 𝑝 0 )) < 𝑝 ∗ for every 𝑚 ≥ 𝑚(𝑛0 ). Thus, it follows from Corollary 3.3.2 that for every 𝑛0 ≥ 𝑁2 and 𝑚 ≥ 𝑚(𝑛0 ) there exist vertices 𝑢 and 𝑣 in the ball of radius 𝑛0 in 𝐺 𝑚 such that   P𝐺𝑝0𝑚 (𝑢 ↔ 𝑣) < exp −(log log 𝑛0 ) 1/2 . 7 Of course many of the proof techniques remain closely inspired by these works! 56 Applying Lemma 3.2.6, we deduce that there exists a constant 𝐶1 such that   0 −1/2 P𝐺𝑝0𝑚 (|𝐾 | ≥ 𝑘) 2 ≤ 𝐶1 𝑛0 𝑝 −2𝑛 𝑘 + exp −(log log 𝑛0 ) 1/2 0 for every 𝑛0 ≥ 𝑁2 , 𝑚 ≥ 𝑚(𝑛0 ), and 𝑘 ≥ 1. Taking 𝑛0 = ⌈𝑐 2 log 𝑘⌉ for an appropriately small constant 𝑐 2 (which makes the first term 𝑂 (𝑘 −1/4 ), say, and hence of lower order than the second term), it follows that there exist positive constants 𝐶2 and 𝑐 3 such that   P𝐺𝑝0𝑚 (|𝐾 | ≥ 𝑘) ≤ 𝐶2 exp −𝑐 3 (log log log 𝑘) 1/2 (3.3.4) for every 𝑛0 ≥ 𝑁2 and 𝑚 ≥ 𝑚(𝑛0 ). Since 𝑝 0 > 𝑝 𝑐 (𝐺), the probability P𝐺𝑝0 (𝑜 ↔ ∞) is positive and it follows from (3.3.4) there exist 𝑘 0 and 𝑚 0 such that P𝐺𝑝0𝑚 (|𝐾 | ≥ 𝑘 0 ) ≤ 1 𝐺 P (𝑜 ↔ ∞) 2 𝑝0 (3.3.5) for every 𝑚 ≥ 𝑚 0 . On the other hand, if 𝑚 is sufficiently large that balls of radius 𝑘 0 are isomorphic in 𝐺 𝑚 and 𝐺 then P𝐺𝑝0𝑚 (|𝐾 | ≥ 𝑘 0 ) ≥ P𝐺𝑝0𝑚 (𝑜 ↔ 𝐵𝑐𝑘 0 ) = P𝐺𝑝0 (𝑜 ↔ 𝐵𝑐𝑘 0 ) ≥ P𝐺𝑝0 (𝑜 ↔ ∞), which contradicts the upper bound of (3.3.5). □ Let us now explain how Proposition 3.3.1 yields a new proof of the 𝑝 𝑐 < 1 theorem as originally established in [DGRSY20]. Recall that G ∗ is the space of all infinite, connected, transitive graphs that are not one-dimensional. Theorem 3.3.3. Every graph 𝐺 ∈ G ∗ satisfies 𝑝 𝑐 (𝐺) < 1. Proof of Theorem 3.3.3 given Proposition 3.3.1. If 𝐺 has exponential growth then sharpness of the phase transition easily implies that 𝑝 𝑐 ≤ (lim𝑟→∞ Gr(𝑟) −1/𝑟 ) < 1 (see also [Hut16; Lyo95]). Since every nonunimodular transitive graph is nonamenable and therefore has exponential growth, it suffices to consider the case that 𝐺 is unimodular. On the other hand, if 𝐺 has polynomial growth, then it is well-known that 𝑝 𝑐 (𝐺) < 1 follows from the fact that 𝑝 𝑐 (Z2 ) < 1 and the structure theory of transitive graphs of polynomial growth as explained in detail in [HT21a, Section 3.4]. We now consider the case that 𝐺 is unimodular and has superpolynomial growth. Let 𝑑 denote the vertex degree of 𝐺 and let 𝑁1 = 𝑁1 (𝑑) and 𝐾 = 𝐾 (𝑑) be the constants from Proposition 3.3.1. It follows by the same reasoning as we gave for (3.3.3) that for every 𝜂 > 0 there exists a constant 57 𝑀 (𝑑, 𝜂) such that 𝑏(𝑚, 𝑝) ≤ 𝜂 for every 𝑚 ∈ L (𝐺, 20) with 𝑚 ≥ 𝑀 and every 𝑝 ≥ 1/𝑑. Since we also have trivially that Gr(𝑛) ≥ 𝑛 for every 𝑛 ≥ 1 since 𝐺 is infinite, it follows that there exists a constant 𝑁2 = 𝑁2 (𝑑) such that (log log 𝑛) −1/2 + 𝐾 · Burn(𝑛, 𝑝) ≤ 1 for every 𝑝 ≥ 1/𝑑 and 𝑛 ≥ 𝑁2 . (3.3.6) Let 𝑛0 = 𝑛0 (𝑑) = 𝑁1 ∨ 𝑁2 . Since P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑝 𝑑 (𝑢,𝑣) for every 𝑢, 𝑣 ∈ 𝑉 and 𝑝 ∈ [0, 1], there exists a constant   1 (log log 𝑛0 ) 1/2 𝑝 0 = 𝑝 0 (𝑑) = ∨ exp − 𝑑 𝑛0 satisfying 1/𝑑 ≤ 𝑝 0 < 1 such that P 𝑝0 (𝑢 ↔ 𝑣) ≥ exp(−(log log 𝑛0 ) 1/2 ) for every 𝑢, 𝑣 ∈ 𝐵𝑛0 . Since 𝑛0 = 𝑁1 ∨ 𝑁2 , it follows from Corollary 3.3.2 and (3.3.6) that  
𝑝 𝑐 (𝐺) ≤ Spr 𝑝 0 ; 2(log log 𝑛0 ) −1/2 + 𝐾 Burn(𝑛0 , 𝑝 0 ) ≤ Spr 𝑝 0 ; 2 . The claim follows since the right hand side is strictly less than one. 3.4 □ Making connections via sharp threshold theory In this section we describe a powerful new way to extend point-to-point connection lower bounds from one scale to another, which we call the “snowballing method”. We develop this method in Section 3.4, then apply it to prove the implication (F) of Proposition 3.3.1 in Section 3.4. In Section 3.4 we will also explain how the method allows us to conclude the proof of locality for unimodular graphs satisfying a mild uniform superpolynomial growth assumption. Snowballing We now begin to develop the snowballing method. This method is primarily encapsulated through the following proposition, whose proof is the main goal of this section, but the intermediate lemmas used in its proof can be used to prove results of indepedent interest as discussed in Section 3.7. Given (not necessarily finite) non-empty sets of vertices 𝐴, 𝐵, and Λ in a graph 𝐺 = (𝑉, 𝐸) and a parameter 𝑝 ∈ [0, 1], we define  Λ  Λ 𝜏𝑝 ( 𝐴, 𝐵) := min P 𝑝 𝑎 ← →𝑏 , 𝑎∈𝐴 𝑏∈𝐵 Λ where we recall that {𝑎 ← → 𝑏} denotes the event that 𝑎 is connected to 𝑏 by an open path all of whose vertices belong to Λ. We will also write 𝜏𝑝Λ ( 𝐴) := 𝜏𝑝Λ ( 𝐴, 𝐴) and 𝜏𝑝 ( 𝐴, 𝐵) := 𝜏𝑝𝑉 ( 𝐴, 𝐵). We also use the notion of distance 𝛿( 𝑝, 𝑞) between two parameters 𝑝, 𝑞 ∈ (0, 1) as defined in Section 3.3. 58 Proposition 3.4.1 (Snowballing). For each 𝑑 ≥ 1 and 𝐷 < ∞ there exist positive constants 𝑐 1 = 𝑐 1 (𝐷) and ℎ0 = ℎ0 (𝑑, 𝐷), and universal positive constants 𝑐 2 and 𝑐 3 such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a unimodular transitive graph with vertex degree 𝑑, let 𝐴1 , . . . , 𝐴𝑛 be non-empty sets of vertices in 𝐺, and suppose that 0 < 𝑝 1 < 𝑝 2 < 1 are such that there is at most one infinite cluster P 𝑝 -almost surely for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. Let ℎ ≥ (min𝑖 | 𝐴𝑖 |) −1 and let 𝑟 be a positive integer with ℎ𝑟 ≥ 1 such that P 𝑝 (Piv[1, ℎ𝑟]) < ℎ for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. If 𝑝 1 ≥ 1/𝑑, 𝛿 = 𝛿( 𝑝 1 , 𝑝 2 ) ≤ 𝐷, and ℎ ≤ ℎ0 then the implication   3 4 ℎ𝑐1 𝛿 ≤ 𝑐 3 𝑛−1 and 𝜏𝑝Λ1 ( 𝐴𝑖 ∪ 𝐴𝑖+1 ) ≥ 4ℎ𝑐1 𝛿 for every 𝑖 = 1, . . . , 𝑛 − 1   𝐵2𝑟 (Λ) Λ Λ ⇒ 𝜏𝑝2 ( 𝐴1 , 𝐴𝑛 ) ≥ 𝑐 2 𝜏𝑝1 ( 𝐴1 )𝜏𝑝1 ( 𝐴𝑛 ) (3.4.1) holds for every non-empty set of vertices Λ in 𝐺. In particular, taking Λ = 𝑉 yields the implication   3 4 ℎ𝑐1 𝛿 ≤ 𝑐 3 𝑛−1 and 𝜏𝑝1 ( 𝐴𝑖 ∪ 𝐴𝑖+1 ) ≥ 4ℎ𝑐1 𝛿 for every 𝑖 = 1, . . . , 𝑛 − 1   ⇒ 𝜏𝑝2 ( 𝐴1 , 𝐴𝑛 ) ≥ 𝑐 2 𝜏𝑝1 ( 𝐴1 )𝜏𝑝1 ( 𝐴𝑛 ) (3.4.2) whenever 𝑝 1 ≥ 1/𝑑, 𝛿 = 𝛿( 𝑝 1 , 𝑝 2 ) ≤ 𝐷, and ℎ ≤ ℎ0 . Remark 3.4.1. The fact that we can always find an integer 𝑟 such that P 𝑝 (Piv[1, ℎ𝑟]) < ℎ for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ] can be deduced from the fact that there is at most one infinite cluster P 𝑝 -almost surely for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ] by an easy compactness argument. Remark 3.4.2. The Λ = 𝑉 case of this lemma stated in (3.4.2) already allows us to easily deduce that 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺) when the transitive graphs in the sequence (𝐺 𝑛 )𝑛≥1 all satisfy a uniform 10 superpolynomial growth lower bound of the form Gr(𝑟) ≥ 𝑟 𝑐(log log 𝑟) . This is explained in detail in Section 3.4. Working within finite domains as in (3.4.1) will be useful when we apply Proposition 3.4.1 at low-growth scales in Section 3.6. The basic idea underlying Proposition 3.4.1, which is inspired by earlier works including [CMT22; DKT21], is that one can use the universal two-arm estimates derived from the work of Aizenman, Kesten, and Newman [AKN87b] as reviewed in Section 3.2 to bound the maximum influence of an edge on certain connection events, which can then be used as an input in Talagrand’s sharp threshold theorem [Tal94]. Compared to those works, our primary additional insight is that these methods can be made vastly more efficient (especially in the high-growth case) by working with ghost field connection events instead of more obvious connection events. Intuitively, these ghost field connection events are “smoother” than ordinary connection events, making it easier to bound the maximum influence of an edge. Moreover, the influence bound we get by using the two-ghost 59 inequality of [Hut20a] gets better as the size of the relevant sets increases, so that we get extremely strong sharp threshold estimates when the graph has high growth. Given a set of vertices 𝐴 in a graph 𝐺 and a parameter ℎ ∈ [0, 1], the ghost field of intensity ℎ on 𝐴 is the random subset G 𝐴 of 𝐴 in which each vertex is included independently at random with probability8 ℎ. We denote the law of G 𝐴 by Gℎ𝐴 . We record the following reformulation of the two-ghost inequality of [Hut20a]. Lemma 3.4.2. Let 𝐺 be a unimodular transitive graph of vertex degree 𝑑. There exists a constant 𝐶 = 𝐶 (𝑑) such that √︄ 1− 𝑝 ℎ Gℎ𝐴 ⊗ Gℎ𝐵 ⊗ P 𝑝 ({𝑥 ↔ G 𝐴 } ∩ {𝑦 ↔ G𝐵 } ∩ {𝑥 ↮ 𝑦} ∩ {𝑥 ↮ ∞ or 𝑦 ↮ ∞}) ≤ 𝐶 𝑝 for every ℎ ∈ [0, 1], every pair of neighbouring vertices 𝑥, 𝑦 ∈ 𝑉 (𝐺), every two sets of vertices 𝐴, 𝐵 ⊆ 𝑉 (𝐺), and every 𝑝 ∈ (0, 1). Proof. It suffices without loss of generality to consider the case 𝐴 = 𝐵 = 𝑉 since the relevant probability is increasing in 𝐴 and 𝐵. In this case, the probability is the same as if we had a single ghost field instead of two independent ghost fields, since the restrictions of a ghost field to the clusters of 𝑥 and 𝑦 are independent when these clusters are disjoint. This version of the lemma then follows easily from Theorem 3.2.5. Alternatively, one can deduce the desired estimate from [Hut20a, Theorem 1.6]. (The only difference is that in that paper the ghost fields are parameterised by 1 − 𝑒 −ℎ and are random sets of edges rather than vertices. As with Theorem 3.2.5, this is not a problem since the edge version of the statement is stronger than the vertex version.) □ The following lemma states roughly that the probability that two low-intensity ghost fields are connected in a region Λ undergoes a sharp threshold with respect to the percolation parameter 𝑝. This rough statement has two caveats: as we increase the percolation parameter, we also have to increase the ghost field intensity and thicken the region Λ. Although the argument using ghost field connections is new, the way we adapt it to run inside a given domain Λ (when Λ is not the whole vertex set) is inspired by the analysis of [CMT22, Section 5]. Lemma 3.4.3 (Sharp threshold for ghost connections). For each 𝑑 ≥ 1 and 𝐷 < ∞ there exists a positive constant 𝑐 = 𝑐(𝑑, 𝐷) such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a unimodular 8 In the literature one often takes this probability to be 1 − 𝑒 −ℎ , which makes certain calculations more convenient. The distinction makes little difference since 1 − 𝑒 −ℎ = ℎ ± 𝑂 (ℎ2 ) as ℎ → 0. 60 transitive graph with vertex degree 𝑑, let 𝐴, 𝐵 be non-empty sets of vertices in 𝐺, and suppose that 0 < 𝑝 1 < 𝑝 2 < 1 are such that either (i) there is at most one infinite cluster P 𝑝 -almost surely for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ], or (ii) 𝐵 has finite complement. If 𝑝 1 ≥ 1/𝑑 and 𝛿( 𝑝 1 , 𝑝 2 ) ≤ 𝐷 then the implication     𝐵𝑟 (Λ) Λ Gℎ𝐴 ⊗ Gℎ𝐵 ⊗ P 𝑝1 (G 𝐴 ← → G𝐵 ) ≥ ℎ𝑐𝛿( 𝑝1 ,𝑝2 ) ⇒ Gℎ𝐴𝑐 ⊗ Gℎ𝐵𝑐 ⊗ P 𝑝2 (G 𝐴 ←−−→ G𝐵 ) ≥ 1 − ℎ𝑐𝛿( 𝑝1 ,𝑝2 ) holds for every ℎ ≤ 1/𝑑 and every set Λ ⊆ 𝑉, where 𝑟 is the minimum positive integer such that P 𝑝 (Piv[1, ℎ𝑟]) < ℎ for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. Remark 3.4.3. To prove Proposition 3.4.1 we will apply this lemma only under the hypothesis (i). The version with hypothesis (ii) can be used as part of an alternative of the joint continuity of 𝜃 ( 𝑝, 𝐺) in the supercritical region, as discussed in Section 3.7. Before proving Lemma 3.4.3 we first recall Talagrand’s inequality [Tal94], which (in combination with Russo’s formula [Rus78]) states that there exists a universal positive constant 𝑐 such that if 𝐴 ⊆ {0, 1} 𝐸 is an increasing event in a finite product space then   −1 2 1 𝑑 P 𝑝 (A) ≥ 𝑐P 𝑝 (A)(1 − P 𝑝 (A)) · 𝑝(1 − 𝑝) log log 𝑑𝑝 𝑝(1 − 𝑝) 𝑝(1 − 𝑝) max𝑒∈𝐸 P 𝑝 (Piv𝑒 [A]) (3.4.3) for every 𝑝 ∈ (0, 1). Proof of Lemma 3.4.3. We may assume without loss of generality that 𝐴, 𝐵, and Λ are finite, exhausting by finite sets and taking a limit otherwise. Since we want to apply Talagrand’s inequality to the inhomogeneous9 product measure Gℎ𝐴 ⊗ Gℎ𝐵 ⊗ P 𝑝 , we will first encode a random variable with this law as a function of i.i.d. random bits. Define   log(1 − 𝑝 1 ) , 𝑞 1 := 1 − (1 − 𝑝 1 ) 1/𝑚 𝐸 , and 𝑞 2 := 1 − (1 − 𝑝 2 ) 1/𝑚 𝐸 . 𝑚 𝐸 := log((𝑑 − 1)/𝑑) 9 The fact that Talagrand’s inequality for homogeneneous product measures implies a version for inhomogeneous measures with parameters close to zero was already observed in [DT22, Appendix A]. In our setting we have some parameters that may be close to zero and others that may be close to 1. 61 The assumption that 𝑝 1 ≥ 1/𝑑 ensures that 𝑚 𝐸 ≥ 1, while it follows from the definitions that (1 − 𝑞 1 ) 𝑚 𝐸 = (1 − 𝑝 1 ), that (1 − 𝑞 2 ) 𝑚 𝐸 = (1 − 𝑝 2 ), and that !    −1 log(1 − 𝑝 1 ) 1 2𝑑 − 1 2 ≤ 𝑞 1 = 1 − exp ≤ . log(1 − 𝑝 1 ) ≤ 𝑑 log((𝑑 − 1)/𝑑) 𝑑 𝑑2 (Here we used only that 𝑥/2 ≤ ⌊𝑥⌋ ≤ 𝑥 for 𝑥 ≥ 1.) We also define   log ℎ 𝑚𝐺 = , log 𝑞 1 which satisfies 𝑚 𝐺 ≥ 1 since ℎ ≤ 1/𝑑 ≤ 𝑞 1 . For each 𝑞 ∈ (0, 1), let P̄𝑞 be the law of a random variable bits = (bits 𝐴 , bits𝐵 , bits𝜔 ) ∈ {0, 1} 𝐴×{1,...,𝑚 𝐺 } × {0, 1} 𝐵×{1,...,𝑚 𝐺 } × {0, 1} 𝐸×{1,...,𝑚 𝐸 } =: {0, 1}Ω , whose constituent random bits are independent Bernoulli random variables of parameter 𝑞. Given bits = (bits 𝐴 , bits𝐵 , bits𝜔 ) ∈ {0, 1}Ω , we define (G 𝐴 , G𝐵 , 𝜔) as a function of bits by G 𝐴 (𝑎) = 𝑚𝐺 Ö bits 𝐴 (𝑎, 𝑖), G𝐵 (𝑏) = 𝑖=1 𝑚𝐺 Ö bits𝐵 (𝑏, 𝑖), and 𝜔(𝑒) = 1 − 𝑖=1 𝑚𝐸 Ö (1 − bits𝜔 (𝑒, 𝑖)), 𝑖=1 so that the triple (G 𝐴 , G𝐵 , 𝜔) has law G𝑞𝐴𝑚𝐺 ⊗ G𝑞𝐵𝑚𝐺 ⊗ P1−(1−𝑞) 𝑚𝐸 . The choice of parameter 𝑚 𝐺 ensures that 𝑞 1𝑚 𝐺 ≥ ℎ and hence that Λ Λ P̄𝑞1 (G 𝐴 ← → G𝐵 ) ≥ Gℎ𝐴 ⊗ Gℎ𝐵 ⊗ P 𝑝1 (G 𝐴 ← → G𝐵 ). Note moreover that if A ⊆ {0, 1} 𝐴 × {0, 1} 𝐵 × {0, 1} 𝐸 is an increasing event, 𝑒 ∈ 𝐸 is an edge, and 1 ≤ 𝑘 ≤ 𝑚 𝐸 then (𝑒, 𝑘) is a closed pivotal for the event {bits : (G 𝐴 , G𝐵 , 𝜔) ∈ A} if and only if 𝑒 is a closed pivotal for the event A in the configuration 𝜔, so that (1 − 𝑞) P̄𝑞 (𝑒, 𝑘) pivotal for {bits : (G 𝐴 , G𝐵 , 𝜔) ∈ A}
= (1 − 𝑞) 𝑚 𝐸 G𝑞𝐴𝑚𝐺 ⊗ G𝑞𝐵𝑚𝐺 ⊗ P1−(1−𝑞) 𝑚𝐸 (Piv𝑒 [A]). (3.4.4) On the other hand, an element (𝑥, 𝑘) of 𝐴 × {1, . . . , 𝑚 𝐺 } can only possibly be an open pivotal if bits 𝐴 (𝑥, 𝑗) = 1 for every 𝑗 ∈ {1, . . . , 𝑚 𝐺 }. Similar considerations also apply with 𝐴 replaced by 𝐵, so that we obtain the coarse bound
𝑞 P̄𝑞 (𝑥, 𝑘) pivotal for {bits : (G 𝐴 , G𝐵 , 𝜔) ∈ A} ≤ 𝑞 𝑚 𝐺 for every 𝑥 ∈ 𝐴 ⊔ 𝐵 and 1 ≤ 𝑘 ≤ 𝑚 𝐺 . 62 (3.4.5) Let ℓ := ⌊ℎ−1 ⌋ and for each 𝑖 = 1, . . . , ℓ define the event 𝐵𝑖𝑟 ℎ (Λ) E𝑖 := {G 𝐴 ←−−−−→ G𝐵 }. We want to bound the maximum influence of an element of (𝐸 ×{1, . . . , 𝑚 𝐸 }) ⊔ ( 𝐴×{1, . . . , 𝑚 𝐺 }) ⊔ (𝐵 × {1, . . . , 𝑚 𝐺 }) on the event E𝑖 . For elements of ( 𝐴 × {1, . . . , 𝑚 𝐺 }) ⊔ (𝐵 × {1, . . . , 𝑚 𝐺 }) it will suffice to use the trivial bound of (3.4.5), so that it remains only to bound the pivotality probability max𝑒 Gℎ𝐴′ ⊗ Gℎ𝐵′ ⊗ P 𝑝 (Piv𝑒 [E𝑖 ]) where (1 − 𝑝) = (1 − 𝑞) 𝑚 𝐸 and ℎ′ = 𝑞 𝑚 𝐺 . Following [CMT22], we will do this not for every 𝑖 but instead show that an influence bound of the desired form must hold for an average choice of 𝑖. (Note that if we are working directly in the case Λ = 𝑉 then this issue does not arise.) More precisely, we claim that for each 𝑞 ∈ [𝑞 1 , 𝑞 2 ] there exists a set 𝐼 (𝑞) ⊆ {1, . . . , ℓ} with |𝐼 | ≥ 1/(3ℎ) such that max max 𝑞(1 − 𝑞)Gℎ𝐴′ ⊗ Gℎ𝐵′ ⊗ P 𝑝 (Piv𝑒 [E𝑖 ]) ≤ 𝐶𝑑 (ℎ′) 1/2 , 𝑖∈𝐼 𝑒∈𝐸 (3.4.6) where 𝐶𝑑 is a constant depending only on the degree 𝑑. There are two separate cases to consider: Edges both of whose endpoints belong to 𝐵 (𝑖−1)𝑟 ℎ (Λ) (bulk edges) and edges with at least one endpoint not in 𝐵 (𝑖−1)𝑟 ℎ (Λ) (boundary edges). Bulk edges. First consider an edge 𝑒 both of whose endpoints 𝑥 and 𝑦 belong to 𝐵 (𝑖−1)𝑟 ℎ (Λ). If 𝜔 is such that 𝑒 is pivotal for the event E𝑖 then at least one of the following two events must occur: (i) The endpoints 𝑥 and 𝑦 of 𝑒 belong to distinct 𝜔-clusters, one of which intersects G 𝐴 but not G𝐵 and the other of which intersects G𝐵 but not G 𝐴 ; at least one of these clusters must be finite almost surely by the hypotheses of the lemma, which allows us to bound the relevant probability using the two-ghost inequality. (ii) The endpoints 𝑥 and 𝑦 of 𝑒 are both 𝜔-connected to the boundary of 𝐵𝑖𝑟 ℎ (Λ) but are not 𝜔-connected to each other within 𝐵𝑖𝑟 ℎ (Λ), so that Piv[1, 𝑟 ℎ] (𝑥) and Piv[1, 𝑟 ℎ] (𝑦) both hold. Thus, it follows by the two-ghost inequality as stated in Lemma 3.4.2 and the definition of 𝑟 that   (1 − 𝑝) 1/2 ′ 1/2 ′ 12 (ℎ ) + ℎ ≤ 𝐶 (ℎ ) , (3.4.7) 𝑞(1 − 𝑞) P̄𝑞 (Piv𝑒 [E𝑖 ]) ≤ 𝑞(1 − 𝑞) 𝐶1 2 𝑝 1/2 for every 𝑞 ∈ [𝑞 1 , 𝑞 2 ] and 1 ≤ 𝑖 ≤ ℓ, where 𝐶1 and 𝐶2 are constants depending only on 𝑑 and we used that 𝑝 ≥ 1/𝑑. Boundary edges. An edge 𝑒 not having both its endpoints in 𝐵𝑖𝑟 ℎ (Λ) cannot possibly be pivotal for E𝑖 ; we need only bound P̄𝑞 (Piv𝑒 [E𝑖 ]) for edges 𝑒 with both endpoints in 𝐵𝑖𝑟 ℎ (Λ) and with 63 at least one endpoint not in 𝐵 (𝑖−1)𝑟 ℎ (Λ). Rather than bound this maximum influence uniformly in 𝑖, we will bound it on average. Fix 𝑞 ∈ [𝑞 1 , 𝑞 2 ] and, for each 𝑖 ∈ {1, . . . , ℓ}, pick an edge 𝑒𝑖 = 𝑒𝑖 (𝑞) ∈ 𝐵𝑖𝑟 ℎ (Λ) with at least one endpoint not in 𝐵 (𝑖−1)𝑟 ℎ (Λ) that maximises P̄𝑞 (Piv𝑒𝑖 [E𝑖 ]) over all such edges. Notice that the events Piv𝑒𝑖 [E𝑖 ] ∩ {𝜔(𝑒𝑖 ) = 1} for 𝑖 ∈ {1, . . . , ℓ} are pairwise disjoint. Indeed, if Piv𝑒𝑖 [E𝑖 ] ∩ {𝜔(𝑒𝑖 ) = 1} occurs then 𝑒𝑖 must belong to every path connecting G 𝐴 to G𝐵 in 𝐵𝑖𝑟 ℎ (Λ), and such a path cannot possibly include 𝑒 𝑗 if 𝑗 > 𝑖. It follows in particular that ℓ ℓ ∑︁ ∑︁ 𝑞 P̄𝑞 (Piv𝑒𝑖 [E𝑖 ]) = P̄𝑞 (Piv𝑒𝑖 [E𝑖 ] ∩ {𝜔(𝑒𝑖 ) = 1}) ≤ 1. 𝑖=1 𝑖=1 1 such By Markov’s inequality, it follows that we can find a subset 𝐼 (𝑞) ⊆ {1, . . . , ℓ} with |𝐼 | ≥ 3ℎ that max 𝑞 P̄𝑞 (Piv𝑒𝑖 [E𝑖 ]) ≤ 6ℎ ≤ 6(ℎ′) 1/2 𝑖∈𝐼 as required. This concludes the proof of (3.4.6). 𝐷 Now, the assumption that 𝛿( 𝑝 1 , 𝑝 2 ) ≤ 𝐷 implies that (1 − 𝑝 2 ) ≥ (1 − 𝑝 1 ) 𝑒 and hence that 𝐷 𝐷 1 − 𝑞 2 ≥ (1 − 𝑞 1 ) 𝑒 ≥ ((2𝑑 − 1)/(2𝑑)) 𝑒 > 0, so that 𝑞 1 and 𝑞 2 are bounded away from 0 and 1 by constants depending only on 𝑑 and 𝐷. As such, Talagrand’s inequality (which is valid to use in our setting since all our events only depend on the status of finitely many bits) together with (3.4.6) yields that   P̄𝑞 (E𝑖 ) 1 1 1 𝑑 log ≥ 𝑐 1 log ≥ 𝑐 3 log ≥ 𝑐 2 𝑚 𝐺 log 𝑚 /2 𝑑𝑞 𝑞1 ℎ 𝐶𝑑 𝑞 𝐺 1 − P̄𝑞 (E𝑖 ) for every 𝑞 1 ≤ 𝑞 ≤ 𝑞 2 and every 𝑖 ∈ 𝐼 (𝑞), where 𝑐 1 , 𝑐 2 , and 𝑐 3 are positive constants depending only on 𝑑 and 𝐷. Since ℓ ≤ ℎ−1 and the derivative is non-negative for every 𝑖, we can sum over 𝑖 to obtain that   ℓ P̄𝑞 (E𝑖 ) 𝑐3 1 ∑︁ 𝑑 1 log ≥ log ℓ 𝑖=1 𝑑𝑞 3 ℎ 1 − P̄𝑞 (E𝑖 ) for every 𝑞 1 ≤ 𝑞 ≤ 𝑞 2 . Integrating this differential inequality yields that     ℓ  P̄𝑞2 (E𝑖 ) P̄𝑞1 (E𝑖 ) 1 ∑︁ 𝑐3 1 log − log ≥ |𝑞 2 − 𝑞 1 | log ℓ 𝑖=1 3 ℎ 1 − P̄𝑞2 (E𝑖 ) 1 − P̄𝑞1 (E𝑖 ) and hence that there exists 𝑖 ∈ {1, . . . , ℓ} such that           P̄𝑞2 (E𝑖 ) P̄𝑞1 (E𝑖 ) 1 1 max log , log ≥ max log , − log 1 − P̄𝑞2 (E𝑖 ) P̄𝑞1 (E𝑖 ) 1 − P̄𝑞2 (E𝑖 ) 1 − P̄𝑞1 (E𝑖 ) 𝑐2 1 ≥ |𝑞 2 − 𝑞 1 | log . 6 ℎ 64 ωp2 ωp2 ωp1 Figure 3.1: Schematic illustration of the event whose probability is estimated in Lemma 3.4.4 when Λ = 𝑉: If any two points in 𝐴 have a reasonable probability to be connected in 𝜔 𝑝1 , then it is unlikely that 𝑋 is connected to a weak ghost field on 𝐴 in 𝜔 𝑝2 and 𝑌 is connected to a weak ghost field on 𝐴 in 𝜔 𝑝1 but that 𝑋 and 𝑌 are not connected in 𝜔 𝑝2 . The claim follows easily from this together with the inequality log( (𝑑−1)/𝑑) log( (𝑑−1)/𝑑) |𝑞 2 − 𝑞 1 | = |(1 − 𝑝 2 ) 1/𝑚 𝐸 − (1 − 𝑝 1 ) 1/𝑚 𝐸 | ≥ (1 − 𝑝 2 ) log(1− 𝑝1 ) − (1 − 𝑝 1 ) log(1− 𝑝1 )     𝑑−1 𝑑−1 log(1 − 𝑝 2 ) = − 1 log 1 − exp ≥ 𝑐 4 𝛿( 𝑝 1 , 𝑝 2 ), 𝑑 log(1 − 𝑝 1 ) 𝑑 which holds by calculus with the constant 𝑐 4 depending only on 𝑑 and 𝐷. □ Our next goal is to run a chaining-like argument but with ghost fields. In this analogy, the previous lemma can be thought of as an existence statement whereas the next lemma can be thought of as a uniqueness statement. (Note that our proof of the next lemma relies crucially on the previous lemma.) We work with the standard monotone coupling (𝜔 𝑝 ) 𝑝∈[0,1] of Bernoulli bond percolation, Λ write {𝐴 ← → 𝐵} for the event that 𝐴 is connected to 𝐵 by a path that is contained in Λ and open in 𝑝 Λ 𝜔 𝑝 , and write {𝐴 ← → 𝐵} for the event that 𝐴 is not connected to 𝐵 by any such path. 𝑝 Lemma 3.4.4 (Gluing ghost connections). For each 𝑑 ≥ 1 and 𝐷 < ∞ there exist positive constants 𝑐 1 = 𝑐 1 (𝑑, 𝐷) and 𝑐 2 = 𝑐 2 (𝑑, 𝐷) such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a unimodular transitive graph with vertex degree 𝑑, let 𝐴, 𝑋, and 𝑌 be non-empty sets of vertices in 𝐺, and suppose that 0 < 𝑝 1 < 𝑝 2 < 1 are such that there is at most one infinite cluster P 𝑝 -almost surely for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. If 𝑝 1 ≥ 1/𝑑 and 𝛿 = 𝛿( 𝑝 1 , 𝑝 2 ) ≤ 𝐷 then the implication " # " #   𝐵𝑟 (Λ) Λ 𝐵𝑟 (Λ) 𝑝2 𝑝1 𝑝2 𝜏𝑝Λ1 ( 𝐴) ≥ ℎ𝑐1 𝛿 ⇒ Gℎ𝐴 ⊗ P 𝑋 ←−−→ G 𝐴 and 𝑌 ←−→ G 𝐴 but 𝑋 ←−−→ 𝑌 ≤ 3ℎ𝑐2 𝛿 65 3 holds for every ℎ ≤ 1/𝑑 and every set Λ ⊆ 𝑉, where 𝑟 is the minimum positive integer such that P 𝑝 (Piv[1, ℎ𝑟]) < ℎ for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. An illustration of the event whose probability this lemma estimates is given in Figure 3.1. Remark 3.4.4. Several of the calculations in the following proof can be simplified significantly if one allows the constants to depend on how small 1 − 𝑝 1 is. Getting the constants to be independent of the choice of 𝑝 1 is important when using our methods to deduce that 𝑝 𝑐 < 1 for transitive graphs of superlinear volume growth. 4 4 Proof of Lemma 3.4.4. It suffices to prove the claim with 3ℎ𝑐2 𝛿 replaced by max{ℎ𝑐1 𝛿 , 3ℎ𝑐2 𝛿 }, as the latter can be bounded by the former after an appropriate decrease of the relevant constant. We write 𝑝 4/3 and 𝑝 5/3 for the parameters defined by 𝑝 4/3 = Spr( 𝑝 1 ; 𝛿/3) and 𝑝 5/3 = Spr( 𝑝 1 ; 2𝛿/3), so that 𝑝 1 ≤ 𝑝 4/3 ≤ 𝑝 5/3 ≤ 𝑝 2 . To lighten notation we write 𝜔1 = 𝜔 𝑝1 , 𝜔4/3 = 𝜔 𝑝4/3 , 𝜔5/3 = 𝜔 𝑝5/3 , and 𝜔2 = 𝜔 𝑝2 . We will write ≍, ⪯, and ⪰ for equalities and inequalities holding to within positive multiplicative constants depending only on 𝑑 and 𝐷. We also use the asymptotic big-𝑂 and big-Ω notation with all implicit constants depending only on 𝑑 and 𝐷. Let 𝑐 0 = 𝑐 0 (𝑑, 𝐷) be the constant from Lemma 3.4.3. We will prove the claim with 𝑐 1 = (𝑐 0 ∧ 1)/9. Assume to this end that 𝜏𝑝Λ1 ( 𝐴) ≥ ℎ𝑐1 𝛿 . Let 𝐾 𝑋 (2, 𝑟) be the set of vertices that are connected to 𝑋 by an 𝜔2 -open path in 𝐵𝑟 (Λ) and let 𝐾𝑌 (1, 0) be the set of vertices that are connected to 𝑌 by an 𝜔1 -open path in Λ. Suppose that there exist two disjoint, connected sets of vertices 𝐶 𝑋 ⊇ 𝑋 and 𝐶𝑌 ⊇ 𝑌 such that P(𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 ) > 0 and min{Gℎ𝐴 (G 𝐴 ∩ 𝐶 𝑋 ≠ ∅), Gℎ𝐴 (G 𝐴 ∩ 𝐶𝑌 ≠ ∅)}   𝐵𝑟 (Λ) Λ 𝐴 → 𝑌 | 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 ≥ ℎ𝑐1 𝛿 ; (3.4.8) ≥ Gℎ ⊗ P 𝑋 ←−−−→ G 𝐴 ← 𝑝2 𝑝1 If no such pair of sets exists then the conclusion holds trivially (since we are proving a modified 4 version of the claim with max{ℎ𝑐1 𝛿 , 3ℎ𝑐2 𝛿 } on the right hand side of the implication). For such 𝐶 𝑋 and 𝐶𝑌 we have that   Λ Λ 𝐶𝑋 𝐶𝑌 Gℎ ⊗ Gℎ ⊗ P G𝐶𝑋 ← → G𝐶𝑌 ≥ G𝐶ℎ 𝑋 ⊗ G𝐶ℎ 𝑌 (G𝐶𝑋 ∩ 𝐴 ≠ ∅ and G𝐶𝑌 ∩ 𝐴 ≠ ∅) min P(𝑢 ← → 𝑣) 𝑝1 𝑢,𝑣∈𝐴 Λ = Gℎ𝐴 (G 𝐴 ∩ 𝐶 𝑋 ≠ ∅)Gℎ𝐴 (G 𝐴 ∩ 𝐶𝑌 ≠ ∅) min P(𝑢 ← → 𝑣) 𝑢,𝑣∈𝐴 ≥ ℎ𝑐1 𝛿 · ℎ𝑐1 𝛿 · ℎ𝑐1 𝛿 = ℎ3𝑐1 𝛿 ≥ ℎ 66 𝑐0 3 𝛿 . 𝑝1 𝑝1 Thus, since 𝛿( 𝑝 1 , 𝑝 4/3 ) = 13 𝛿( 𝑝 1 , 𝑝 2 ), it follows from Lemma 3.4.3 (applied with 𝑝 1 and 𝑝 4/3 rather than 𝑝 1 and 𝑝 2 ) that     𝑐0 𝐵𝑟 (Λ) 𝐵𝑟 (Λ) P 𝐶 𝑋 ←−−→ 𝐶𝑌 ≤ G𝐶ℎ𝑐𝑋0 ⊗ G𝐶ℎ𝑐𝑌0 ⊗ P G𝐶𝑋 ←−−−→ G𝐶𝑌 ≤ ℎ 3 𝛿 . (3.4.9) 𝑝 4/3 𝑝 4/3 Let 𝑀 denote the maximum cardinality of a set of edge-disjoint paths from 𝐶 𝑋 to 𝐶𝑌 that are contained in 𝐵𝑟 (Λ) and are open in 𝜔5/3 with the possible exception of their first and last edges. (We stress that 𝐶 𝑋 and 𝐶𝑌 are not random, but are fixed sets satisfying (3.4.8).) By Menger’s theorem, 𝑀 is equal to the minimum size of a set of edges separating 𝐶 𝑋 from 𝐶𝑌 in the subgraph of 𝐵𝑟 (Λ) spanned by those edges that are either 𝜔5/3 -open or have at least one endpoint in 𝐶 𝑋 ∪ 𝐶𝑌 . Observe that 𝑀 is independent of the event {𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 } since 𝑀 depends only on edges with neither endpoint in 𝐶 𝑋 ∪ 𝐶𝑌 while the latter event depends only on edges with at least one endpoint in 𝐶 𝑋 ∪ 𝐶𝑌 . Conditional on 𝜔5/3 , each edge that is open in 𝜔5/3 is closed in 𝜔4/3 with probability 𝑝 5/3 − 𝑝 4/3 . 𝛽 := 𝑝 5/3 It follows that   𝐵𝑟 (Λ) P 𝐶 𝑋 ←−−→ 𝐶𝑌 | 𝑀 ≤ 𝑁 ≥ 𝛽 𝑁 𝑝 4/3 for every 𝑁 ≥ 0 and hence by (3.4.9) and the aforementioned independence that 𝑐0 P(𝑀 ≤ 𝑁 | 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 ) = P (𝑀 ≤ 𝑁) ≤ 𝛽−𝑁 ℎ 3 𝛿 (3.4.10) for every 𝑁 ≥ 0. Let 𝐾𝑌 (5/3, 𝑟) be the set of vertices that are connected to 𝑌 by an 𝜔5/3 -open path in 𝐵𝑟 (Λ). Conditioned on the event {𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 and 𝐾𝑌 (1, 0) = 𝐶𝑌 } and the value of 𝑀, the size of the boundary |𝜕𝐾 𝑋 (2, 𝑟) ∩ 𝜕𝐾𝑌 (5/3, 𝑟)| stochastically dominates a sum of 𝑀 Bernoulli random variables of parameter 𝑝 5/3 − 𝑝 1 . 𝛼1 := 1 − 𝑝1 Indeed, if we take some maximal-cardinality set of paths as in the definition of 𝑀, then the edge adjacent to 𝐶𝑌 of each path in the set is open in 𝜔5/3 with this probability, and the size of the relevant boundary is at least the number of these edges that are open in 𝜔5/3 . Letting 𝑍 be a sum of 𝑁 Bernoulli random variables each of parameter 1 − 𝛼1 , we have that there exist constants 𝑐 3 and 67 𝑐 4 depending only on 𝑑 and 𝐷 such that   𝛼1 𝑁 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 , 𝑀 ≥ 𝑁 P |𝜕𝐾 𝑋 (2, 𝑟) ∩ 𝜕𝐾𝑌 (5/3, 𝑟)| ≤ 2       2 − 𝛼1 𝛼1 ≤P 𝑍≥ 𝑁 =P 𝑍 ≥ 1+ E𝑍 2 2(1 − 𝛼1 )     2 − 𝛼1 2 − 𝛼1 𝛼1 ≤ exp − log − 𝑁 (3.4.11) 2 2 − 2𝛼1 2 for every 𝑁 ≥ 1, where the final inequality follows from the Chernoff bound P(𝑍 ≥ (1 + 𝑥)E𝑍) ≤ (𝑒 −𝑥 (1 + 𝑥) 1+𝑥 ) −E𝑍 , which holds for all sums of independent Bernoulli random variables. We will now apply (3.4.10) and (3.4.11) with   𝑐0 𝛿 1 𝑁 := log , 12 log(1/𝛽) ℎ to prove the inequality 

P 𝜕𝐾 𝑋 2, 𝑟 ∩ 𝜕𝐾𝑌 5/3, 𝑟 ≤ 𝑐0 so that 𝑐0 𝛽−𝑁 ℎ 3 𝛿 ≤ ℎ 6 𝛿 𝑐 0 𝛼1 𝛿 1 log 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 48 log(1/𝛽) ℎ  ≤ 2ℎΩ(𝛿 ) . (3.4.12) 4 To do this, we will make repeated use of the elementary estimates 𝑝 5/3 𝛽 = (1 − 𝑝 5/3 ) 𝑒 − 𝛿/3 − (1 − 𝑝 5/3 ) = (1 − 𝑝 5/3 )((1 − 𝑝 5/3 ) 𝑒 − 𝛿/3 −1 − 1) ≥ (1 − 𝑝 5/3 )((1 − 𝑝 5/3 ) −Ω(𝛿) − 1) ⪰ 𝛿(1 − 𝑝 5/3 ) log 1 1 − 𝑝 5/3 (3.4.13) and 𝛼1 = 1 − (1 − 𝑝 5/3 ) 1−𝑒 −2 𝛿/3 = 1 − (1 − 𝑝 5/3 ) Θ(𝛿) , (3.4.14) where we recall that the implicit constant appearing here depends only on 𝑑 and 𝐷. First suppose that 𝑁 ≥ 1, so that the rounding in the definition of 𝑁 reduces its size by a factor of at most 1/2. In this case, (3.4.10) and (3.4.11) yield that there exist positive constants 𝐶1 , 𝑐 5 and 𝑐 6 depending only on 𝑑 and 𝐷 such that  

𝑐 0 𝛼1 𝛿 1 P 𝜕𝐾 𝑋 2, 𝑟 ∩ 𝜕𝐾𝑌 5/3, 𝑟 ≤ log 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 48 log(1/𝛽) ℎ     𝑐0 𝑐0 𝛿 2 − 𝛼1 1 2 − 𝛼1 𝛼1 𝛿 ≤ ℎ 6 + exp − log − log . (3.4.15) 24 log(1/𝛽) 2 2 − 2𝛼1 2 ℎ 68 We claim that   𝑐0 𝛿 2 − 𝛼1 2 − 𝛼1 𝛼1 log − ⪰ 𝛿4 . 24 log(1/𝛽) 2 2 − 2𝛼1 2 (3.4.16) We prove this by a case analysis according to whether 1 − 𝑝 5/3 ≤ 𝑒 −1/𝛿 . If 1 − 𝑝 5/3 ≥ 𝑒 −1/𝛿 , it follows from (3.4.13) that log 1/𝛽 = 𝑂 (𝛿) (where we stress again that the implicit constants depend only on 𝑑 and 𝐷). On the other hand, since 𝑝 1 ≥ 1/𝑑, we have that 𝛼1 ⪰ 𝛿, and together with the elementary inequality 𝛼12 2 − 𝛼1 𝛼1 2 − 𝛼1 log − ≥ 2 2 − 2𝛼1 2 8 this yields that if 1 − 𝑝 5/3 ≥ 𝑒 −1/𝛿 then   𝑐0 𝛿 2 − 𝛼1 𝛼1 𝛿 2 − 𝛼1 log − ⪰ · 𝛿2 ⪰ 𝛿3 24 log(1/𝛽) 2 2 − 2𝛼1 2 𝛿 as claimed. On the other hand, if 1 − 𝑝 5/3 ≤ 𝑒 −1/𝛿 then 1 − 𝛼1 ≥ 1 − 𝑒 −Ω(1) ⪰ 1, so that 2 − 𝛼1 2 − 𝛼1 𝛼1 1 1 log − ⪰ log ⪰ 𝛿 log . 2 2 − 2𝛼1 2 1 − 𝛼1 1 − 𝑝 5/3 We have under the same assumption that log(1/𝛽) ⪯ log 1 , 1 − 𝑝 5/3 and it follows that if 1 − 𝑝 5/3 ≤ 𝑒 −1/𝛿 then   2 − 𝛼1 𝑐0 𝛿 2 − 𝛼1 𝛼1 log − ⪰ 𝛿2 ⪰ 𝛿3 . 24 log(1/𝛽) 2 2 − 2𝛼1 2 This completes the proof of (3.4.16), which together with (3.4.15) yields the claimed inequality (3.4.12) in the case that 𝑁 ≥ 1. Now suppose that 𝑁 = 0, so that   𝑐 0 𝛼1 𝛿 1 P 𝜕𝐾 𝑋 2, 𝑟 ∩ 𝜕𝐾𝑌 5/3, 𝑟 ≤ 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 log 48 log(1/𝛽) ℎ   𝑐0

= P 𝜕𝐾 𝑋 2, 𝑟 ∩ 𝜕𝐾𝑌 5/3, 𝑟 = 0 𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 ≤ ℎ 3 𝛿 + (1 − 𝛼1 ),

(3.4.17) where, as in (3.4.11), the first term on the right hand side of the second line bounds the probability that 𝑀 = 0 and the second bounds the probability of the appropriate event conditional on 𝑀 ≥ 1. We will once again prove (3.4.12) via case analysis according to whether or not 1 − 𝑝 5/3 ≥ 𝑒 −1/𝛿 . −2 If 1 − 𝑝 5/3 ≥ 𝑒 −1/𝛿 then log(1/𝛽) = 𝑂 (𝛿), and since 𝑁 = 0 it follows that ℎ = 𝑒 −𝑂 (𝛿 ) . As such, 69 4 in this case there exists a positive constant 𝑐 5 such that ℎ𝑐5 𝛿 ≥ 1/2, and the desired inequality (3.4.12) follows trivially since probabilities are bounded by 1. Otherwise, 1 − 𝑝 5/3 ≤ 𝑒 −1/𝛿 and −1 1 − 𝛼1 ≤ (1 − 𝑝 5/3 ) Ω(𝛿) . Since 𝑁 = 0, we also have that ℎ ≥ (1 − 𝑝 5/3 ) 𝑂 (𝛿 ) and hence that 2 ℎ 𝛿 = (1 − 𝛼1 ) Ω(1) . As such, (3.4.17) is stronger than the claimed inequality (3.4.12) when 𝑁 = 0 regardless of the value of 1 − 𝑝 5/3 . This completes the proof of the inequality (3.4.12). Since the sets 𝐶 𝑋 and 𝐶𝑌 were arbitrary sets such that P(𝐾 𝑋 (2, 𝑟) = 𝐶 𝑋 , 𝐾𝑌 (1, 0) = 𝐶𝑌 ) > 0 and that satisfy (3.4.8), it follows from (3.4.12) that   𝐵𝑟 (Λ) 𝐵𝑟 (Λ) 4 Λ 𝐴 Gℎ ⊗ P 𝑋 ←−−→ G 𝐴 ← → 𝑌 but 𝑋 ←−−→ 𝑌 ≤ ℎ𝑐1 𝛿 + 2ℎΩ(𝛿 ) 𝑝2 𝑝1 𝑝2   𝐵𝑟 (Λ) 1 𝑐 0 𝛼1 𝛿 log but 𝑋 ←−−→ 𝑌 , (3.4.18) + P |𝜕𝐾 𝑋 (2, 𝑟) ∩ 𝜕𝐾𝑌 (5/3, 𝑟)| ≥ 𝑝2 48 log(1/𝛽) ℎ where the first term ℎ𝑐1 𝛿 accounts for the possibility that 𝐾 𝑋 (2, 𝑟) and 𝐾𝑌 (1, 0) are equal to some sets 𝐶 𝑋 and 𝐶𝑌 that do not satisfy (3.4.8). It remains to bound the final term appearing on the right hand side of (3.4.18). Notice that we can replace the set 𝐾 𝑋 (2, 𝑟) appearing in (3.4.18) by the set 𝐾˜ 𝑋 of vertices that are connected to 𝑋 by an 𝜔2 -open path using only edges that have both endpoints in 𝐵𝑟 (Λ) and neither endpoint in 𝐾𝑌 (5/3, 𝑟), since the two sets 𝐾 𝑋 (2, 𝑟) and 𝐾˜ 𝑋 are 𝐵𝑟 (Λ) equal when 𝑋 ←−−→ 𝑌 . Now let 𝐶 𝑋 ⊇ 𝑋 and 𝐶𝑌 ⊇ 𝑌 be connected sets of vertices that are disjoint 𝑝2 from each other such that P( 𝐾˜ 𝑋 = 𝐶 𝑋 , 𝐾𝑌 (5/3, 𝑟) = 𝐶𝑌 ) > 0 and 𝑐 0 𝛼1 𝛿 1 |𝜕𝐶 𝑋 ∩ 𝜕𝐶𝑌 | ≥ log ; 48 log(1/𝛽) ℎ If no such sets exist then the last term on the right hand side of (3.4.18) is zero. Each edge that is closed in 𝜔5/3 is open in 𝜔2 with probability 𝑝 2 − 𝑝 5/3 𝛼2 = , 1 − 𝑝 5/3 and we have by independence that   𝐵𝑟 (Λ) P 𝑋 ←−−→ 𝑌 | 𝐾˜ 𝑋 = 𝐶 𝑋 , 𝐾𝑌 (5/3, 𝑟) = 𝐶𝑌 ≤ (1 − 𝛼2 ) |𝜕𝐶𝑋 ∩𝜕𝐶𝑌 | 𝑝2     3 𝑐 0 𝛼1 𝛿 1 1 log log ≤ ℎΩ(𝛿 ) , ≤ exp − 48 log(1/𝛽) 1 − 𝛼2 ℎ where the final inequality follows by a similar calculation used to prove (3.4.16) above. Thus, summing over all choices of 𝐶 𝑋 and 𝐶𝑌 , we obtain that   𝐵𝑟 (Λ) 3 𝑐 𝛼 𝛿 1 0 1 P 𝑋 ←−−→ 𝑌 𝜕 𝐾˜ 𝑋 ∩ 𝜕𝐾𝑌 (5/3, 𝑟) ≥ log ≤ ℎΩ(𝛿 ) , (3.4.19) 𝑝2 48 log(1/𝛽) ℎ and the claim follows from (3.4.18) and (3.4.19). 70 □ We now apply Lemmas 3.4.3 and 3.4.4 to prove Proposition 3.4.1. Ë𝑛 𝐴𝑖 Proof of Proposition 3.4.1. To lighten notation, for each 𝜌 ∈ [0, 1] define Q 𝜌 := 𝑖=1 G 𝜌 , and for each 𝑖 write G𝑖 := G 𝐴𝑖 . Let 𝑐 1 = 𝑐 1 (𝑑, 𝐷) be the constant from Lemma 3.4.3 and let 𝑐 2 = 𝑐 2 (𝑑, 𝐷) and 𝑐 3 = 𝑐 3 (𝑑, 𝐷) be the constants from Lemma 3.4.4 (that are called 𝑐 1 and 𝑐 2 in the statement of that lemma). Define 𝑐 = 𝑐(𝑑, 𝐷) = (2𝐷) −5 (𝑐 1 ∧ 1)(𝑐 2 ∧ 1)(𝑐 3 ∧ 1). Let ℎ ≥ (min𝑖 | 𝐴𝑖 |) −1 , noting that Gℎ𝐴𝑖 (G𝑖 ≠ ∅) ≥ 1 − (1 − (min𝑖 | 𝐴𝑖 |) −1 ) | 𝐴𝑖 | ≥ 1 − 1/𝑒 ≥ 1/2 for every 𝑖, and let 𝑟 be the minimum positive integer such that P 𝑝 (Piv[1, ℎ𝑟]) < ℎ for every 𝑝 ∈ [ 𝑝 1 , 𝑝 2 ]. Fix 𝑢 ∈ 𝐴1 and 𝑣 ∈ 𝐴𝑛 and suppose that 𝜏𝑝Λ1 ( 𝐴𝑖 ∪ 𝐴𝑖+1 ) ≥ 4ℎ𝑐𝛿 4 for all 1 ≤ 𝑖 ≤ 𝑛 − 1. We want to bound from below the probability under P 𝑝2 that 𝑢 and 𝑣 are connected inside 𝐵2𝑟 (Λ). Let 1 ≤ 𝑖 ≤ 𝑛 − 1 be arbitrary. Note that  Λ Qℎ ⊗ P G𝑖 ← → G𝑖+1  𝑝1 4 ≥ Qℎ (G𝑖 ≠ ∅)Qℎ (G𝑖+1 ≠ ∅)𝜏𝑝Λ1 ( 𝐴𝑖 , 𝐴𝑖+1 ) ≥ 4ℎ𝑐𝛿 · and similarly that   1 Λ Qℎ ⊗ P 𝑢 ← → G1 ≥ 𝜏𝑝Λ1 ( 𝐴1 ) 𝑝1 2   Λ and Qℎ ⊗ P G𝑛 ← →𝑣 ≥ 𝑝1  2 1 ≥ ℎ𝑐1 𝛿/2 , (3.4.20) 2 1 Λ 𝜏 ( 𝐴𝑛 ). 2 𝑝1 (3.4.21) Let 𝑝 3/2 = Spr( 𝑝 1 ; 𝛿/2), so that 𝛿( 𝑝 1 , 𝑝 3/2 ) = 12 𝛿. Using (3.4.20), Lemma 3.4.3 implies that 𝐵𝑟 (Λ) Qℎ𝑐 ⊗ P(G𝑖 ←−−→ G𝑖+1 ) ≥ 1 − ℎ𝑐1 𝛿/2 ≥ 1 − ℎ𝑐𝛿 . 𝑝 3/2 (3.4.22) Since we also have by choice of 𝑐 that 𝑟 (Λ) 𝜏𝑝𝐵3/2 ( 𝐴𝑖 ) ≥ 𝜏𝑝Λ1 ( 𝐴𝑖 ) ≥ 𝜏𝑝Λ1 ( 𝐴𝑖 ∪ 𝐴𝑖+1 ) ≥ 4ℎ𝑐𝛿 ≥ (ℎ𝑐1 ) 𝑐2 𝛿/2 , 4 we may apply Lemma 3.4.4 (with [ 𝑝 1 , 𝑝 2 , ℎ, 𝑋, 𝑌 , 𝐴, Λ] := [ 𝑝 3/2 , 𝑝 2 , ℎ𝑐1 , {𝑢}, G𝑖+1 , 𝐴𝑖 , 𝐵𝑟 (Λ)]) to deduce that if ℎ𝑐1 ≤ 1/𝑑 then   𝐵2𝑟 (Λ) 𝐵𝑟 (Λ) 𝐵2𝑟 (Λ) 3 3 Qℎ𝑐 ⊗ P 𝑢 ←−−−→ G𝑖 ←−−→ G𝑖+1 but 𝑢 ←−−−→ G𝑖+1 ≤ 4(ℎ𝑐1 ) 𝑐3 (𝛿/2) = 4ℎ𝑐4 𝛿 , (3.4.23) 𝑝2 𝑝 3/2 𝑝2 where 𝑐 4 = 𝑐 4 (𝑑, 𝐷) = (𝑐 1 · 𝑐 3 )/8. (Note that using ℎ𝑐1 instead of ℎ changes the hypotheses on 𝑟, but this is not a problem since the hypotheses on 𝑟 associated to ℎ𝑐1 are weaker than 71 those associated to ℎ.) A similar application of Lemma 3.4.4 (with [ 𝑝 1 , 𝑝 2 , ℎ, 𝑋, 𝑌 , 𝐴, Λ] := [ 𝑝 3/2 , 𝑝 2 , ℎ𝑐1 , {𝑢}, {𝑣}, 𝐴𝑛 , 𝐵𝑟 (Λ)]) yields that if ℎ𝑐1 ≤ 1/𝑑 then   𝐵2𝑟 (Λ) 𝐵2𝑟 (Λ) 𝐵𝑟 (Λ) 3 (3.4.24) Qℎ𝑐1 ⊗ P 𝑢 ←−−−→ G𝑛 ←−−→ 𝑣 but 𝑢 ←−−−→ 𝑣 ≤ 4ℎ𝑐4 𝛿 . 𝑝2 𝑝2 𝑝 3/2 Now, we have by a union bound that  𝐵2𝑟 (Λ)   𝐵𝑟 (Λ) 𝐵𝑟 (Λ) 𝐵𝑟 (Λ) 𝐵𝑟 (Λ) 𝑝 3/2 𝑝 3/2 𝑝 3/2 𝑝 3/2 P 𝑢 ←−−−→ 𝑣 ≥ Qℎ𝑐1 ⊗ P 𝑢 ←−−→ G1 ←−−→ . . . ←−−→ G𝑛 ←−−→ 𝑣 𝑝2 − 𝑛−1 ∑︁  𝐵2𝑟 (Λ)  𝐵𝑟 (Λ) 𝐵2𝑟 (Λ) 𝑝 3/2 𝑝2 Qℎ𝑐1 ⊗ P 𝑢 ←−−−→ G𝑖 ←−−→ G𝑖+1 but 𝑢 ←−−−→ G𝑖+1 𝑝2 𝑖=1   𝐵2𝑟 (Λ) 𝐵𝑟 (Λ) 𝐵2𝑟 (Λ) 𝑝2 𝑝 3/2 𝑝2  − Qℎ𝑐1 ⊗ P 𝑢 ←−−−→ G𝑛 ←−−→ 𝑣 but 𝑢 ←−−−→ 𝑣 . (3.4.25) Using (3.4.21), (3.4.22), and the Harris-FKG inequality to bound the first term and (3.4.23) and (3.4.24) to control the error terms, we obtain that there exists a universal positive constant 𝑐 5 and positive constants 𝑐 6 and 𝐶 depending only on 𝑑 and 𝐷 such that if ℎ𝑐 ≤ 1/𝑑 then  𝐵2𝑟 (Λ)   𝑛−1 𝜏𝑝Λ ( 𝐴𝑛 ) 3 1 1 − ℎ𝑐𝛿 − 𝑛 · 4ℎ𝑐4 𝛿 2 2 h i 3 ≥ 𝑐 5 1 − 𝐶𝑛ℎ𝑐6 𝛿 𝜏𝑝Λ ( 𝐴1 )𝜏𝑝Λ ( 𝐴𝑛 ), P 𝑢 ←−−−→ 𝑣 ≥ 𝑝2 𝜏𝑝Λ1 ( 𝐴1 )  where we used that 𝜏𝑝Λ1 ( 𝐴1 )𝜏𝑝Λ1 ( 𝐴𝑛 ) ≥ ℎ2𝑐𝛿 ≥ ℎ𝑐1 𝑐3 𝛿 /2 to absorb the error term into the prefactor. The proposition follows easily since the vertices 𝑢 ∈ 𝐴1 and 𝑣 ∈ 𝐴𝑛 were arbitrary. □ 4 3 4 Graphs of high growth and the implication (F) We now apply Proposition 3.4.1 to prove the implication (F) of the main induction step Proposition 3.3.1. We state the implication without including the burn-in term in 𝛿0 ; this will not cause problems since the statement is stronger without this term than with it. Proposition 3.4.5 (The implication (F)). For each 𝑑 ∈ N there exists a constant 𝑁 = 𝑁 (𝑑) ≥ 16 such that the following holds. Let 𝐺 be an infinite, connected, unimodular transitive graph with vertex degree 𝑑, let 𝑝 0 ∈ (0, 1), and let 𝑛0 ≥ 16. Let 𝛿0 = (log log 𝑛0 ) −1/2 , define sequences (𝑛𝑖 )𝑖≥1 and (𝛿𝑖 )𝑖≥1 recursively by   ◦3 ◦3 𝑛𝑖 := exp log (𝑛0 ) + 𝑖 log 9 and 𝛿𝑖 := (log log 𝑛𝑖 ) −1/2 72 and let ( 𝑝𝑖 )𝑖≥1 be an increasing sequence of probabilities satisfying 𝑝𝑖+1 ≥ Spr( 𝑝𝑖 ; 𝛿𝑖 ) for each 𝑖 ≥ 0. Let 𝑛−1 := (log 𝑛0 ) 1/2 . For each 𝑖 ≥ 0 define the statement !   Full-Space(𝑖) = P 𝑝𝑖 (𝑢 ↔ 𝑣) ≥ exp −(log log 𝑛𝑖 ) 1/2 for all 𝑢, 𝑣 ∈ 𝐵𝑛𝑖 and for each 𝑖 ≥ 1 define the statement ! Corridor(𝑖) = 𝜅 𝑝𝑖 (𝑒 [log 𝑚] 10  , 𝑚) ≥ exp −(log log 𝑛𝑖 )  1/2 for every 𝑚 ∈ L (𝐺, 20) ∩ [𝑛𝑖−2 , 𝑛𝑖−1 ] . If 𝑛0 ≥ 𝑁 and 𝑝 0 ≥ 1/𝑑 then the implication h i h i Full-Space(𝑖) ∧ Corridor(𝑖 + 1) =⇒ Full-Space(𝑖 + 1) ∨ ( 𝑝𝑖+1 ≥ 𝑝 𝑐 ) (F) holds for every 𝑖 ≥ 0. Proof of Proposition 3.4.5. Fix 𝑖 ≥ 0 and suppose that Full-Space(𝑖) and Corridor(𝑖 + 1) both 10 9 hold. If 𝑛𝑖 ∈ L (𝐺, 20), then 𝑒 (log 𝑛𝑖 ) ≥ 2𝑛𝑖+1 = 2𝑒 (log 𝑛𝑖 ) whenever 𝑁 is larger than some 10 universal constant, so that if 𝑢, 𝑣 ∈ 𝐵𝑛𝑖+1 then 𝑑 (𝑢, 𝑣) ≤ 2𝑛𝑖+1 ≤ 𝑒 (log 𝑛𝑖 ) and   10 1/2 P 𝑝𝑖+1 (𝑢 ↔ 𝑣) ≥ 𝜅 𝑝𝑖+1 𝑒 [log 𝑛𝑖 ] , 𝑛𝑖 ≥ 𝑒 −(log log 𝑛𝑖+1 ) for every 𝑢, 𝑣 ∈ 𝐵𝑛𝑖+1 . That is, Corridor(𝑖 + 1) trivially implies Full-Space(𝑖 + 1) whenever 𝑛𝑖 ∈ L (𝐺, 20). Now suppose that 𝑛𝑖 ∉ L (𝐺, 20) and suppose that Full-Space(𝑖) holds, so that   Gr(𝑛𝑖 ) ≥ exp((log(𝑛𝑖1/3 )) 20 ) =: ℎ and 𝜏𝑝𝑖 (𝐵𝑛 (𝑢𝑖 )) ≥ exp −(log log 𝑛𝑖 ) 1/2 . Fix two arbitrary vertices 𝑢, 𝑣 ∈ 𝐵𝑛𝑖+1 and let 𝑢 = 𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 = 𝑣 be the vertices in a geodesic from 𝑢 to 𝑣. It follows from the Harris-FKG inequality that for all 𝑗,   𝜏𝑝𝑖 (𝐵𝑛 (𝑢 𝑗 ) ∪ 𝐵𝑛 (𝑢 𝑗+1 )) ≥ 𝜏𝑝𝑖 (𝐵𝑛+1 (𝑢 𝑗 )) ≥ 𝑝𝑖2 𝜏𝑝𝑖 (𝐵𝑛+1 (𝑢 𝑗 )) ≥ 𝑑 −2 exp −(log log 𝑛 𝑗 ) 1/2 . Let 𝑐 1 , 𝑐 2 , 𝑐 3 and ℎ0 = ℎ0 (𝑑) be the constants from Proposition 3.4.1 applied with 𝐷 = 1 (so that 𝑐 1 , 𝑐 2 , and 𝑐 3 are universal) and let 𝑁1 = 𝑁1 (𝑑) be sufficiently large that exp(−(log 𝑛) 20 ) ≤ ℎ0 for every 𝑛 ≥ 𝑁1 . There exists a constant 𝑁2 = 𝑁2 (𝑑) ≥ 𝑁1 such that if 𝑛𝑖 ≥ 𝑛0 ≥ 𝑁2 then   𝑐 1 (log 𝑛𝑖 ) 20 −1 𝑐 1 𝛿𝑖3 ≤ 𝑐 3 𝑛𝑖+1 ℎ = exp − 20 3/2 3 (log log 𝑛𝑖 ) and for all 𝑗, 𝜏𝑝𝑖 (𝐵𝑛 (𝑢 𝑗 ) ∪ 𝐵𝑛 (𝑢 𝑗+1 )) ≥ 𝑑 −2  exp −(log log 𝑛𝑖 ) 73 1/2    4 𝑐 1 (log 𝑛𝑖 ) 20 ≥ 4 exp − 20 = 4ℎ𝑐1 𝛿𝑖 . 2 3 (log log 𝑛𝑖 ) Thus, if 𝑛0 ≥ 𝑁2 then Proposition 3.4.1 (applied with 𝐷 = 1 and 𝐴𝑖 = 𝐵𝑛 (𝑢𝑖 )) implies that for all 𝑗,   P 𝑝𝑖+1 (𝑢 ↔ 𝑣) ≥ 𝜏𝑝𝑖+1 (𝐵𝑛𝑖 (𝑢), 𝐵𝑛𝑖 (𝑣)) ≥ 𝑐 2 𝜏𝑝𝑖 (𝐵𝑛𝑖 (𝑢))𝜏𝑝𝑖 (𝐵𝑛𝑖 (𝑣)) ≥ 𝑐 2 exp −2(log log 𝑛𝑖 ) 1/2 , where the final inequality follows from the assumption that Full-Space(𝑖) holds. Since 𝑢 and 𝑣 were arbitrary vertices in 𝐵𝑛𝑖+1 , it follows that there exists a constant 𝑁3 = 𝑁3 (𝑑) ≥ 𝑁2 such that if 𝑛𝑖 ≥ 𝑛0 ≥ 𝑁3 then Full-Space(𝑖 + 1) holds as claimed. □ As mentioned in the introduction, Proposition 3.4.5 already allows us to conclude the proofs of Theorems 3.3.3 and 8.1.1 under a mild uniform superpolynomial growth assumption. Corollary 3.4.6. Let 𝐺 be an infinite, connected, unimodular transitive graph. If log Gr(𝑟) > (log 𝑟) 20 for all sufficiently large 𝑟 then 𝑝 𝑐 (𝐺) < 1. Corollary 3.4.7. Let (𝐺 𝑛 )𝑛≥1 be a sequence of infinite, connected, unimodular transitive graphs converging to some transitive graph 𝐺, and suppose that there exists 𝑅 such that log Gr(𝑟; 𝐺 𝑛 ) > (log 𝑟) 20 for every 𝑟 ≥ 𝑅 and 𝑛 ≥ 1. Then 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺) Proof of Corollaries 3.4.6 and 3.4.7. Observe that if 𝐺 satisfies log Gr(𝑟; 𝐺) > (log 𝑟) 20 for every 𝑟 ≥ 𝑛0 then the statement Corridor(𝑖) holds vacuously for every 𝑖 since L (𝐺, 20) ∩ [𝑛0 , ∞) is empty. Thus, these two corollaries follow from Proposition 3.4.5 by the same argument used to deduce Theorems 3.3.3 and 8.1.1 from Proposition 3.3.1. (In fact the proof is slightly simpler since one no longer needs to control the burn-in.) □ Remark 3.4.5 (Weaker growth conditions and the gap conjecture). The proof of Corollaries 3.4.6 and 3.4.7 extends straightforwardly to (sequences of) graphs satisfying much weaker growth conditions, such as log Gr(𝑟) ≥ 𝑐 log 𝑟 (log log 𝑟) 10 . (3.4.26) It is plausible that this class (and indeed the class treated by Corollaries 3.4.6 and 3.4.7) includes every transitive graph of superpolynomial growth, so that the methods of this section would suffice to prove locality and non-triviality of 𝑝 𝑐 for unimodular transitive graphs of superpolynomial growth. (One would still need the remainder of the paper to handle sequences of graphs of polynomial growth converging to a graph of superpolynomial growth, such as the Cayley graphs of the free step-𝑠 nilpotent groups, which converge to trees as 𝑠 → ∞.) Indeed, one formulation of Grigorchuk’s gap conjecture (see [Gri14]) states that there exist universal positive constants 𝑐 and 𝛾 𝛾 such that if 𝐺 is a Cayley graph of superpolynomial growth then Gr(𝑟) ≥ 𝑒 𝑐𝑟 for every 𝑟 ≥ 1, and it seems reasonable to extend this conjecture to transitive graphs. Thus, the lower bound (3.4.26) 74 required for our arguments to work is much smaller than what might plausibly hold universally for all transitive graphs of superpolynomial growth. On the other hand, the best known bound for the gap conjecture, due to Shalom and Tao [ST10b], states that there exists a universal constant 𝑐 such that log Gr(𝑟) ≥ 𝑐 log 𝑟 (log log 𝑟) 𝑐 (3.4.27) for every Cayley graph of superpolynomial growth and every 𝑟 ≥ 1. (The authors claim their proof should yield this estimate with the constant 𝑐 = 0.01, but do not carry out the necessary bookkeeping in the paper.) Even if the strong form of the gap conjecture is false, there does not seem to be any reason to believe that the Shalom-Tao bound (3.4.27) is optimal, so that (3.4.26) may very well hold for all transitive graphs of superpolynomial growth. 3.5 Quasi-polynomial growth I: Building disjoint tubes In this section we prove the geometric facts that we will later use to analyze percolation in the low-growth (a.k.a. quasi-polynomial growth) regime. Let 𝑑 ≥ 1, let 𝐺 ∈ G𝑑∗ , and let 𝐷 ≥ 1 be a fixed parameter. We recall that the set of low growth scales L (𝐺, 𝐷) is defined to be n o 𝐷 1/3 L (𝐺, 𝐷) = 𝑛 ≥ 1 : log Gr(𝑚) ≤ (log 𝑚) for all 𝑚 ∈ [𝑛 , 𝑛] . It would suffice for all our applications to take e.g. 𝐷 = 20; we keep 𝐷 as a parameter for now to emphasize that the analysis carried out in this section and Section 3.6 works for arbitrarily large 𝐷. Ð Recall that a tube is defined to be a set of the form 𝐵𝑟 (𝛾) = 𝑖 𝐵𝑟 (𝛾𝑖 ) where 𝛾 is a path and 𝑟 ∈ (0, ∞) is a parameter we call the thickness of the tube; we call len(𝛾) the length of the tube. We will also sometimes write 𝐵(𝛾, 𝑟) = 𝐵𝑟 (𝛾) to avoid writing large expressions in the subscript. (Strictly speaking the length and thickness of a tube depends on the pair (𝛾, 𝑟) used to represent it, but we will not belabour this point further.) We would like to show that whenever 𝑛 is in the low growth regime, for all suitable sets ( 𝐴, 𝐵) of vertices at scale 𝑛, we can find many disjoint tubes from 𝐴 to 𝐵 that are reasonably thick and not unreasonably long. Definition 3.5.1. Let 𝐺 be a connected transitive graph. Given 𝑘, 𝑟, ℓ ≥ 1 (which need not be integers) and 𝑛 ≥ 1, we say that 𝐺 has (𝑘, 𝑟, ℓ)-plentiful radial tubes at scale 𝑛 if there exists a family of paths Γ in 𝐺 with |Γ| ≥ 𝑘 satisfying all of the following properties: • Every path 𝛾 ∈ Γ starts in the sphere 𝑆𝑛 and ends in the sphere 𝑆4𝑛 ; • For every two distinct paths 𝛾, 𝛾 ′ ∈ Γ, the tubes 𝐵(𝛾, 𝑟) and 𝐵(𝛾 ′, 𝑟) are disjoint. 75 Figure 3.2: Schematic illustration of radial and annular tubes. Left: The (𝑘, 𝑟, ℓ)-plentiful radial tubes condition means that we can find 𝑘 disjoint tubes crossing the annulus that all have thickness 𝑟 and length at most ℓ. Right: the (𝑘, 𝑟, ℓ)-plentiful annular tubes condition means that for any two crossings of the annulus, we can find 𝑘 disjoint tubes connecting the two crossings that all have thickness 𝑟 and length at most ℓ; these tubes are not required to stay inside the annulus. • Every path 𝛾 ∈ Γ has length at most ℓ; (In particular, the parameter 𝑘 controls the number of tubes, the parameter 𝑟 controls the thickness of the tubes, and the parameter ℓ controls the length of the tubes.) Given 𝑚 ≥ 𝑛 ≥ 1, we say that a set of vertices 𝐴 ⊆ 𝑉 is an (𝑛, 𝑚) crossing if it contains a path from 𝑆𝑛 to 𝑆 𝑚 . We say that 𝐺 has (𝑘, 𝑟, ℓ)-plentiful annular tubes at scale 𝑛 if for every pair of sets of vertices ( 𝐴, 𝐵) that are both (𝑛, 3𝑛) crossings10 , there exists a family of paths Γ in 𝐺 with |Γ| ≥ 𝑘 satisfying all of the following properties: • Every path 𝛾 ∈ Γ starts in 𝐴 and ends in 𝐵; • For every two distinct paths 𝛾, 𝛾′ ∈ Γ, the tubes 𝐵(𝛾, 𝑟) and 𝐵(𝛾 ′, 𝑟) are disjoint; • Every path 𝛾 ∈ Γ has length at most ℓ. We say that 𝐺 has (𝑘, 𝑟, ℓ)-plentiful tubes at scale 𝑛 if 𝐺 has both (𝑘, 𝑟, ℓ)-plentiful annular tubes and (𝑘, 𝑟, ℓ)-plentiful radial tubes at scale 𝑛. See Figure 3.2 for an illustration. Given parameters 10 The constant 3 that appears here could safely be replaced by any constant strictly larger than 2. constant to be strictly larger than 2 to not cause problems with our application of Proposition 3.5.9. 76 We need the 𝑐, 𝜆 > 0, we say that 𝐺 has (𝑐, 𝜆)-polylog-plentiful tubes at scale 𝑛 if it has (𝑘, 𝑟, ℓ)-plentiful tubes with 𝑘 = (log 𝑛) 𝑐𝜆 , 𝑟 = 𝑛(log 𝑛) −𝜆/𝑐 , and ℓ = 𝑛(log 𝑛) 𝜆/𝑐 . (NB: These tubes are very thick!) Remark 3.5.1. The property of having (𝑘, 𝑟, ℓ)-plentiful tubes at scale 𝑛 gets stronger as 𝑘 and 𝑟 increase and gets weaker as ℓ increases; the property of having (𝑐, 𝜆)-polylog-plentiful tubes at scale 𝑛 gets stronger as 𝑐 increases but has no obvious monotonicity in the parameter 𝜆. There is of course a trade-off between the number of tubes and their thicknesses, since e.g. if the (𝑛, 3𝑛) crossings 𝐴 and 𝐵 are segments of geodesics from the origin between distances 𝑛 and 3𝑛 then we cannot have more than 2𝑛/𝑟 disjoint tubes of thickness 𝑟 connecting 𝐴 and 𝐵. In our applications in Section 3.6 we will want to use families of tubes that have thickness 𝑛(log 𝑛) −𝑂 (1) and length 𝑛(log 𝑛) 𝑂 (1) , which is why we phrase Proposition 3.5.2 in terms of polylog-plentiful tubes. As will be clear later in the section, our constructions allow for many other possible trade-offs between 𝑘 and 𝑟 which may be useful in future applications. Ideally, we would like to say that there is some constant 𝑐 ∈ (0, 1) such that for every 𝜆, 𝐺 has (𝑐, 𝜆)-polylog-plentiful tubes at every sufficiently large scale 𝑛 ∈ L (𝐺, 𝐷). This is, however, slightly stronger than what we have been able to prove. (Fortunately it is also stronger than we need for our applications!) We instead establish the slightly weaker statement that for every scale 𝑛 ∈ L (𝐺, 𝐷), there exists a large interval of scales not much smaller than 𝑛 on which we have plentiful tubes. We now give the precise statement, the proof of which takes up the rest of this section. Proposition 3.5.2 (Quasi-polynomial growth yields a large range of consecutive scales with polylog-plentiful tubes). For each 𝐷 ∈ [1, ∞), 𝜆 ∈ [1, ∞), and 𝑑 ≥ 1 there exist positive constants 𝑐(𝑑, 𝐷) ∈ (0, 1) and 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆) such that the following holds. Let 𝐺 be an infinite, connected, unimodular transitive graph with vertex degree 𝑑 that is not one-dimensional. For each integer 𝑛 ≥ 𝑛0 with 𝑛 ∈ L (𝐺, 𝐷) there exist integers 𝑛1/3 ≤ 𝑚 1 ≤ 𝑚 2 ≤ 𝑛 with 𝑚 2 ≥ 𝑚 11+𝑐 such that 𝐺 has (𝑐, 𝜆)-polylog-plentiful tubes at every scale 𝑚 1 ≤ 𝑚 ≤ 𝑚 2 . The constant 𝑐 = 𝑐(𝑑, 𝐷) can be thought of as an “exchange rate”, governing the cost to trade between the number of tubes and their thicknesses, while 𝜆 is the parameter we vary to make this trade. It is very important that the constant 𝑐 does not depend on 𝜆! Remark 3.5.2. No non-trivial statements about plentiful annular tubes can be made without some kind of growth upper bound (or other geometric assumption), since the 3-regular tree does not have plentiful (𝑘, 𝑟, ℓ)-plentiful annular tubes for any 𝑘, 𝑟, ℓ > 1. Similarly, the assumption that the graph is not one-dimensional is needed since the line graph Z does not have (𝑘, 𝑟, ℓ)-plentiful 77 radial or annular tubes for any 𝑘, 𝑟, ℓ > 1. (On the other hand, the assumption of unimodularity is redundant since it is implied by the existence of large scales with subexponential growth [Hut20a, Section 5.1].) To prove this proposition, we split into two cases according to the rate of change of growth in the given interval. More precisely, we will split according to whether or not there exists 𝑛 in a suitable initial segment of the interval such that 𝐺 satisfies the small-tripling condition Gr(3𝑛) ≤ 35 Gr(𝑛) at scale 𝑛, where the constant 5 could be replaced by any other constant strictly larger than 4. (This is related to the fact that four is the critical dimension for two independent random walks to intersect infinitely often almost surely.) Our proofs in the two cases are completely different from one another. In the first case, when there does exist such an 𝑛, we will build our disjoint tubes by applying the structure theory of transitive graphs of polynomial volume growth [BGT12a; TT21a; EH23d]. Informally, this theory guarantees that, for a large interval of scales, our graph looks approximately like the Cayley graph of a finitely presented group whose relations are generated by cycles of diameter much smaller than the given scale. We will use these techniques to prove the following proposition. Proposition 3.5.3 (Slow tripling yields plentiful tubes). For each 𝑑 ≥ 1 and 𝜅 < ∞ there exist positive constants 𝑐 = 𝑐(𝑑, 𝜅), 𝐶 = 𝐶 (𝑑, 𝜅), and 𝑛0 = 𝑛0 (𝑑, 𝜅) such that if 𝐺 is an infinite, connected, transitive graph of vertex degree 𝑑 that is not one-dimensional and 𝑛 ≥ 𝑛0 is such that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛), then there exists a set 𝐴 ⊂ [𝑛, ∞) with | 𝐴| ≤ 𝐶 such that for each 𝑘 ≥ 1, 𝐺 Ð has (𝑐𝑘, 𝑐𝑘 −1 𝑚, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes at every scale 𝑚 ≥ 𝐶 𝑘𝑛 such that 𝑚 ∉ 𝑎∈𝐴 [𝑎, 2𝑘𝑎]. In the second case, when there does not exist such a scale 𝑛 with small tripling, we will prove that the desired plentiful tubes condition holds using random walks. More specifically, we will apply an estimate of [BDKY15], which can be thought of as a “quantitative weak elliptic Harnack inequality” for graphs of subexponential growth. Under our low-growth assumption, this inequality implies that two random walks on 𝐺 started at distance 𝑛′ ∈ [𝑛1/3 , 𝑛0.9 ], say, can be coupled to coincide with good probability by the time they reach distance 𝑛′ (log Gr(𝑛′)) 𝑂 (1) . On the other hand, the assumption that the growth is rapidly increasing (in an at-least-five-dimensional fashion) lets us prove that two independent pairs of these coupled random walks are unlikely to have their tubes intersect. (This will be proven using fairly standard random walk techniques, most notably the isoperimetric inequality of Coulhon and Saloff-Coste [CS93] and the heat kernel bounds resulting from this inequality together with the work of Morris and Peres [MP05].) Unfortunately these 78 walks will have length of order (𝑛′) 2 (log Gr(𝑛′)) 𝑂 (1) , which is larger than we want by a factor of 𝑛′. This can be fixed by a simple coarse-graining argument, using the diffusivity of random walks on low-growth graphs, to replace the random walk by a shorter path with essentially the same tube around it. We will use these techniques to prove the following proposition. Proposition 3.5.4 (Fast tripling and quasi-polynomial growth yield plentiful tubes). Let 𝐺 be an infinite connected unimodular transitive graph with vertex degree 𝑑, let 𝐷, 𝜆 ≥ 1 and let 𝜀 > 0. There exist positive constants 𝑐 = 𝑐(𝑑, 𝐷, 𝜀) and 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆, 𝜀) such that if 𝑛 ≥ 𝑛0 satisfies Gr(𝑚) ≤ 𝑒 (log 𝑚) 𝐷 and Gr(3𝑚) ≥ 35 Gr(3𝑚) for every 𝑛1−𝜀 ≤ 𝑚 ≤ 𝑛1+𝜀 then 𝐺 has (𝑐, 𝜆)-polylog-plentiful tubes at scale 𝑛. We prove Proposition 3.5.3 in Section 3.5 and Proposition 3.5.4 in Section 3.5. Before doing this, we note that Proposition 3.5.2 follows easily from these two propositions. Proof of Proposition 3.5.2 given Propositions 3.5.3 and 3.5.4. Let 𝑐 1 (𝑑, 5), 𝐶 (𝑑, 5), and 𝑛1 (𝑑, 5) be the constants from Proposition 3.5.3 with 𝜅 := 5. Let 𝑐 2 (𝑑, 𝐷, 0.1) and 𝑛2 (𝑑, 𝐷, 𝜆, 0.1) be the constants from Proposition 3.5.4 with 𝜀 := 0.1. We may assume that 𝑐 1 ∨ 𝑐 2 ≤ 1/2 and 𝐶 ≥ 2. Suppose that some 𝑛 ∈ L (𝐺, 𝐷) is large with respect to 𝑑, 𝐷, 𝜆, satisfying in particular 𝑛 ≥ (𝑛1 ∨ 𝑛2 ) 3 . If there exists 𝑛′ ∈ [𝑛1/3 , 𝑛10/11 ] such that Gr(3𝑛′) ≤ 35 Gr(𝑛′), then we may apply Proposition 3.5.3 with 𝑘 := (log 𝑛) 𝜆 to obtain an interval of scales [𝑚 1 , 𝑚 2 ] ⊆ [𝑛1/3 , 𝑛] with 𝑚 2 ≥ 𝑚 11.1 on which 𝐺 has (1/(2𝐶), 𝜆)-polylog-plentiful tubes on every scale. Otherwise, since each 𝑛′ ∈ [𝑛0.4 , 𝑛0.8 ] satisfies [(𝑛′) 0.9 , (𝑛′) 1.1 ] ⊆ [𝑛1/3 , 𝑛10/11 ], we can apply Proposition 3.5.4 to each such scale to obtain that 𝐺 has (𝑐 1 , 𝜆)-plentiful tubes at every scale in the interval [𝑛0.4 , 𝑛0.8 ]. This is easily seen to imply the claim in either case. □ Using the structure theory of approximate groups The goal of this subsection is to prove Proposition 3.5.3. Let us first give some relevant context. Given a graph 𝐺, a vertex 𝑣 ∈ 𝑉 (𝐺), and a radius 𝑟 ≥ 0, we define the exposed sphere 𝑆𝑟∞ (𝑣) to be the set of vertices 𝑢 ∈ 𝑆𝑟 (𝑣) such that there exists an infinite self-avoiding path started at 𝑢 that never returns to 𝐵𝑟 (𝑣) after its first step. When 𝐺 is transitive, we set 𝑆𝑟∞ := 𝑆𝑟∞ (𝑜). In [CMT22], the authors applied results of Timar [Tim07] to obtain geometric control of exposed spheres in transitive graphs of polynomial growth using the fact that (by Gromov’s theorem [Gro81b] and Trofimov’s theorem [Tro84b]) these graphs are quasi-isometric to Cayley graphs of nilpotent groups, which are finitely presented. 79 In this section, we will run a similar argument to build our disjoint tubes under the hypotheses of Proposition 3.5.3. While these hypotheses suffice to guarantee polynomial growth by the results of [BGT12a; TT21a] (discussed in detail below), a key technical difference between our analysis and that of [CMT22] is that we must run all our arguments in a more finitary, quantitative way; we need to build our disjoint tubes at a scale not much larger than the scale where we are assumed to witness relative polynomial growth. This requires us to engage more deeply with the structure theory of approximate groups than was necessary in [CMT22]. Indeed, rather than using Gromov’s theorem and Trofimov’s theorem, we will instead apply finitary versions of these theorems due to Breuillard, Green, and Tao [BGT12a] and Tessera and Tointon [TT21a]. Moreover, we will also apply a new structure theoretic result proven in our paper [EH23d], which can be thought of as a “uniform” version of the statement that groups of polynomial growth are finitely presented. Remark 3.5.3. It will be convenient in this section to let Γ denote a group. This will not conflict with the Γ used to denote a set of disjoint tubes, which we will not use in this section. Structure theory. We now state the main structure-theoretic results we will use after reviewing some relevant definitions. We begin with Tessera and Tointon’s finitary structure theorem for vertex-transitive graphs of low growth [TT21a]; this theorem builds on Breuillard, Green, and Tao’s structure theorem for approximate groups [BGT12a] as well as Carolino’s extension of this theorem to locally compact approximate groups [Car15]. Recall that a function 𝜙 : 𝑉1 → 𝑉2 between the vertex sets of two graphs 𝐺 1 = (𝑉1 , 𝐸 1 ) and 𝐺 2 = (𝑉2 , 𝐸 2 ) is said to be an (𝛼, 𝛽)quasi-isometry (a.k.a. rough isometry) if 𝛼−1 𝑑 (𝑥, 𝑦) − 𝛽 ≤ 𝑑 (𝜙(𝑥), 𝜙(𝑦)) ≤ 𝛼𝑑 (𝑥, 𝑦) + 𝛽 for every 𝑥, 𝑦 ∈ 𝑉1 and every vertex 𝑧 ∈ 𝑉2 is within distance at most 𝛽 of 𝜙(𝑉1 ); note that the second property holds automatically if 𝜙 is surjective. Given a transitive graph 𝐺 and a subgroup 𝐻 ⊆ Aut(𝐺), we write 𝐺/𝐻 for the associated quotient graph. If 𝐻 is a normal subgroup of Aut(𝐺) then the action of Aut(𝐺) on 𝐺 descends to a transitive action of Aut(𝐺) on 𝐺/𝐻 (see [TT21a, Section 3]), and we write Aut(𝐺)𝐺/𝐻 for the image of Aut(𝐺) in Aut(𝐺/Γ) induced by this action. Note that we view Aut(𝐺) as a topological group where convergence is given by pointwise convergence (see [TT21a, §4]). Theorem 3.5.5 (Finitary structure theory of transitive graphs of polynomial growth). For each 𝐾 ≥ 1 there exist constants 𝑛0 = 𝑛0 (𝐾) and 𝐶 = 𝐶 (𝐾) such that the following holds. Let 𝐺 be a connected (locally finite) (vertex-)transitive graph with a distinguished vertex 𝑜, and suppose that 80 there exists 𝑛 ≥ 𝑛0 such that Gr(3𝑛) ≤ 𝐾 Gr(𝑛). Then there exists a compact normal subgroup 𝐻 ⊳ Aut(𝐺) such that: 1. Every fibre of the projection 𝜋 : 𝐺 → 𝐺/𝐻 has diameter at most 𝐶𝑛. 2. Aut(𝐺)𝐺/𝐻 can be canonically identified with Aut(𝐺)/𝐻. 3. Aut(𝐺)/𝐻 has a nilpotent normal subgroup 𝑁 of rank, step and index at most 𝐶. 4. The set 𝑆 = {𝑔 ∈ Aut(𝐺)𝐺/𝐻 : 𝑑𝐺/𝐻 (𝑔(𝐻𝑜), 𝐻𝑜) ≤ 1} is a finite symmetric generating set for Aut(𝐺)/𝐻. 5. Every vertex stabiliser of the action of Aut(𝐺)/𝐻 on 𝐺/𝐻 has cardinality at most 𝐶. 6. If for each 𝑣 ∈ 𝐺 we let 𝑔𝑣 ∈ Aut(𝐺)/𝐻 be such that 𝑔𝑣 (𝜋(𝑒)) = 𝜋(𝑣) then 𝑣 ↦→ 𝑔𝑣 is a (1, 𝐶𝑛)-quasi-isometry from 𝐺 to Cay(Aut(𝐺)/𝐻, 𝑆). 7. If Gr′ denotes the growth function of the Cayley graph Cay(Aut(𝐺)/𝐻, 𝑆) then Gr(𝑚 2 ) Gr′ (𝑚 2 ) 𝐶 Gr(𝑚 2 + 𝐶𝑛) ≤ ′ ≤ 𝐶 Gr(𝑚 1 + 𝐶𝑛) Gr (𝑚 1 ) Gr(𝑚 1 ) for every 𝑚 1 , 𝑚 2 ∈ N. 8. The growth bound Gr(𝑚 2 ) ≤ 𝐶 (𝑚 2 /𝑚 1 ) 𝐶 Gr(𝑚 1 ) holds for every 𝑚 2 ≥ 𝑚 1 ≥ 𝑚. Proof of Theorem 3.5.5. The first five items of the theorem are essentially equivalent to [TT21a, Theorem 2.3] (although that theorem is slightly more general as it allows one to replace Aut(𝐺) with any other transitive group of automorphisms of 𝐺). In their original statement of the theorem, Tessera and Tointon do not explicitly identify the rough isometry from 𝐺 to Cay(Aut(𝐺)/𝐻, 𝑆), but the fact we can take it to be of the form above is implicit in their proof. Item 7 is implied by [TT21a, Proposition 9.1], while Item 8 follows from Item 7 together with [BT16, Theorem 1.1]. □ Remark 3.5.4. The set 𝑆 has size equal to the union of the stabilizers of the vertices {𝑢 ∈ 𝐺/𝐻 : 𝑢 ∼ 𝑣}, so that |𝑆| ≤ 𝐶 (deg(𝑜) + 1). Remark 3.5.5. For 𝐾 = 3𝜅 with 𝜅 an integer, the growth bound Gr(𝑚 2 ) ≤ 𝐶 (𝑚 2 /𝑚 1 ) 𝐶 Gr(𝑚 1 ) can be improved to the sharp bound Gr(𝑚 2 ) ≤ 𝐶 (𝑚 2 /𝑚 1 ) 𝜅 Gr(𝑚 1 ) under the stronger assumption that the graph satisifes an absolute growth bound of the form Gr(𝑛) ≤ 𝜀 𝐾 𝑛 𝜅+1 at a sufficiently large scale and for a sufficiently small constant 𝜀 𝐾 > 0 [TT21a, Corollary 1.5]. This strong bound is not implied by the small-tripling condition (which is the relevant condition for our applications) as 81 shown in [Tao17a, Example 1.11]. (Indeed, this example suggests that the small-tripling condition Gr(3𝑚) ≤ 3𝜅 Gr(𝑚) should not imply any bound on the limiting growth dimension stronger than 𝑂 (𝜅 2 ). Optimal bounds on growth implied by small tripling will be established in forthcoming work of Tessera and Tointon.) Uniform finite presentation. We next state our theorem on uniform finite presentation proven in [EH23d]. Given a set of elements 𝐴 in a group Γ, we define ⟨⟨𝐴⟩⟩ to be the normal subgroup of Γ generated by 𝐴 and define 𝐴¯ = 𝐴 ∪ {id} ∪ 𝐴−1 . Consider a group Γ with a finite generating set 𝑆, let 𝐹𝑆 be the free group on 𝑆 and let 𝜋 : 𝐹𝑆 → 𝐺 be the associated group homomorphism with kernel 𝑅. Since Γ  𝐹𝑆 /𝑅, we can think of the sequence of quotients (Γ𝑟 )𝑟 ≥1 defined by Γ𝑟 := 𝐹𝑆 /⟨⟨𝑆¯𝑟 ∩ 𝑅⟩⟩ as being finitely presented approximations to Γ, since Γ𝑟 admits a finite presentation Γ𝑟 = ⟨𝑆 | 𝑅𝑟 ⟩ = 𝐹𝑆 /⟨⟨𝑅𝑟 ⟩⟩ with 𝑅𝑟 = 𝑆¯𝑟 ∩ 𝑅 ⊆ 𝑆¯𝑟 . These approximations have the property that the Cayley graphs Cay(Γ𝑟 , 𝑆) and Cay(Γ, 𝑆) have isomorphic ((𝑟/2) − 1)-balls. We record this fact in the following lemma, which is taken from [EH23d, Lemma 5.6]. (Although it is stated there only in the case that 𝑟 is a power of 2, the same proof works for arbitrary 𝑟.) Lemma 3.5.6. Let Γ be a group with a finite generating set 𝑆. For all 𝑖 ≥ 1, the quotient map Γ𝑟 → Γ induces a map of the associated Cayley graphs that restricts to an isomorphism between the balls of radius ⌊𝑟/2⌋ − 1. The main result of [EH23d] can be stated as follows. Theorem 3.5.7 ([EH23d], Theorem 1.1). For each 𝐾, 𝑑 < ∞ there exist constants 𝑛0 = 𝑛0 (𝐾) and 𝐶 = 𝐶 (𝐾, 𝑑) such that if Γ is a group and 𝑆 is a finite generating set for Γ with |𝑆| ≤ 𝑑 whose growth function Gr satisfies Gr(3𝑛) ≤ 𝐾 Gr(𝑛) for some integer 𝑛 ≥ 𝑛0 then n o # 𝑘 ∈ N : 𝑘 ≥ log2 𝑛 and ⟨⟨𝑅2𝑘+1 ⟩⟩ ≠ ⟨⟨𝑅2𝑘 ⟩⟩ ≤ 𝐶. For our purposes, the main output of Theorems 3.5.5 and 3.5.7 is that if the small tripling condition Gr(3𝑛) ≤ 𝐾 Gr(𝑛) holds at some sufficiently large 𝑛, then at “most” scales 𝑟 ≥ 𝑛 the graph “looks like” the Cayley graph of a finitely presented group with relations generated by words of length much smaller than 𝑟. Using the structure theory to build disjoint tubes. It remains to apply Theorems 3.5.5 and 3.5.7 to prove Proposition 3.5.3. When working with a group Γ and a finite generating set 𝑆 of Γ, we will continue to use the notation (Γ𝑟 )𝑟 ≥1 and (𝑅𝑟 )𝑟 ≥1 as defined above. 82 Let us introduce some more definitions. Let 𝐺 = (𝑉, 𝐸) be a graph, and consider the set {0, 1} 𝐸 with addition modulo 2 as a vector space over Z2 . If 𝐺 = Cay(Γ, 𝑆) for some group Γ = ⟨𝑆|𝑅⟩ with finite generating set 𝑆 and (not necessarily finitely generated) relation group 𝑅 ⊳ 𝐹𝑆 , then we can identify every oriented cycle started at the origin with a word in 𝑅. Under this identification, we see that if 𝑅 is generated as a normal subgroup by words of length at most 𝑟 in 𝐹𝑆 , then every cycle in 𝐺 can be written as a mod-2 sum of cycles of length at most 𝑟. Note that this leads to a notion of finite presentation for graphs that are not Cayley graphs, namely that their cycle space is generated as a Z2 -vector space by the cycles of some finite length 𝑟 < ∞; see [Tim07] for further details. We will rely crucially on the following lemma, which is essentially due to Timár [Tim07]. Recall that an infinite graph 𝐺 = (𝑉, 𝐸) is one-ended if for every finite set of vertices 𝑊 ⊆ 𝑉, the graph 𝐺\𝑊 has exactly one infinite component; groups of polynomial growth are one-ended if and only if they are not one-dimensional. (Note that when 𝐺 is the Cayley graph of an infinite finitely-generated group, the property of being one-ended is independent of the choice of finite generating set, so that one can sensibly refer to a group as being one-ended without specifying a generating set.) Lemma 3.5.8. Let 𝐺 be an infinite, connected, one-ended transitive graph. If 𝑟 ∈ N has the property that every cycle in 𝐺 is equal to a sum of cycles of length at most 𝑟, then for every 𝑘, 𝑛 ∈ N Ð with 𝑟 ≤ 𝑘 ≤ 𝑛 and every 𝑢, 𝑣 ∈ 𝑆𝑛∞ there exists a path from 𝑢 to 𝑣 that is contained in 𝑥∈𝑆∞𝑛 𝐵2𝑘 (𝑥) and has length at most 3𝑘 |𝐵3𝑛 | /|𝐵 𝑘 |. Proof. This statement is implicit in the proofs of [CMT22, Lemma 2.1 and 2.7], and is an easy consequence of [Tim07, Theorem 5.1]. □ We will also need two more elementary geometric facts. The first states that if a path travels from a sphere 𝑆𝑟 to 𝑆2𝑟+1 , then it must pass through the exposed sphere 𝑆𝑟∞ . Proposition 3.5.9 ([FGO15, Proposition 5]). Let 𝐺 be an infinite connected transitive graph and let 𝑟 ∈ N. Every path that starts in 𝑆𝑟 and ends in 𝑆2𝑟+1 contains a vertex in 𝑆𝑟∞ . The next lemma lets us pass disjoint tubes through quasi-isometries with all relevant quantities changing in a controlled way. Lemma 3.5.10. Let 𝐺 = (𝑉, 𝐸) and 𝐺 ′ = (𝑉 ′, 𝐸 ′) be two graphs and let 𝜙 : 𝑉 → 𝑉 ′ be an (𝛼, 𝛽)quasi isometry for some 𝛼, 𝛽 ≥ 1. Let 𝑢, 𝑣 ∈ 𝑉 and suppose that 𝑥, 𝑦 ∈ 𝑉 ′ satisfy 𝑑 (𝑥, 𝜙(𝑢)) ≤ 𝛽 83 and 𝑑 (𝑦, 𝜙(𝑣)) ≤ 𝛽. For each path 𝛾 ′ from 𝑥 to 𝑦 in 𝐺 ′, there exists a path 𝛾 from 𝑢 to 𝑣 in 𝐺 such that len(𝛾) ≤ 10𝛼(len(𝛾 ′) + 𝛽) and 𝜙(𝐵𝑟 (𝛾)) ⊆ 𝐵𝛼𝑟+(5𝛼2 +2) 𝛽 (𝛾 ′) for every 𝑟 ≥ 0. Proof of Lemma 3.5.10. Let 𝜓 : 𝑉 ′ → 𝑉 be a function such that 𝜓(𝑥) = 𝑢, 𝜓(𝑦) = 𝑣, and 𝑑 (𝜙(𝜓(𝑧)), 𝑧) ≤ 𝛽 for every 𝑧 ∈ 𝑉; such a function exists by the definition of an (𝛼, 𝛽)-quasiisometry and our assumptions on 𝑢, 𝑣, 𝑥, and 𝑦. Let ℓ = ⌈len(𝛾 ′)/⌈𝛽⌉⌉, and let the sequence ℓ be defined by 𝑛 = ⌈𝛽⌉𝑖 for 𝑖 < ℓ and 𝑛 = len(𝛾 ′). For each 0 ≤ 𝑖 ≤ ℓ let 𝑥 = 𝛾 ′ and let (𝑛𝑖 )𝑖=0 𝑖 ℓ 𝑖 𝑛𝑖 𝑢𝑖 = 𝜓(𝑥𝑖 ), so that 𝑢 0 = 𝑢 and 𝑢 ℓ = 𝑣. For each 0 ≤ 𝑖 < ℓ, the points 𝑥𝑖 and 𝑥𝑖+1 have distance at most ⌈𝛽⌉ ≤ 2𝛽 in 𝐺 ′, so that 𝜙(𝑢𝑖 ) and 𝜙(𝑢𝑖+1 ) have distance at most 4𝛽 in 𝐺 ′. Since 𝜙 is an (𝛼, 𝛽)-quasi-isometry, it follows that 𝑢𝑖 and 𝑢𝑖+1 have distance at most 5𝛼𝛽 in 𝐺. Let 𝛾 be formed by concatenating geodesics between 𝑢𝑖 and 𝑢𝑖+1 for each 0 ≤ 𝑖 < ℓ. The path 𝛾 clearly has length at most 5𝛼𝛽⌈len(𝛾 ′)/⌈𝛽⌉⌉ ≤ 10𝛼(len(𝛾 ′) + 𝛽). Moreover, given 𝑟 ≥ 1, each point in 𝐵𝑟 (𝛾) has distance at most 5𝛼𝛽 + 𝑟 from one of the points 𝑢𝑖 , whose image under 𝜙 has distance at most 𝛽 from one of the points of 𝛾 ′. Using the definition of an (𝛼, 𝛽)-quasi-isometry again, this implies that every point in 𝜙(𝐵𝑟 (𝛾)) has distance at most 𝛼(5𝛼𝛽 + 𝑟) + 2𝛽 from a point of 𝛾 ′, which is equivalent to the desired set inclusion. □ We now have everything we need to prove Proposition 3.5.3. Proof of Proposition 3.5.3. Fix 𝜅 < ∞ and let 𝑛0 = 𝑛0 (𝜅) and 𝐶1 = 𝐶1 (𝜅) be the constants from Theorem 3.5.5 applied with 𝐾 = 3𝜅 . Suppose that 𝐺 is an infinite, connected, transitive graph of vertex degree 𝑑 that is not one-dimensional and that 𝑛 ≥ 𝑛0 is such that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛). Let 𝐻 be the normal subgroup of Aut(𝐺) that is guaranteed to exist by Theorem 3.5.5, let 𝑆 be the generating set for Aut(𝐺)/𝐻 from Item 4 of Theorem 3.5.5, and write Γ := Aut(𝐺)/𝐻. Letting Gr′ denote the growth function of the Cayley graph Cay(Γ, 𝑆), we have by Items 7 and 8 of Theorem 3.5.5 that Gr′ (3𝑛) 𝐶1 Gr(3𝑛 + 𝐶1 𝑛) ≤ ≤ 𝐶12 (3 + 𝐶1 ) 𝐶1 . Gr′ (𝑛) Gr(𝑛) Moreover, as discussed in Remark 3.5.4, the generating set 𝑆 has at most 𝐶1 (𝑑 + 1) elements. Thus, if we take 𝑛1 = 𝑛1 (𝜅) ≥ 𝑛0 and 𝐶2 = 𝐶2 (𝜅, 𝑑) to be the constants from Theorem 3.5.7 applied with 𝐾 = 𝐶12 (3 + 𝐶1 ) 𝐶1 and 𝑑 = 𝐶1 (𝑑 + 1), we have that if 𝑛 ≥ 𝑛1 then n o # 𝑖 ∈ N : 𝑖 ≥ log2 𝑛 and ⟨⟨𝑅2𝑖+1 ⟩⟩ ≠ ⟨⟨𝑅2𝑖 ⟩⟩ ≤ 𝐶2 , (3.5.1) 84 where 𝑅 𝑘 is the set of relations of Γ that have word length at most 𝑘 and ⟨⟨𝑅 𝑘 ⟩⟩ is the smallest normal subgroup of the free group 𝐹𝑆 generated by 𝑅 𝑘 . Let 𝐺 ′ = (𝑉 ′, 𝐸 ′) be the Cayley graph Cay(Γ, 𝑆), let 𝜙 : 𝑉 → 𝑉 ′ be the (1, 𝐶1 𝑛)-quasi-isometry that is guaranteed to exist by Theorem 3.5.5 (which we may assume maps 𝑜 to the identity of Γ, which we denote by id), and let 𝜓 : 𝑉 ′ → 𝑉 be such that 𝑑 (𝜙(𝜓(𝑥)), 𝑥) ≤ 𝐶1 𝑛 for every 𝑥 ∈ 𝑉 ′, such a function being guaranteed to exist since 𝜙 is a (1, 𝐶1 𝑛)-quasi-isometry. This function 𝜓 is easily seen to be a (1, 𝐶3 𝑛)-quasi-isometry for an appropriate choice of constant 𝐶3 = 𝐶3 (𝜅). For each 𝑘 ≥ 1 let 𝐺 ′𝑘 be the Cayley graph Cay(Γ𝑘 , 𝑆) of the group Γ𝑘 defined by Γ𝑘 = ⟨𝑆 | 𝑅 𝑘 ⟩. Fix 𝑛 ≥ 𝑛1 and consider the set 𝐴 = {2𝑖 : 𝑖 ≥ log2 𝑛 and ⟨⟨𝑅2𝑖+5 ⟩⟩ ≠ ⟨⟨𝑅2𝑖 ⟩⟩}. This set contains at most five times as many elements as the set considered in (3.5.1), so that | 𝐴| ≤ 5𝐶2 . Fix 𝑘 ≥ 𝑛 and suppose that 𝑚 ≥ 2𝑘𝑛 does not belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴, so that if we define 𝑎(𝑚) = sup{𝑎 ∈ 𝐴 : 𝑎 ≤ 𝑚} then 𝑎(𝑚) ≤ 𝑚/2𝑘. The definition of 𝐴 ensures that ⟨⟨𝑅2𝑎(𝑚) ⟩⟩ = ⟨⟨𝑅16𝑚 ⟩⟩ and hence by Lemma 4.5.6 that the Cayley graphs 𝐺 ′ and 𝐺 ′𝑎(𝑚) have isomorphic (8𝑚 − 1)-balls. For each 𝑟 < 4𝑚 we write (𝑆𝑟∞ ) ′ for the exposed spheres in 𝐺 ′ and 𝐺 ′𝑎(𝑚) , which can be identified by Proposition 3.5.9. We also identify the balls 𝐵𝑟′ (𝑥) in 𝐺 ′ and 𝐺 ′𝑎(𝑚) for points 𝑥 of distance at most 4𝑚 from the identity and all 𝑟 ≤ 𝑚, where the prime on 𝐵𝑟′ (𝑥) reminds us that we are working with 𝐺 ′ rather than 𝐺. Now, since Γ𝑎(𝑚) has its group of relations generated by its relations of length at most 𝑎(𝑚) ≤ 𝑚/2𝑘, ∞ ) ′ there it follows from Lemma 3.5.8 that for each 𝑚/2𝑘 ≤ 𝑚 1 ≤ 𝑚 2 ≤ 3𝑚 and each 𝑢, 𝑣 ∈ (𝑆 𝑚 2 Ð exists a path from 𝑢 to 𝑣 in 𝐺 ′𝑎(𝑚) that is contained in 𝑥∈(𝑆∞𝑚 ) ′ 𝐵′2𝑚1 (𝑥) and has length at most 2 3𝑚 1 Gr(3𝑚 2 )/Gr(𝑚 1 ), where all sets and growth functions are identical in the two groups Γ𝑎(𝑚) and Γ by the restrictions placed on 𝑘 and 𝑚. Since these paths are entirely contained within the ball for which 𝐺 ′ and 𝐺 ′𝑎(𝑚) are identical, they exist in 𝐺 ′ also. Moreover, since 𝑚 2 ≥ 𝑚 1 ≥ 𝑚/2𝑘 ≥ 𝑛, we have by Item 9 of Theorem 3.5.5 as above that   𝐶1 Gr′ (3𝑚 2 ) 2 𝐶1 𝑚 2 ≤ 𝐶1 (3 + 𝐶1 ) Gr′ (𝑚 1 ) 𝑚1 so that the length of this path is at most 3𝐶12 (3 + 𝐶1 ) 𝐶1 (𝑚 2 /𝑚 1 ) 𝐶1 𝑚 1 =: 𝐶4 (𝑚 2 /𝑚 1 ) 𝐶1 𝑚 1 . (Everything discussed in this paragraph is still under the assumption that 𝑚 ≥ 2𝑘𝑛 does not belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴.) We now use the existence of these paths in 𝐺 ′ to guarantee the desired plentiful tube conditions in the original graph 𝐺. More concretely, we will prove that there exist positive constants 𝑐 = 𝑐(𝜅, 𝑑), 85 𝐶 = 𝐶 (𝜅, 𝑑), and 𝑛2 = 𝑛2 (𝜅) ≥ 𝑛1 such that if 𝑛 ≥ 𝑛2 then 𝐺 has (𝑐𝑘, 𝑐𝑘, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes on each scale 𝑚 ≥ 𝐶 𝑘𝑛 such that 𝑚 does not belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴. We begin by constructing annular tubes. It suffices to construct tubes in the case that the two (𝑚, 3𝑚) crossings are both the vertex sets of paths 𝜂1 , 𝜂2 from 𝑆 𝑚 to 𝑆3𝑚 since any crossing contains the vertex set of such a path. Fix 𝑚 ≥ 2𝑘𝑛 and two paths 𝜂1 and 𝜂2 from 𝑆 𝑚 to 𝑆3𝑚 in 𝐺, and suppose that 𝑚 does not belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴. Apply Lemma 3.5.10 with each of these paths and the (1, 𝐶3 𝑛)-quasi-isometry 𝜓 : 𝑉 ′ → 𝑉 (taking 𝑢 and 𝑣 to be the endpoints of 𝜂𝑖 , 𝑥 to be 𝜙(𝑢) and 𝑦 to be 𝜙(𝑣)) to obtain two paths 𝜂′1 and 𝜂′2 in 𝐺 ′. Using the (1, 𝐶1 𝑛)-quasi-isometry property of 𝜙, each of these paths starts at distance at most 𝑚 + 𝐶1 𝑛 from 𝜙(𝑜) = id and ends at distance at least 3𝑚 − 𝐶1 𝑛 from id. If 𝑚 ≥ 9𝐶1 𝑛 then both paths start at distance at most 10 9𝑚 26 from id and end at distance at least 9 𝑚 from id. As such, it follows from Proposition 3.5.9 that if ∞ ) ′ in 𝐺 for each integer 𝑚 ≥ 9𝐶1 𝑛 then the paths 𝜂′1 and 𝜂′2 both intersect the exposed sphere (𝑆 𝑚 2 12 10 𝑖 ∈ 𝐼 := Z ∩ [ 9 𝑚, 9 𝑚]. (The only property of these numbers we will need is that 12 > 10 and 12 < 26/2.) Fix 𝑚 1 = ⌈𝑚/2𝑘⌉ ≤ 𝑚 and for each integer 𝑖 ∈ 𝐼 let 𝑥𝑖 be a point of 𝜂′1 belonging to (𝑆𝑖∞ ) ′ and let 𝑦𝑖 be a point of 𝜂′2 belonging to (𝑆𝑖∞ ) ′. If 𝑚 ≥ max{9𝐶1 𝑛, 2𝑘𝑛} then, since 𝑚 was assumed not to belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴, it follows from Lemma 3.5.8 as discussed above Ð that for each such 𝑖 there exists a path 𝛾𝑖′ from 𝑥𝑖 to 𝑦𝑖 that is contained in (𝑆𝑖∞ ) ′ 𝐵′2𝑚1 (𝑥) and has length at most 𝐶4 (𝑖/𝑚 1 ) 𝐶1 𝑚 1 . Since 𝑖 ≤ 3𝑚, the length of this path can be bounded by 𝐶5 𝑘 𝐶1 −1 𝑚 for an appropriate constant 𝐶5 = 𝐶5 (𝜅). Moreover, if 𝑟 ≥ 1 and 𝑖 and 𝑗 are two integers 𝑖, 𝑗 ∈ 𝐼 satisfying |𝑖 − 𝑗 | > 4𝑚 1 + 2𝑟 then the tubes of radius 𝑟 around 𝛾𝑖′ and 𝛾 ′𝑗 in 𝐺 ′ are disjoint since Ð Ð they are contained in disjoint annuli (𝑆𝑖∞ ) ′ 𝐵′2𝑚1 +𝑟 (𝑥) and (𝑆∞𝑗 ) ′ 𝐵′2𝑚1 +𝑟 (𝑥). Since Lemma 3.5.10 guaranteed that the paths 𝜂′1 and 𝜂′2 have images under 𝜓 contained in the 4𝐶1 𝑛-neighbourhoods of 𝜂1 and 𝜂2 respectively, we can for each 𝑖 ∈ 𝐼 find points 𝑢𝑖 and 𝑣 𝑖 in 𝜂1 and 𝜂2 respectively such that 𝑑 (𝑥𝑖 , 𝜙(𝑢𝑖 )) and 𝑑 (𝑦𝑖 , 𝜙(𝑣 𝑖 )) are at most 6𝐶1 𝑛. Applying Lemma 3.5.10 to each of the paths 𝛾𝑖′ (with the quasi-isometry 𝜙, the points 𝑢𝑖 and 𝑣 𝑖 in 𝐺, the points 𝑥𝑖 and 𝑦𝑖 in 𝐺 ′, and the quasi-isometry constants 𝛼 = 1 and 𝛽 = 6𝐶1 𝑛), it follows that there exist constants 𝐶6 = 𝐶6 (𝜅) and 𝐶7 = 𝐶7 (𝜅) such that for each 𝑖 ∈ 𝐼 there exists a path 𝛾𝑖 from 𝑢𝑖 to 𝑣 𝑖 of length at most 𝐶6 𝑘 𝐶1 −1 such that if 𝑖, 𝑗 ∈ 𝐼 satisfy |𝑖 − 𝑗 | ≥ 𝐶7 𝑚 1 then the tubes of radius 𝑚 1 around 𝛾𝑖 and 𝛾 𝑗 are disjoint. The claim about annular tubes follows easily by taking a 𝐶7 𝑚 1 -separated subset of 𝐼 (i.e., a subset of 𝐼 in which all distinct pairs of integers have distance at least 𝐶7 𝑚 1 ), since such a set may be taken to have size at least 𝐼/𝐶7 𝑚 1 ≥ 𝑐 1 𝑘 for some positive constant 𝑐 1 = 𝑐 1 (𝜅) whenever 𝑛 is larger than some constant 𝑛2 = 𝑛2 (𝜅) ≥ 𝑛1 . (The freedom to increase 𝑛1 to 𝑛2 lets us make sure that every real number we round down is at least 1, so that rounding cannot reduce any 86 relevant quantities by more than a factor of 1/2.) We now briefly argue that the same construction also yields radial tubes crossing a shifted annulus. Run the above construction again with 𝜂1 and 𝜂2 taken to be the two portions crossing from 𝑆 𝑚 to 𝑆12𝑚 of a doubly-infinite geodesic passing through 𝑜, so that every point in 𝜂1 has distance at least 2𝑚 from every point in 𝜂2 , but with various constants changed appropriately since we are now working with (𝑚, 12𝑚) crossings instead of (𝑚, 3𝑚) crossings. In particular, the interval 𝐼 can now be taken to be Z ∩ [8𝑚, 10𝑚], with various other constants changing to reflect this change. Consider the point 𝑢 in 𝜂1 that has distance 9𝑚 from 𝑜. When we perform the above construction to build paths between 𝜂1 and 𝜂2 , each path 𝛾𝑖 starts at distance at most 𝑚 from 𝑢 and ends at distance at least 11𝑚 from 𝑢. Thus, since 𝐺 is transitive, the family of paths we have constructed verifies the (𝑐𝑘, 𝑐𝑘, 𝐶 𝑘 𝐶 𝑚)-plentiful radial tubes condition holds at the scale 𝑚 as desired, for some constants 𝑐 = 𝑐(𝜅, 𝑑) and 𝐶 = 𝐶 (𝜅, 𝑑). As before, this works under the assumption that 𝑛 ≥ 𝑛2 for some 𝑛2 = 𝑛2 (𝜅) and that 𝑚 ≥ 𝐶 𝑘𝑛 does not belong to [𝑎, 2𝑘𝑎] for any 𝑎 ∈ 𝐴. □ Using random walk trajectories In this section we prove Proposition 3.5.4, which verifies the plentiful tubes condition for graphs that have quasi-polynomial absolute growth but a fast rate of relative growth over an appropriate range of scales. As discussed above, we will construct the required collections of disjoint tubes by modifying certain conditioned random walk trajectories. To avoid parity issues, we work with lazy random walks throughout the section. We will spend most of the section proving general bounds on the behaviour of random walk on some scale in terms of the growth of the graph at that scale, specializing to the setting of Proposition 3.5.4 only at the very end of the proof. Given a graph 𝐺 and a vertex 𝑢 of 𝐺, let P𝑢 denote the law of the lazy random walk started from 𝑢, which at each step either stays in place with probability 1/2 or else crosses a uniform random edge emanating from its current position, and let the heat kernel 𝑝 𝑡 (𝑢, 𝑣) be defined by 𝑝 𝑡 (𝑢, 𝑣) = P𝑢 (𝑋𝑡 = 𝑣). We begin by recalling two important facts about random walks on graphs of quasi-polynomial growth that will be used in the proof: the Varopoulos-Carne inequality [Var85; Car85], which implies near-diffusive estimates on the rate of escape, and the total variation inequality of [Yad23, Chapter 7.5], which implies that two walks started from different vertices can be coupled to coalesce by the time they reach a distance that is near-linear in their starting distance. Diffusive estimates from Varopoulos-Carne. We now state the Varopoulos-Carne inequality, which gives Gaussian-like bounds on the 𝑛-step transition probabilities between two specific ver- 87 tices; see e.g. [LP16c, Chapter 13.2] for a modern treatment. This inequality does not require transitivity, and holds for the random walk on any graph. Theorem 3.5.11 (Varopoulos-Carne). Let 𝐺 = (𝑉, 𝐸) be a (locally finite) graph. Then √︄   deg(𝑣) 𝑑 (𝑢, 𝑣) 2 𝑝 𝑡 (𝑢, 𝑣) ≤ 2 exp − , deg(𝑢) 2𝑡 for every 𝑡 ≥ 1 and every 𝑢, 𝑣 ∈ 𝑉. The Varopoulos-Carne inequality easily implies that the random walk on a graph of quasipolynomial growth is very unlikely to be at a distance much larger than 𝑡 1/2 (log 𝑡) 𝑂 (1) from its starting point. Corollary 3.5.12. Let 𝐺 = (𝑉, 𝐸) be a (locally finite) transitive graph and let 𝑜 be a vertex of 𝐺. Then    2 𝑛 P𝑜 max 𝑑 (𝑜, 𝑋𝑘 ) ≥ 𝑛 ≤ 2(𝑡 + 1) Gr(𝑛) exp − 0≤𝑘 ≤𝑡 2𝑡 for every 𝑡, 𝑛 ≥ 1. Proof of Corollary 3.5.12. If max0≤𝑘 ≤𝑡 𝑑 (𝑜, 𝑋𝑘 ) ≥ 𝑛 then there exists 0 ≤ 𝑘 ≤ 𝑡 such that 𝑋 𝑘 has distance exactly 𝑛 from 𝑜. Since the number of points at distance 𝑛 from 𝑜 it at most Gr(𝑛), the claim follows from Theorem 3.5.11 by taking a union bound over the possible values of 𝑘 and 𝑋𝑘 . □ Remark 3.5.6. For transitive graphs of polynomial growth, Varopoulos-Carne implies a displace√︁ √ ment upper bound of the form 𝑡 log 𝑡 while the true displacement is of order 𝑡 with high probability. (This sharp upper bound on the displacement can be proven using a (highly nontrivial) improvement of the Varopoulos-Carne inequality due to Hebisch and Saloff-Coste [HS93].) As such, our reliance on Varopoulos-Carne leads to all of the estimates in this section having polylog terms that are known to be unnecessary for transitive graphs of polynomial growth and are presumably non-optimal for transitive graphs of quasi-polynomial growth also. Coupling from low growth. We now explain how low growth can be used to couple two walks to coalesce within time not much larger than quadratic in their starting distance; we will eventually concatenate these pairs of coupled random walks to build annular tubes. We first recall some 88 relevant definitions. Given two probability measures 𝜇 and 𝜈 on a countable set Ω, the total variation distance ∥𝜇 − 𝜈∥ TV between 𝜇 and 𝜈 is defined by ∥𝜇 − 𝜈∥ TV = sup |𝜇( 𝐴) − 𝜈( 𝐴)| = 𝐴⊆Ω 1 ∑︁ |𝜇(𝜔) − 𝜈(𝜔)|. 2 𝜔∈Ω The total variation distance is indeed a distance in the sense that it defines a metric on the space of probability measures on Ω. The total variation distance is related to coupling (and to the theory of optimal transport) by the variational formula n o ∥𝜇 − 𝜈∥ TV = inf P(𝑋 ≠ 𝑌 ) : 𝑋, 𝑌 random variables with 𝑋 ∼ 𝜇 and 𝑌 ∼ 𝜈 . In particular, given two vertices 𝑥 and 𝑦 in a graph, we can couple the lazy random walks started at 𝑥 and 𝑦 to coincide at time 𝑚 with probability 1 − ∥P𝑥 (𝑋𝑚 = ·) − P𝑦 (𝑋𝑚 = ·)∥ TV . Note that if the two walks coincide at time 𝑚 then we can trivially couple them to remain equal at all subsequent times, so that ∥P𝑥 (𝑋𝑚 = ·) − P𝑦 (𝑋𝑚 = ·)∥ TV is a decreasing function of 𝑚 when 𝑥 and 𝑦 are fixed. These couplings will be used when we construct annular tubes using pairs of coupled random walks. Given a (locally finite) transitive graph 𝐺 = (𝑉, 𝐸), the Shannon entropy 𝐻𝑡 of the 𝑡th step of the lazy random walk is defined to be h i ∑︁ 𝐻𝑡 = −E𝑜 log 𝑝 𝑡 (𝑜, 𝑋𝑡 ) = − 𝑝 𝑡 (𝑜, 𝑥) log 𝑝 𝑡 (𝑜, 𝑥). 𝑥∈𝑉 Since the Shannon entropy of any random variable taking values in a set of size 𝑛 is at most log 𝑛, the quantity 𝐻𝑡 satisfies the trivial inequality 𝐻𝑡 ≤ log Gr(𝑡). The following extremely useful inequality11 relates the total variation distance to the increments of the Shannon entropy; versions of this inequality have been rediscovered independently in the works [EK10; BDKY15; Oza18] as discussed in detail in [Yad23, Chapter 7.5]. Theorem 3.5.13. If 𝐺 is a (locally finite) transitive graph and 𝑜 is a vertex of 𝐺 then 1 ∑︁ ∥P𝑜 (𝑋𝑡 = ·) − P𝑥 (𝑋𝑡−1 = ·) ∥ 2TV ≤ 𝐻𝑡 − 𝐻𝑡−1 deg(𝑜) 𝑥∼𝑜 for every 𝑡 ≥ 1. To apply this inequality, we will need to bound the entropy in terms of the growth. While we always have the trivial bound 𝐻𝑡 ≤ log Gr(𝑡), it is also possible to bound the entropy in terms of 11 known in some circles as “the cool inequality”. 89 [log Gr(𝑡 1/2 )] 2 , which is a significantly better bound when the growth is much larger on scale 𝑡 than scale 𝑡 1/2 . (While this bound is worse than the trivial bound when the growth is subexponential and sufficiently regular, it better fits into our philosophy of understanding the behaviour of the random walk at some scale from the growth of the graph at that scale alone.) Lemma 3.5.14. For each 𝑑 ≥ 1 there exists a constant 𝐶 = 𝐶 (𝑑) such that if 𝐺 is a (locally finite) transitive graph of degree 𝑑 then 
2 𝐻𝑡 ≤ 𝐶 log Gr 𝑡 1/2 for every 𝑡 ≥ 1. Proof of Lemma 3.5.14. We may assume that the diameter of 𝐺 is at least 𝑡 1/2 , the claim being trivial otherwise since 𝐻𝑡 ≤ log |𝑉 |. Recall that if 𝑋 and 𝑌 are two random variables defined on the same probability space, the conditional entropy 𝐻 (𝑋 | 𝑌 ) is defined to be the expected entropy of the conditional law of 𝑋 given 𝑌 . Bayes’ rule for the conditional entropy states that 𝐻 (𝑋) = 𝐻 (𝑌 ) + 𝐻 (𝑋 |𝑌 ) − 𝐻 (𝑌 |𝑋) ≤ 𝐻 (𝑌 ) + 𝐻 (𝑋 |𝑌 ). Applying this inequality with 𝑋 = 𝑋𝑡 and 𝑌 = 1 (𝑋𝑡 ∈ 𝐵𝑟 ), and using that the entropy of a random variable supported on a set of size 𝑁 is at most log 𝑁, we obtain that  2 𝑟 𝐻𝑡 ≤ log 2+log Gr(𝑟)+[log Gr(𝑡)]P𝑜 (𝑋𝑡 ∉ 𝐵𝑟 ) ≤ log 2+log Gr(𝑟)+2(𝑡+1) Gr(𝑟) exp − log Gr(𝑡) 2𝑡 for every 𝑟, 𝑡 ≥ 1. Using the fact that the growth is submultiplicative and that Gr(𝑛) ≤ 𝑑 𝑛+1 , we obtain that if 𝑛 := ⌊𝑡 1/2 ⌋ divides 𝑟 then   𝑟 𝑟2 𝑟 2 log Gr(𝑛) − , 𝐻𝑡 ≤ log 2 + log Gr(𝑛) + 2𝑑 (𝑡 + 1) exp 𝑛 𝑛 2𝑡 and the claim follows by taking 𝑟 to be a multiple of 𝑛 closest to 𝐶 log[𝑡 Gr(𝑛)] for an appropriately large constant 𝐶 = 𝐶 (𝑑). (Note that log[𝑡 Gr(𝑛)] and log Gr(𝑛) are of the same order since the diameter of 𝐺 is at least 𝑡 1/2 and hence Gr(𝑛) ≥ 𝑛.) □ This inequality easily implies the following simple bound on the total variation distance in terms of the growth, yielding in particular that the total variation distance is small whenever 𝑡 is much larger than 𝑑 (𝑥, 𝑦) 2 log Gr(𝑡 1/2 ) 2 . 90 Corollary 3.5.15. For each 𝑑 ≥ 1 there exists a constant 𝐶 = 𝐶 (𝑑) such that if 𝐺 = (𝑉, 𝐸) is a (locally finite) transitive graph of degree 𝑑 then ∥P𝑥 (𝑋𝑡 = ·) − P𝑦 (𝑋𝑡 = ·)∥ TV ≤ 𝐶 log Gr(𝑡 1/2 ) 𝑑 (𝑥, 𝑦) 𝑡 1/2 for every 𝑡 ≥ 1 and 𝑥, 𝑦 ∈ 𝑉. Proof of Corollary 3.5.15. It follows by a standard computation that the total variation distance between a Binomial(𝑛, 1/2) distribution and a Binomial(𝑛 + 1, 1/2) distribution is of order 𝑛−1/2 . Indeed, if we let 𝜇 be the Binomial(𝑛, 1/2) distribution and let 𝜈 be the Binomial(𝑛 + 1, 1/2) distribution then 𝜇 is absolutely continuous with respect to 𝜈 with density 𝜇(𝑘) 2(𝑛 − 𝑘 + 1) = 𝜈(𝑘) 𝑛+1 for every 0 ≤ 𝑘 ≤ 𝑛 + 1, and we have by an easy computation (using e.g. Jensen’s inequality and the linearity of the variance) that 𝑛+1 𝑛+1 1 ∑︁ 𝑛 − 2𝑘 + 1 1 ∑︁ 𝜇(𝑘) − 1 𝜈(𝑘) = 𝜈(𝑘) = 𝑂 (𝑛−1/2 ) ∥𝜇 − 𝜈∥ TV = 2 𝑘=0 𝜈(𝑘) 2 𝑘=0 𝑛+1 as claimed. Since the conditional laws of the lazy random walks 𝑋𝑡 and 𝑋𝑡+1 are the same given that the number of non-lazy steps are the same, it follows that ∥P𝑥 (𝑋𝑡 = ·) − P𝑥 (𝑋𝑡−1 = ·) ∥ TV ≤ 𝐶1 𝑡 −1/2 for every 𝑡 ≥ 1 and 𝑥 ∈ 𝑉, where 𝐶1 is a universal constant. Putting this together with Theorem 3.5.13 yields that if 𝑥 and 𝑦 are neighbouring vertices on a transitive graph of degree 𝑑 then √︁ P𝑥 (𝑋𝑡 = ·) − P𝑦 (𝑋𝑡 = ·) TV ≤ 4𝑑 (𝐻𝑡 − 𝐻𝑡−1 ) + 𝐶1 𝑡 −1/2 for every 𝑡 ≥ 1. Since the left hand side is increasing in 𝑡 and, by Lemma 3.5.14, there exists 𝑡/2 ≤ 𝑘 ≤ 𝑡 with 𝐻 𝑘 − 𝐻 𝑘−1 ≤ 2𝑡 𝐻𝑡 ≤ 𝐶𝑡2 log Gr(𝑡 1/2 ) 2 for some constant 𝐶2 = 𝐶2 (𝑑), it follows that √︂ 4𝐶2 𝑑 −1/2 P𝑥 (𝑋𝑡 = ·) − P𝑦 (𝑋𝑡 = ·) TV ≤ 𝐶1 𝑡 + log Gr(𝑡 1/2 ) 2 𝑡 for every pair of adjacent vertices 𝑥 and 𝑦 and every 𝑡 ≥ 1. The analogous bound for arbitrary pairs of vertices follows from this and the triangle inequality for the total variation distance. □ Hitting probabilities of balls. We now want to argue that tubes around independent random walks started at distant vertices are likely to be disjoint under the assumption that our graph “looks 91 at least five dimensional” on all relevant scales. In fact we will prove more general versions of these estimates in which “five” is replaced by an arbitrary constant 𝜅 > 4. We begin by noting the following simple analytic consequence of the results of Coulhon and Saloff-Coste [CS93] and Morris and Peres [MP05], which lets us convert growth bounds into bounds on the heat kernel 𝑝 𝑡 (𝑢, 𝑣). Lemma 3.5.16. For each integer 𝑑 ≥ 1 there exists a positive constant 𝑐 = 𝑐(𝑑) ∈ (0, 1] such that if 𝐺 = (𝑉, 𝐸) is an infinite unimodular transitive graph with vertex degree 𝑑 then 𝑝 𝑡 (𝑢, 𝑣) ≤ 1   Gr 𝑐𝑡 1/2 log Gr 𝑡 1/2
 −1/2  for every integer 𝑡 ≥ 4 and every pair of vertices 𝑢, 𝑣 in 𝐺. Proof of Lemma 3.5.16. It suffices to prove an inequality of the form 𝑝 2𝑡 (𝑢, 𝑣) ≤ 1   𝑐 Gr 𝑐𝑡 1/2 log Gr 𝑡 1/2
 −1/2  (3.5.2) for every integer 𝑡 ≥ 4 and every 𝑢, 𝑣 ∈ 𝑉, where 𝑐 = 𝑐(𝑑) is a positive constant depending only on the degree. Indeed, odd values of 𝑡 can then be handled using the inequality 𝑝 2𝑡+1 (𝑢, 𝑣) ≤ 1 ∑︁ 𝑝 2𝑡 (𝑢, 𝑣′) ≤ max 𝑝 2𝑡 (𝑢, 𝑣′), ′ 𝑣 ∼𝑣 𝑑 𝑣 ′ ∼𝑣 while the constant outside of the growth function can be absorbed into the constant inside the growth function using the inequality Gr(3𝑛𝑚) ≥ 𝑛 Gr(𝑚), which holds for all positive integers 𝑛, 𝑚 in any infinite transitive graph as an elementary consequence of the triangle inequality. We now prove an estimate of the form (3.5.2). Let the inverse growth function Gr−1 be defined by Gr−1 (𝑥) := inf{𝑛 : Gr(𝑛) ≥ 𝑥} and recall that the isoperimetric profile of 𝐺 is the function Φ : [1, ∞) → [0, 𝑑] defined by   |𝜕𝑊 | Φ(𝑥) := inf : 𝑊 ⊆ 𝑉 (𝐺) and 0 < |𝑊 | ≤ 𝑥 . (3.5.3) |𝑊 | For transitive unimodular graphs, the isoperimetric profile and the growth are related by the inequality 1 , (3.5.4) Φ(𝑥) ≥ −1 2 Gr (2𝑥) 92 which was proven for Cayley graphs by Coulhon and Saloff-Coste [CS93] and extended to unimodular transitive graphs by Saloff-Coste [Sal95] and Lyons, Morris, and Schramm [LMS08]; we use the statement given in [LP16c, Theorem 10.46]. To make use of this inequality, we will apply the results of Morris and Peres [MP05], which imply that there exists a constant 𝑐 1 (𝑑) ∈ (0, 1) such that 1 n ∫ o 𝑝 2𝑡 (𝑢, 𝑣) ≤ 𝑦 1 𝑐 1 sup 𝑦 : 1 𝑥Φ(4𝑥) d𝑥 ≤ 𝑐 𝑡 1 2 for every integer 𝑡 ≥ 1 and every 𝑢, 𝑣 ∈ 𝑉. Using (3.5.4) to estimate the integral that appears here, we have for each 𝑦 ∈ [1, ∞) that ∫ 𝑦 ∫ 𝑦 ∫ 𝑦 4 Gr−1 (8𝑥) 2 1 1 −1 2 d𝑥 ≤ d𝑥 ≤ 4 Gr (8𝑦) d𝑥 = 4 Gr−1 (8𝑦) 2 log(𝑦). 2 𝑥 𝑥 𝑥Φ(4𝑥) 1 1 1 Thus, to prove an estimate of the form (3.5.2) it suffices to verify that " # 12 1 © 𝑐1𝑡 ª if 𝑦 ≥ 1 satisfies 𝑦 ≤ Gr ­ ® 1 8 4 log Gr(𝑡 2 ) « ¬ then 4 Gr−1 (8𝑦) 2 log(𝑦) ≤ 𝑐 1 𝑡. (3.5.5) √︁ This follows straightforwardly by noting that 8𝑦 ≤ Gr( 𝑐 1 𝑡/(4 log 𝑦)) whenever 𝑦 satisfies the upper bound on the left hand side of (3.5.5). □ Lemma 3.5.16 has the following elementary corollary, which we will apply only in situations where
log Gr 𝑡 1/2 is much smaller than 𝑛−1 𝑡 1/2 . Corollary 3.5.17 (Leaving a ball). For each integer 𝑑 ≥ 1 and real number 𝜅 ≥ 1 there exists a constant 𝐶 = 𝐶 (𝑑, 𝜅) such that if 𝐺 = (𝑉, 𝐸) is an infinite, connected, unimodular transitive graph with vertex degree 𝑑 and 𝑛, 𝑡 ≥ 1 are integers such that Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑛 ≤ 𝑚 ≤ 12 𝑡 1/2 then  
oi 𝜅 𝑛2 𝜅/2 P𝑢 (𝑋𝑡 ∈ 𝐵𝑛 (𝑣)) ≤ 𝐶 log max 2 , Gr 𝑛 𝑡 𝑛 h n 𝑡 for every pair of vertices 𝑢 and 𝑣 in 𝐺. (Note that this corollary holds vacuously when 𝑡 ≤ 𝑛2 .) Remark 3.5.7. This estimate is quite similar to that appearing in e.g. [Lyo+20]; the important distinction is that we only assume the tripling condition Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for 𝑚 = 𝑂 (𝑡 1/2 ) rather than for all sufficiently large scales. 93 Proof of Corollary 3.5.17. Since max𝑢 P𝑢 (𝑋𝑡 ∈ 𝐵𝑛 (𝑣)) is a decreasing function of 𝑡, it suffices to prove the claim under the slightly stronger assumption that Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑛 ≤ 𝑚 ≤ 𝑡 1/2 . Fix 𝑛, 𝑡 ≥ 1 and let 𝑐 be the constant from Lemma 3.5.16. We have by submultiplicativity of the growth function that  h  Gr(𝑡 1/2 ) ≤ Gr 𝑐𝑡 1/2 log Gr 𝑡 1/2 i −1/2  𝑐 −1 [ log Gr ( 𝑡 1/2 )] 1/2 and hence that    √︃ h  i −1/2    𝑐 [ log Gr ( 𝑡 1/2 )] −1/2
1/2 1/2 1/2 1/2 . Gr 𝑐𝑡 log Gr 𝑡 ≥ Gr 𝑡 = exp 𝑐 log Gr 𝑡 Applying Lemma 3.5.16, it follows that if 𝑟 := 𝑐𝑡 1/2 h  log Gr 𝑡 1/2 i −1/2 (3.5.6) ≤ 𝑛 then   1/2 1 2𝑡 𝑝 𝑡 (𝑢, 𝑣) ≤ ≤ exp −𝑐 Gr(r) 𝑛 for every 𝑢, 𝑣 ∈ 𝑉. It follows by a union bound that if 𝑟 ≤ 𝑛 then   1/2 2𝑡 Gr(𝑛), P𝑢 (𝑋𝑡 ∈ 𝐵𝑛 (𝑣)) ≤ exp −𝑐 𝑛 which is stronger than the desired inequality. Now suppose that 𝑟 ≥ 𝑛. The assumption Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑛 ≤ 𝑚 ≤ 𝑡 1/2 guarantees that Gr(𝑟) ≥ Gr(𝑛) ⌊logÖ 3 (𝑟/𝑛)⌋ 𝑖=1  𝑟 𝜅 Gr(3𝑖 𝑛) Gr(𝑛). ≥ 3𝑛 Gr(3𝑖−1 𝑛) We deduce from Lemma 3.5.16 that there exists a constant 𝐶 such that √︃
𝜅     √︃  1/2   3𝑛 log Gr 𝑡 ©    ª
1 1 1/2 ­ ® 𝑝 𝑡 (𝑢, 𝑣) ≤ ≤ min ­ , exp −𝑐 log Gr 𝑡 ® 1/2   Gr (𝑟) Gr(𝑛) 𝑐𝑡     « ¬    2  𝜅/2 o n
𝑛 𝑡 1 log max 2 , Gr 𝑛 , ≤𝐶 𝑡 Gr(n) 𝑛 where the second inequality follows since min{𝐴𝑥 𝜅 , 𝑒 −𝑥 } ≤ 𝐴(1 ∨ log(1/𝐴)) 𝜅 for every 𝐴, 𝑥 > 0 (as can be checked by case analysis according to whether 𝑥 ≥ log(1/𝐴)). □ We next analyze the probability of hitting a ball whose radius is much smaller than its distance from the starting point. For transitive graphs of polynomial growth, a similar estimate without the logarithmic term can be proven by a similar calculation as in [Hut20i, Lemma 4.4]. 94 Lemma 3.5.18 (Hitting a distant ball). For each integer 𝑑 ≥ 1 and real number 𝜅 > 2 there exists a constant 𝐶 = 𝐶 (𝑑, 𝜅) such that if 𝐺 is an infinite, connected, unimodular transitive graph with vertex degree 𝑑 and 𝑛, 𝑡 ≥ 1 are integers such that 𝑡 ≥ 𝑛2 and Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑛 ≤ 𝑚 ≤ 𝑡 1/2 then   𝜅−2  
 (3𝜅+2)/2 𝑛 P𝑢 (hit 𝐵𝑛 (𝑣) before time 𝑡) ≤ 𝐶 log max 𝑑 (𝑢, 𝑣), Gr 2𝑛 𝑑 (𝑢, 𝑣) for every pair of vertices 𝑢 and 𝑣 with 𝑑 (𝑢, 𝑣) ≥ 2𝑛. Proof of Lemma 3.5.18. For each 1 ≤ 𝑠 ≤ 𝑡, let 𝐴𝑠 be the event that the random walk hits 𝐵𝑛 (𝑣) between times 𝑠 and 𝑡. (We will optimize over the choice of 𝑠 at the end of the proof.) It follows from Corollary 3.5.17 that there exist constants 𝐶1 and 𝐶2 depending only on 𝑑 and 𝜅 such that h i E𝑢 #{𝑠 ≤ 𝑘 ≤ 2𝑡 : 𝑋 𝑘 ∈ 𝐵2𝑛 (𝑣)} ≤ 𝐶1 2𝑡  ∑︁ 𝑘=𝑠   
𝜅 𝑛2 𝜅/2 𝑘 log max 2 , Gr 2𝑛 𝑘 𝑛 h n𝑠
oi 𝜅 𝑛 𝜅 , (3.5.7) ≤ 𝐶2 log max 2 , Gr 2𝑛 𝑛 𝑠 (𝜅−2)/2  where the second inequality follows by calculus. On the other hand, it follows from Corollary 3.5.12 and a straightforward calculation that  P𝑤 𝑋𝜏+𝑘 ∈ 𝐵2𝑛 for every 𝑘 ≤  1 𝑛2 ≥ 8 log Gr(𝑛) 2 2 for every 𝑤 ∈ 𝐵𝑛 (𝑣), and since 𝑡 + 8 log𝑛Gr(𝑛) ≤ 2𝑡 it follows by the strong Markov property that i h E𝑢 #{𝑠 ≤ 𝑘 ≤ 2𝑡 : 𝑋 𝑘 ∈ 𝐵2𝑛 (𝑣)} | 𝐴𝑠 ≥ 𝑛2 . 16 log Gr(𝑛) (3.5.8) Putting together the estimates (3.5.7) and (3.5.8) yields that h i h n𝑠 E𝑢 #{𝑠 ≤ 𝑘 ≤ 2𝑡 : 𝑋 𝑘 ∈ 𝐵2𝑛 (𝑣)}
oi 𝜅+2 𝑛 𝜅−2 h i ≤ 𝐶3 log max 2 , Gr 2𝑛 P𝑢 ( 𝐴𝑠 ) ≤ , (𝜅−2)/2 𝑛 𝑠 E𝑢 #{𝑠 ≤ 𝑘 ≤ 2𝑡 : 𝑋 𝑘 ∈ 𝐵2𝑛 (𝑣)} | 𝐴𝑠 while, since every point in 𝐵𝑛 (𝑣) has distance at least 𝑑 (𝑢, 𝑣)/2 from 𝑢, it follows from the Varopoulos-Carne inequality and a union bound as in the proof of Corollary 3.5.12 that   𝑑 (𝑢, 𝑣) 2 P𝑢 (hit 𝐵𝑛 (𝑣) before time 𝑠) ≤ 2𝑠 Gr(𝑛) exp − . 8𝑠 95 Putting together these estimates yields that  
oi 𝜅+2 𝑛 𝜅−2 𝑑 (𝑢, 𝑣) 2 + 2𝑠 Gr(𝑛) exp − P𝑢 (hit 𝐵𝑛 (𝑣) before time 𝑡) ≤ 𝐶3 log max 2 , Gr 2𝑛 , 8𝑠 𝑛 𝑠 (𝜅−2)/2 h n𝑠 and the claimed inequality follows by taking 𝑠 = 𝑐′ 𝑑 (𝑢, 𝑣) 2 (log Gr(𝑛)) −1 for an appropriately small constant 𝑐′. □ Disjoint tubes from coarse-grained random walks. As noted above, a naive construction of disjoint tubes using random walks is not appropriate for our plentiful tubes condition, since the two walks will couple at a time roughly quadratic in their starting distance rather than roughly linear. To circumvent this issue, we will instead consider tubes around certain coarse-grained versions of the random walk defined through what we call ironing, where we replace portions of the random walk with geodesics between their endpoints. This process will also be useful when we analyze intersections between random walk tubes, as the ironing process allows us to circumvent overcounting issues that would arise in a naive first-moment argument. We now define the ironing procedure formally; see fig. 3.3 for an illustration. Let 𝐺 be a graph. For every pair of distinct vertices 𝑢, 𝑣 ∈ 𝑉 (𝐺), fix a geodesic 𝜁 (𝑢, 𝑣) in 𝐺 from 𝑢 to 𝑣. (The choice of 𝜁 is irrelevant to our arguments; we need only that it is done deterministically for every pair of vertices before we start running any random walks.) Fix 𝑟 > 0 and let 𝛾 be a finite path in 𝐺. We define a sequence (𝜏𝑖 )𝑖≥0 recursively as follows: Let 𝜏0 = 0. For each 𝑖 ≥ 0, if 𝑑 (𝛾𝜏𝑖 , 𝛾 𝑘 ) < 𝑟 for every 𝜏𝑖 ≤ 𝑘 ≤ len(𝛾) we set 𝜏𝑖+1 = len(𝛾) and stop. Otherwise, we set 𝜏𝑖+1 to be the minimal time 𝑘 after 𝜏𝑖 that 𝑑 (𝛾𝜏𝑖 , 𝛾 𝑘 ) ≥ 𝑟. We define the crease number cr(𝛾) = cr𝑟 (𝛾) to be the number of non-zero terms in this sequence, so that 𝜏cr(𝛾) = len(𝛾), call the points {𝛾𝜏𝑖 : 0 ≤ 𝑖 ≤ cr𝑟 (𝛾)} crease points, and define the ironed path iron(𝛾) = iron𝑟 (𝛾) by concatenating geodesics between crease points iron𝑟 (𝛾) := 𝜁 (𝛾𝜏0 , 𝛾𝜏1 ) ◦ 𝜁 (𝛾𝜏1 , 𝛾𝜏2 ) ◦ · · · ◦ 𝜁 (𝜏cr(𝛾)−1 , 𝜏cr(𝛾) ). Thus, the ironed path iron(𝛾) is a finite path in 𝐺 which has the same start and end points as 𝛾, has length at most 𝑟 · cr(𝛾), and satisfies the containment of tubes 𝐵𝑟 (iron𝑟 (𝛾)) ⊆ 𝐵2𝑟 (𝛾) and 𝐵𝑟 (𝛾) ⊆ 𝐵2𝑟 ({𝛾𝜏𝑖 : 0 ≤ 𝑖 ≤ cr𝑟 (𝛾)}) ⊆ 𝐵2𝑟 (iron𝑟 (𝛾)). For graphs of quasi-polynomial growth, we can use the Varopoulos-Carne inequality to show that the length of an ironed random walk iron𝑟 (𝑋 𝑡 ) is of order at most 𝑟 −1 𝑡 (log 𝑡) 𝑂 (1) with high √ probability when 𝑟 = 𝑂 ( 𝑡). We write 𝑋 𝑡 for the path formed by the first 𝑡 steps of the random walk. 96 Figure 3.3: Schematic illustration of the ironing procedure applied to a path: The original path is in black, from left to right. The black dots are crease points. A new crease point is formed every time the path leaves a fixed-radius ball centered at the previous crease point. The straight blue line segments are geodesics between consecutive crease points (together with the final point of the walk). The ironed path is formed by concatenating these geodesics. Lemma 3.5.19. Let 𝐺 be a (locally finite) transitive graph and let 𝑢 be a vertex of 𝐺. Then  2  𝑡 𝑟 𝑡 ≤ 2𝑡𝑚 Gr(𝑟) exp − P𝑢 cr𝑟 (𝑋 ) > 𝑚 2𝑚 for every 𝑟, 𝑡, 𝜆 ≥ 1. 𝑡 ) > 𝑡/𝑚 to hold, there must exist 0 ≤ 𝑖 ≤ 𝑡 such that Proof. In order for the inequality cr𝑟 ((𝑋𝑖 )𝑖=0 𝑑 (𝑋𝑖 , 𝑋𝑖+𝑚 ) ≥ 𝑟. As such, the claim follows from Corollary 3.5.12 and a union bound. □ We now analyze intersections between independent random walk tubes. Given two vertices 𝑢 and 𝑣, we write P𝑢 ⊗ P𝑣 for the law of a pair of independent lazy random walks 𝑋 and 𝑌 started at 𝑢 and 𝑣 respectively. Lemma 3.5.20 (Intersections of random walk tubes). For each integer 𝑑 ≥ 1 and real number 𝜅 > 4 there exists a constant 𝐶 = 𝐶 (𝑑, 𝜅) such that if 𝐺 = (𝑉, 𝐸) is an infinite, connected, unimodular transitive graph with vertex degree 𝑑 and 𝑛, 𝑡 ≥ 1 are integers such that 𝑡 ≥ 𝑟 2 and Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑟 ≤ 𝑚 ≤ 𝑡 1/2 then   P𝑢 ⊗ P𝑣 there exist 0 ≤ 𝑖, 𝑗 ≤ 𝑡 such that 𝑑 (𝑋𝑖 , 𝑌 𝑗 ) ≤ 𝑟   𝜅−4  
 (3𝜅+4)/2 𝑡 𝑟 ≤𝐶 log max 𝑑 (𝑢, 𝑣), Gr 4𝑟 , 𝑑 (𝑢, 𝑣) 𝑑 (𝑢, 𝑣) 2 97 for every 𝑢, 𝑣 ∈ 𝑉 with 𝑑 (𝑢, 𝑣) ≥ 4𝑟. Remark 3.5.8. Although it will suffice for all our applications, we note that this bound is very wasteful when 𝑡 is much larger than 𝑑 (𝑢, 𝑣) 2 . A more careful analysis would use that 𝑌𝜏𝜅 is typically very far from 𝑢 when 𝑘 is large. Proof of Lemma 3.5.20. Let 𝐴 be the event that there exists 𝑤 ∈ 𝑉 with 𝑑 (𝑢, 𝑤) ≥ 12 𝑑 (𝑢, 𝑣) and 1 ≤ 𝑖, 𝑗 ≤ 𝑡 such that 𝑑 (𝑋𝑖 , 𝑤), 𝑑 (𝑌 𝑗 , 𝑤) ≤ 𝑟. It suffices by symmetry to prove that there exists a constant 𝐶 = 𝐶 (𝑑, 𝜅) such that   𝜅−2 
 (3𝜅+4)/2 𝑡  𝑟 P𝑢 ⊗ P𝑣 ( 𝐴) ≤ 𝐶 2 log max 𝑑 (𝑢, 𝑣), Gr 4𝑟 . 𝑑 (𝑢, 𝑣) 𝑟 Let 𝜏0 , . . . , 𝜏cr𝑟 (𝑌 𝑡 ) be the stopping times used to define the ironed walk iron𝑟 (𝑌 𝑡 ), and observe that if 𝐴 holds then there must exist 0 ≤ 𝑘 ≤ cr𝑟 (𝑌 𝑡 ) and 0 ≤ 𝑖 ≤ 𝑡 such that 𝑑 (𝑌𝜏𝑘 , 𝑣) ≥ 1 1 𝑑 (𝑢, 𝑣) − 𝑟 ≥ 𝑑 (𝑢, 𝑣) 2 4 and 𝑑 (𝑋𝑖 , 𝑌𝜏𝑘 ) ≤ 2𝑟. Thus, it follows from Lemma 3.5.18 and a union bound that there exists a constant 𝐶1 = 𝐶1 (𝑑, 𝜅) such that P𝑢 ⊗ P𝑣 ( 𝐴)      1 𝑡 𝑡 𝑡 ≤ P𝑣 cr𝑟 (𝑌 ) > + max P𝑢 hit 𝐵2𝑟 (𝑤) before time 𝑡 : 𝑤 ∈ 𝑉, 𝑑 (𝑢, 𝑤) ≥ 𝑑 (𝑢, 𝑣) 𝑚 𝑚 4  2   𝜅−2 
 (3𝜅+2)/2 𝑟 𝑟 𝑡  ≤ 2𝑡𝑚 Gr(𝑟) exp − log max 𝑑 (𝑢, 𝑣), Gr 4𝑟 + 𝐶1 2𝑚 𝑚 𝑑 (𝑢, 𝑣) and the claim follows by taking 𝑚 = 𝑐′𝑟 2 (log max{𝑑 (𝑢, 𝑣), Gr(𝑟)}) −1 for an appropriately small constant 𝑐′ = 𝑐′ (𝑑, 𝜅). □ We now have everything we need to prove Proposition 3.5.4. Proof of Proposition 3.5.4. We will prove the claim concerning annular tubes (which is harder); the changes to the proof needed to establish the claim concerning radial tubes are straightforward and will be explained briefly at the end of the proof. We will prove a general condition for 𝐺 to have (𝑘, 𝑟, ℓ)-plentiful annular tubes on scale 𝑛 in terms of the growth of 𝐺, which we specialize to give the claim about scales of quasi-polynomial growth at the end of the proof. Fix 𝑛 ≥ 1, 𝑘 ≤ 𝑛/2, 𝑟 ≤ 𝑛, 𝑡 ≥ 𝑛 and two (𝑛, 3𝑛) crossings 𝐴 and 𝐵 as in 98 the definition of plentiful annular tubes, so that 𝐴 and 𝐵 each contain points at every distance from 𝑜 between 𝑛 and 3𝑛. We will carry out our analysis under the hypotheses that Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for every 𝑟 ≤ 𝑚 ≤ 𝑡 (3.5.9) and 1 for every 𝑥, 𝑦 ∈ 𝐵3𝑛 . (3.5.10) 4 We will return to the question of when suitable 𝑡 and 𝑟 satisfying these hypotheses can be chosen at the end of the proof. ∥P𝑥 (𝑋𝑡 = ·) − P𝑦 (𝑋𝑡 = ·)∥ TV ≤ Since 𝐴 and 𝐵 each contain at least one point at each distance from 𝑜 between 𝑛 and 3𝑛, and since 𝑛/2𝑘 ≥ 1, we may choose for each 1 ≤ 𝑖 ≤ 𝑘 points 𝑎𝑖 ∈ 𝐴 and 𝑏𝑖 ∈ 𝐵 such that 𝑛 𝑛 𝑛 𝑛 ≤ 𝑑 (𝑜, 𝑎𝑖 ) ≤ 𝑛 + (4𝑖 − 3) and 𝑛 + (4𝑖 − 2) ≤ 𝑑 (𝑜, 𝑏𝑖 ) ≤ 𝑛 + (4𝑖 − 1) 𝑛 + (4𝑖 − 4) 2𝑘 2𝑘 2𝑘 2𝑘 for every 1 ≤ 𝑖 ≤ 𝑘, so that the set of points {𝑎𝑖 } ∪ {𝑏𝑖 } is (𝑛/𝑘)-separated (i.e., any two distinct points in the set have distance at least 𝑛/𝑘). For each 𝑖, let Q𝑖 be the joint law of a random walk (𝑋𝑖,𝑚 )𝑚≥0 started at 𝑎𝑖 and a random walk (𝑌𝑖,𝑚 )𝑚≥0 started at 𝑏𝑖 , coupled so that 𝑋𝑡 = 𝑌𝑡 with probability at least 3/4 (such a coupling exists by the hypothesis (3.5.10) imposed on the value of Ë 𝑡), and let Q = Q𝑖 be the law of the collection {𝑋𝑖,𝑚 , 𝑌𝑖,𝑚 : 1 ≤ 𝑖 ≤ 𝑘, 𝑚 ≥ 0} in which the pairs ((𝑋𝑖,𝑚 )𝑚≥0 , (𝑌𝑖,𝑚 )𝑚≥0 ) are sampled independently for each 1 ≤ 𝑖 ≤ 𝑘. For each 1 ≤ 𝑖 ≤ 𝑘, consider the events   16𝑡 𝑡
𝑡
log 𝑡 Gr(𝑟) . A𝑖 = {𝑋𝑖,𝑡 = 𝑌𝑖,𝑡 } and B𝑖 = len iron𝑟 (𝑋 ) , len iron𝑟 (𝑌 ) ≤ 𝑟 The event A𝑖 has probability at least 3/4 for every 1 ≤ 𝑖 ≤ 𝑘 by construction. Meanwhile, Lemma 3.5.19 implies that   16𝑡 𝑐 𝑡 Q(B𝑖 ) ≤ 2P𝑢 cr𝑟 (𝑋 ) > 2 log max{𝑡, Gr(𝑟)} 𝑟 ≤ 4𝑡𝑟 2 [log max{𝑡, Gr(𝑟)}] Gr(𝑟) exp [−8 log max{𝑡, Gr(𝑟)}] , which is less than 1/4 if 𝑡 is larger than some universal constant 𝑡 0 . Now, for each 1 ≤ 𝑖, 𝑗 ≤ 𝑘 let I𝑖, 𝑗 = {𝐵2𝑟 (𝑋𝑖𝑡 ) ∪ 𝐵2𝑟 (𝑌𝑖𝑡 ) has non-empty intersection with 𝐵2𝑟 (𝑋 𝑡𝑗 ) ∪ 𝐵2𝑟 (𝑌 𝑗𝑡 )}. We have by a union bound that if 𝑖 and 𝑗 are distinct then n 𝑛o 𝑡 𝑡 Q(I𝑖, 𝑗 ) ≤ 4 max P𝑢 ⊗ P𝑣 (𝐵2𝑟 (𝑋 ) ∩ 𝐵2𝑟 (𝑌 ) ≠ ∅) : 𝑑 (𝑢, 𝑣) ≥ 𝑘   
 (3𝜅+4)/2 𝑟 𝑘 𝜅−4 𝑡𝑘 2  ≤ 𝐶 2 log max 𝑛, Gr 8𝑟 =: 𝛼 = 𝛼(𝑛, 𝑡, 𝑘, 𝑟). 𝑛 𝑛 99 Thus, if for each 1 ≤ 𝑖 ≤ 𝑘 we define the event C𝑖 = {there exist at most 4𝛼𝑘 values of 1 ≤ 𝑗 ≤ 𝑘 with 𝑗 ≠ 𝑖 such that I𝑖, 𝑗 holds} then Q(C𝑖 ) ≥ 34 by Markov’s inequality. It follows by a union bound that Q(A𝑖 ∩ B𝑖 ∩ C𝑖 ) ≥ 1/4, and hence that if we define  𝑘 E = # 1 ≤ 𝑖 ≤ 𝑘 : A𝑖 ∩ B𝑖 ∩ C𝑖 holds ≥ 4 then Q(E) > 0. Consider the random set of indices 𝐼 = {1 ≤ 𝑖 ≤ 𝑘 : A𝑖 ∩B𝑖 ∩C𝑖 holds but I𝑖, 𝑗 does not hold for any 𝑗 < 𝑖 for which A 𝑗 ∩ B 𝑗 ∩ C 𝑗 holds}. 1 min{𝑘, 𝛼−1 }. Moreover, on this On the event E, the set 𝐼 has size at least ⌈(𝑘/4)/(1 + 4𝛼𝑘)⌉ ≥ 20 event, the set of paths formed by concatenating iron𝑟 (𝑋𝑖𝑡 ) and the reversal of iron𝑟 (𝑌𝑖𝑡 ) for each 𝑖 ∈ 𝐼 have the property that each such path has length at most 32𝑡/𝑟 log 𝑡 Gr(𝑟), and the tubes of thickness 𝑟 around distinct such paths are disjoint. Since this event has positive probability, there must exist a set of paths with this property. Since the sets 𝐴 and 𝐵 were arbitrary, it follows that there exists a positive constant 𝑐 = 𝑐(𝑑, 𝜅) such that 𝐺 has      𝜅−4  32𝑡 𝑛2 −(3𝜅+4)/2 𝑛 log max{𝑡, Gr(𝑟)} -plentiful , 𝑟, 𝑐 min 𝑘, 2 [log max{𝑛, Gr(8𝑟)}] 𝑟𝑘 𝑟 𝑡𝑘 annular tubes on scale 𝑛 for each 1 ≤ 𝑟, 𝑘 ≤ 𝑛/2 and 𝑡 ≥ 𝑛2 satisfying (3.5.9) and (3.5.10). We now specialize to the setting of the proposition. Let 𝐷, 𝜆 ≥ 1 and 0 < 𝜀 < 1 and suppose 𝐷 that 𝑛 ≥ 1 satisfies Gr(𝑚) ≤ 𝑒 (log 𝑚) and Gr(3𝑚) ≥ 35 Gr(𝑚) for every 𝑛1−𝜀 ≤ 𝑚 ≤ 𝑛1+𝜀 . Let 𝑘 = (log 𝑛) 𝜆 , 𝑟 = 𝑛(log 𝑛) −20𝐷𝜆 , and 𝑡 = 𝑛2 (log 𝑛) 2𝐷 . There exists 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆, 𝜀) such that if 𝑛 ≥ 𝑛0 then 𝑛1−𝜀 ≤ 𝑟, 𝑡 1/2 ≤ 𝑛1+𝜀 , so that (3.5.9) holds. Moreover, since log Gr(𝑡 1/2 ) ≤ (log 𝑡 1/2 ) 𝐷 ≤ (1+𝜀) 2 (log 𝑛) 𝐷 , it follows from Corollary 3.5.15 that (3.5.10) holds whenever 𝑛 ≥ 𝑛1 for some constant 𝑛1 = 𝑛1 (𝑑, 𝐷, 𝜆, 𝜀) ≥ 𝑛0 . It follows that there exists a constant 𝐶 such that if 𝑛 ≥ 𝑛1 then 𝐺 has o  n  21 𝑐 min (log 𝑛) 𝜆 , (log 𝑛) (20𝜆− 2 )𝐷−𝜆 , 𝑛(log 𝑛) −20𝐷𝜆 , 𝐶 (log 𝑛) (3+20𝜆)𝐷 -plentiful annular tubes on scale 𝑛. Since the plentiful annular tubes condition is monotone increasing in the number and thickness parameters and monotone decreasing in the length parameter, it follows that there exists a constant 𝑛2 = 𝑛2 (𝑑, 𝐷, 𝜆, 𝜀) ≥ 𝑛1 such that if 𝑛 ≥ 𝑛2 then 𝐺 has (1/40𝐷, 𝜆)polylog-plentiful annular tubes on scale 𝑛 as claimed (where we used the assumption that 𝑛 is large to absorb the constant prefactors into the exponents of the logarithms). 100 For the claim concerning radial tubes, one uses random walks started at 𝑘 equidistant points along a geodesic from 𝑜 to 𝑆𝑛 . Corollary 3.5.17 implies that each of these random walks has a good probability not to belong to 𝐵3𝑛 at times of order 𝑛2 (log Gr(𝑛)) 𝜅 , so that we can obtain tubes from 𝑆𝑛 to 𝑆3𝑛 using segments of the resulting ironed paths. The analysis of the number, thickness, and lengths we can take these walks to have with the resulting tubes being disjoint is similar to the annular case above. (Indeed, the analysis is somewhat simpler since we use single walks instead of coupled pairs of walks. This also makes the dependence on the growth better than in the annular case.) □ 3.6 Quasi-polynomial growth II: Analysis of percolation In this section, we analyze percolation in the low-growth regime, i.e., on scales 𝑛 where Gr(𝑛) ≤ 𝐷 𝑒 [log 𝑛] for a constant 𝐷. Our goal is to show that a lower bound on (full-space) connection probabilities at some low-growth scale 𝑛 implies a lower bound on connection probabilities within a tube at a much larger scale after sprinkling. This analysis will employ both the polylog-plentiful tubes condition from Proposition 3.5.2 and the outputs of the ghost field technology developed in Proposition 3.4.1. Recall that U𝑑∗ is the space of infinite, connected, unimodular transitive graphs of degree 𝑑 that are not one-dimensional. Proposition 3.6.1 (Inductive analysis of percolation in the low-growth regime.). For each 𝑑, 𝐷 ≥ 1 there exist positive constants 𝜆 0 = 𝜆 0 (𝑑, 𝐷) and 𝑐 = 𝑐(𝑑, 𝐷) such that for each 𝜆 ≥ 𝜆 0 there exist constants 𝐾1 = 𝐾1 (𝑑, 𝐷, 𝜆) and 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆) such that if 𝐾 ≥ 𝐾1 and 𝑛 ≥ 𝑛0 then the following holds: If 𝐺 ∈ U𝑑∗ and for each 1 ≤ 𝑏 ≤ 𝑛 we define  𝐾 log log 𝑛 𝛿(𝑏, 𝑛) = 𝛿 𝐾 (𝑏, 𝑛) = min{log 𝑛, log Gr(𝑏)}  1/4 then the implication   1/2 𝑛 ∈ L (𝐺, 𝐷), 𝜅 𝑝1 (𝑛, ∞) ≥ 𝑒 −(log log 𝑛) , P 𝑝1 (Piv[4𝑏, 𝑛1/3 ]) ≤ (log 𝑛) −1 , and 𝛿(𝑏, 𝑛) ≤ 1     𝑐𝜆 1/2 ⇒ 𝑝 2 ≥ 𝑝 𝑐 or 𝜅 𝑝2 𝑒 (log 𝑛) , 𝑛 ≥ 𝑒 −3(log log 𝑛) (3.6.1) holds for every 𝑛 ≥ 𝑛0 , 𝑏 ≤ 18 𝑛1/3 , and 𝑝 2 ≥ 𝑝 1 ≥ 1/𝑑 with 𝑝 2 ≥ Spr( 𝑝 1 , 𝛿(𝑏, 𝑛)). The parameter 𝜆 controls how far a connection probability lower bound is propagated by sprinkling, 𝑐𝜆 namely from scale 𝑛 to scale 𝑒 (log 𝑛) , where 𝑐 is independent of 𝜆. Proposition 3.6.1 morally says 101 that for any choice of 𝜆, if we sprinkle by a sufficient amount at a sufficiently large scale, we 𝑐𝜆 can achieve this propagation from scale 𝑛 to scale 𝑒 (log 𝑛) . (Note that since the conclusion of Proposition 3.6.1 gets stronger as 𝜆 increases, the 𝜆 ≥ 𝜆 0 condition is redundant; we have nevertheless included it because some of our working is simpler if we can assume that 𝜆 is large.) The overall strategy of this section is closely inspired by [CMT22]. A key insight of that paper was that if one is working in the supercritical phase of percolation, then one can sometimes deduce positive information (e.g. a lower bound on set-to-set connection probabilities) from negative information (e.g. an upper bound on set-to-set connection probabilities). Thus, one can analyze the supercritical phase by a case analysis according to whether or not certain point-to-point connection lower bounds hold: both assumptions are useful. This only makes sense because being in the supercritical phase is already a positive hypothesis. For our purposes, we cannot assume that we are in the supercritical phase. However, we can assume that we have a two-point lower bound at some large scale 𝑛, which lets us pretend that we are supercritical when working with scales much smaller than 𝑛. The positive information that this lets us deduce by working with scales smaller than 𝑛 is so strong that it implies set-to-set connection lower bounds even at scales much larger than 𝑛. Besides this, there are two main complications we need to address when adapting the methods of [CMT22] in this section. First, our quasi-polynomial growth and plentiful tubes conditions are rather different than the polynomial growth and connectivity of exposed spheres used in [CMT22], and many details of the argument must change to accommodate this. Second, and more seriously, we must use methods that use the growth upper bound at one scale only, since that is all we can assume; this is very different from [CMT22] where their graphs are assumed to have polynomial growth at all scales and many of the arguments work by inducting up from scale 1. We now begin to work towards the proof of Proposition 3.6.1. We begin by recording various notations and important constants that will be used throughout the proof. First, we let A (𝑏, 𝑛, 𝑝 1 ) = A 𝐷,𝐾 (𝑏, 𝑛, 𝑝 1 ; 𝐺) be the statement on the left hand side of the implication (3.6.1):   −(log log 𝑛) 1/2 1/3 −1 A (𝑏, 𝑛, 𝑝 1 ) = 𝑛 ∈ L (𝐺, 𝐷), 𝜅 𝑝1 (𝑛, ∞) ≥ 𝑒 , P 𝑝1 (Piv[4𝑏, 𝑛 ]) ≤ (log 𝑛) and 𝛿(𝑏, 𝑛) ≤ 1 . We will always assume that 𝛿( 𝑝 1 , 𝑝 2 ) is larger than the quantity 𝛿 := 𝛿(𝑏, 𝑛) = 𝛿 𝐾 (𝑏, 𝑛) introduced in Proposition 3.6.1. We regard the choices of 𝑑 ∈ N, 𝐷 ≥ 1, 𝐾 ≥ 1, and 𝜆 ≥ 20, as well as the graph 𝐺 ∈ U𝑑∗ , as being fixed for the remainder of the section. We write 𝑝 3/2 := Spr( 𝑝 1 ; 𝛿( 𝑝 1 , 𝑝 2 )/2), so that 𝑝 2 = Spr( 𝑝 3/2 ; 𝛿( 𝑝 1 , 𝑝 2 )/2) by the semigroup property of our sprinkling operation, and let 𝑐 1 = 𝑐 1 (𝑑, 𝐷) > 0 and 𝑁 = 𝑁 (𝑑, 𝐷, 𝜆) be the constants guaranteed to exist by Proposition 3.5.2. 102 Inspired by [CMT22], we will split the proof of Proposition 3.6.1 into two cases according to how easy it is to connect points at certain well-chosen intermediate scales. For each 1 ≤ 𝑚 ≤ 𝑛 we define the two-point zone n o 𝑚 tz(𝑚) = tz(𝑚, 𝑛) := sup 𝑟 ≥ 0 : 𝜏𝑝𝐵3/2 (𝐵𝑟 ) ≥ (log 𝑛) −1 , 𝐵 where we recall that 𝜏𝑝𝐵 ( 𝐴) is the quantity defined by 𝜏𝑝𝐵 ( 𝐴) := inf 𝑥,𝑦∈𝐴 P 𝑝 (𝑥 ← → 𝑦). We stress that the two-point zone tz(𝑚) is defined using the intermediate parameter 𝑝 3/2 . Knowing the value of tz(𝑚) for some 𝑚 provides us with both positive information (on scales smaller than tz(𝑚)) and negative information (on scales larger than tz(𝑚)). For each 𝑛 ∈ L (𝐺, 𝐷) with 𝑛 ≥ 𝑁, let 𝑛1/3 ≤ 𝑚 1 ≤ 𝑚 2 ≤ 𝑛1/(1+𝑐1 ) be such that 𝑚 2 ≥ 𝑚 11+𝑐1 and 𝐺 has (𝑐 1 , 𝜆)-polylog-plentiful tubes at every scale 𝑚 1 ≤ 𝑚 ≤ 𝑚 2 (such 𝑚 1 and 𝑚 2 existing by Proposition 3.5.2), and let S (𝑛) = S𝑑,𝐷,𝜆 (𝑛) = {𝑚 ∈ N : 𝑚 11+(𝑐1 /4) ≤ 𝑚 ≤ 𝑚 11+(3𝑐1 /4) }. From now on we will mostly work at scales 𝑚 ∈ S (𝑛), so that 𝐺 has plentiful tubes not just at these scales but at a large range of consecutive scales on either side of every such scale. Let 𝑐 2 = 𝑐 2 (𝑑) be the minimum of the four constants appearing in Proposition 3.4.1 with ‘𝐷’ equal to 1 (known in the statement of that proposition as 𝑐 1 , 𝑐 2 , 𝑐 3 , and ℎ0 ; the proposition becomes weaker if we replace all four constants by their minimum), define 𝑐 3 = 𝑐 3 (𝑑) := 2−9 𝑐 2 , and consider the statement B (𝑛, 𝑝 3/2 ) = B𝑑,𝐷,𝜆,𝐾 (𝑛, 𝑝 3/2 )   𝑚 [log 𝑛] 𝑐3 𝐾 ≥ 𝑚(log 𝑛) 3𝜆/𝑐1 . = there exists 𝑚 ∈ S (𝑛) such that tz 2 We think of B (𝑛, 𝑝 3/2 ) as our “positive assumption” about percolation on scale 𝑛: it means that there is a “good” scale 𝑚 ∈ S (𝑛) such that points in a ball of radius not much smaller than 𝑚 can be connected with reasonable probability within the ball of radius 𝑚/2 when 𝑝 = 𝑝 3/2 . Notational conventions and standing assumptions: Recall that the choices of 𝑑 ∈ N, 𝐷 ≥ 1, 𝐾 ≥ 1, 𝜆 ≥ 20, and 𝐺 ∈ U𝑑∗ are considered to be fixed for the remainder of the section. The constants 𝑁, 𝑐 1 , 𝑐 2 , and 𝑐 3 used to define S (𝑛) and B (𝑛, 𝑝) will be used with the same meaning throughout this section. We also define 𝑐 −1 = 𝑐 −1 (𝑑) to be a positive constant such that Spr( 𝑝; 𝛿) ≥ 𝑝 + 𝑐 −1 𝛿 for every 𝑝 ≥ 1/𝑑 and 0 < 𝛿 ≤ 1, which exists by (3.3.1). Finally, we −5 fix 𝐾0 = 𝐾0 (𝑑, 𝐷) = 211 (𝑐−1 3 ∨ 𝑐 −1 ∨ 1) throughout the section: all subsequent lemmas in this section will include 𝐾 ≥ 𝐾0 as an implicit hypothesis. These conventions do not apply outside of this section (Section 3.6). Proposition 3.6.1 follows trivially from the following two lemmas. 103 Lemma 3.6.2 (Concluding from a positive assumption). There exists a constant 𝜆 0 = 𝜆 0 (𝑑, 𝐷) such that if 𝜆 ≥ 𝜆 0 then there exists a constant 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆) ≥ 𝑁 such that the implication     𝑐1 𝜆/4 1/2 A 𝐷,𝐾 (𝑏, 𝑛, 𝑝 1 ) ∧ B𝑑,𝐷,𝜆,𝐾 (𝑛, 𝑝 3/2 ) ⇒ 𝑝 2 ≥ 𝑝 𝑐 or 𝜅 𝑝2 𝑒 (log 𝑛) , 𝑛 ≥ 𝑒 −3(log log 𝑛) (3.6.2) holds for every 𝑛 ≥ 𝑛0 , 𝑏 ≤ 81 𝑛1/3 , and every pair of probabilities 𝑝 2 ≥ 𝑝 1 ≥ 1/𝑑 with 𝛿( 𝑝 1 , 𝑝 2 ) ≥ 𝛿 𝐾 (𝑏, 𝑛). Lemma 3.6.3 (Concluding from a negative assumption). There exists a constant 𝜆 0 = 𝜆 0 (𝑑, 𝐷) such that if 𝜆 ≥ 𝜆 0 then there exist constants 𝐾1 = 𝐾1 (𝑑, 𝐷, 𝜆) ≥ 𝐾0 and 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝜆) ≥ 𝑁 such that if 𝐾 ≥ 𝐾1 and 𝑛 ≥ 𝑛0 then the implication     𝑐1 𝜆/4 1/2 A 𝐷,𝐾 (𝑏, 𝑛, 𝑝 1 ) ∧ (¬ B𝑑,𝐷,𝜆,𝐾 (𝑛, 𝑝 3/2 )) ⇒ 𝑝 2 ≥ 𝑝 𝑐 or 𝜅 𝑝2 𝑒 (log 𝑛) , 𝑛 ≥ 𝑒 −3(log log 𝑛) (3.6.3) 1 1/3 holds for every 𝑛 ≥ 𝑛0 , 𝑏 ≤ 8 𝑛 , and every pair of probabilities 𝑝 2 ≥ 𝑝 1 ≥ 1/𝑑 with 𝛿( 𝑝 1 , 𝑝 2 ) ≥ 𝛿 𝐾 (𝑏, 𝑛). Concluding from a positive assumption Our next goal is to prove Lemma 3.6.2. We begin with the following lemma, which will later allow us to use the positive assumption B (𝑛, 𝑝) to deduce lower bounds on the corridor function after sprinkling. Note that we are not yet using the polylog-plentiful tubes condition or the assumption B (𝑛, 𝑝), so that the parameter 𝜆 does not appear in this lemma. (Recall our standing assumption throughout this section that 𝐾 ≥ 𝐾0 = 𝐾0 (𝑑, 𝐷).) We define 𝑝 7/4 = Spr( 𝑝 1 ; 43 𝛿( 𝑝 1 , 𝑝 2 )) = Spr( 𝑝 3/2 ; 41 𝛿( 𝑝 1 , 𝑝 2 )), and remind the reader that the two-point zone tz(𝑚) was defined with respect to the parameter 𝑝 3/2 . Lemma 3.6.4. There exist a constant 𝑛0 = 𝑛0 (𝑑, 𝐷) ≥ 𝑁 and a universal constant 𝑐 4 such that if 𝑛 ≥ 𝑛0 then the implication   𝑚   1/2 A 𝐷,𝐾 (𝑏, 𝑛, 𝑝 1 ) ⇒ 𝜅 𝑝7/4 tz [log 𝑛] 𝑐3 𝐾 , 𝑚 ≥ 𝑐 4 𝑒 −2(log log 𝑛) for every 𝑛1/3 ≤ 𝑚 ≤ 𝑛 2 holds for every 𝑛 ≥ 𝑛0 , 𝑏 ≤ 81 𝑛1/3 , and 𝑝 1 ≥ 1/𝑑. Proof of Lemma 3.6.4. Fix 𝑛1/3 ≤ 𝑚 ≤ 𝑛 and a path 𝛾 starting at some vertex 𝑢, ending at some
vertex 𝑣, and with len 𝛾 ≤ tz 𝑚2 (log 𝑛) 𝑐3 𝐾 . Pick a subsequence 𝑢 = 𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 = 𝑣 of 𝛾 with
𝑘 ≤ 5(log 𝑛) 𝑐3 𝐾 and dist(𝑢𝑖 , 𝑢𝑖+1 ) ≤ 41 tz 𝑚2 for every 1 ≤ 𝑖 < 𝑘. (There are rounding issues that
may prevent such a sequence existing when tz 𝑚2 is small, but this is not a problem when 𝑛 is 104 
large.) We claim that 14 tz 𝑚2 ≥ 𝑏. Indeed, the hypothesis 𝑏 ≤ 18 𝑛1/3 guarantees that 4𝑏 ≤ 𝑚/2, and we have by a union bound that 𝐵 𝐵 𝑚/2 (𝐵4𝑏 ) ≥ 𝜏𝑝1𝑚/2 (𝐵4𝑏 ) ≥ 𝜅 𝑝1 (𝑛, ∞) − P 𝑝1 (Piv[4𝑏, 𝑛1/3 ]) 𝜏𝑝3/2 ≥ 𝑒 −(log log 𝑛) 1/2 − [log 𝑛] −1 ≥ 1 −[log log 𝑛] 1/2 𝑒 (3.6.4) 2 𝐵 𝑚/2 (𝐵4𝑏 ) ≥ (log 𝑛) −1 for every 𝑛 larger than some universal constant. It follows in particular that 𝜏𝑝3/2
for every 𝑛 larger than some universal constant 𝑛0 , and hence that 41 tz 𝑚2 ≥ 𝑏 when 𝑛 ≥ 𝑛0 by
maximality of tz 𝑚2 . Let ℎ = 𝑐 2 max{Gr(𝑏) −1 , 𝑛−1/15 }, so that if 𝑛0 ≥ 3 (to guarantee that 𝑛1/15 ≥ log 𝑛) then   𝑐 2 𝐾 log log 𝑛 𝑐 2 (𝛿/4) 3 𝑐 2 (𝛿/4) 4 = (log 𝑛) −2𝑐3 𝐾 ≤ (log 𝑛) −2 ≤ℎ ≤ exp log ℎ · 8 · ℎ log ℎ 2 (3.6.5) by the definition of 𝛿 𝐾 (𝑏, 𝑛) and the assumption that 𝐾 ≥ 𝐾0 . We want to apply Proposition 3.4.1 𝑘 , the superset ‘Λ’ is the tube 𝐵 where ‘𝑛’ is 𝑘, the sets ‘𝐴𝑖 ’ are the balls (𝐵 𝑏 (𝑢𝑖 ))𝑖=1 𝑚/2 (𝛾), the ghost field intensity ‘ℎ’ is equal to ℎ, ‘𝑝 1 ’ is 𝑝 3/2 , and the sprinkling amount ‘𝛿’ is 𝛿/4. To do this, it suffices to verify that if 𝑛 is sufficiently large then ℎ𝑐2 (𝛿/4) ≤ 𝑐 2 𝑘 −1 3 𝐵 and 𝑚/2 𝜏𝑝3/2 (𝛾) (𝐵 𝑏 (𝑢𝑖 )∪𝐵 𝑏 (𝑢𝑖+1 )) ≥ 4ℎ𝑐2 (𝛿/4) 4 for every 1 ≤ 𝑖 ≤ 𝑘 − 1. (The assumption that ℎ is sufficiently small holds automatically since we defined 𝑐 2 to be the minimum of the constants appearing in Proposition 3.4.1 and set ℎ ≤ 𝑐 2 .) The inequality (3.6.5) 3 implies that the first required inequality ℎ𝑐2 (𝛿/4) ≤ 𝑐 2 𝑘 −1 holds whenever 𝑛 is larger than some 3 constant depending only on 𝑑, since ℎ𝑐2 (𝛿/4) ≤ (log 𝑛) −2𝑐3 𝐾 , 𝑘 ≤ 5(log 𝑛) 𝑐3 𝐾 and 𝑐 3 𝐾 ≥ 2. For the second required inequality, note that for every such 𝑖 and for every 𝑢 ∈ 𝐵 𝑏 (𝑢𝑖 ) and 𝑣 ∈ 𝐵 𝑏 (𝑢𝑖+1 ), we have 𝑢, 𝑣 ∈ 𝐵tz(𝑚/2)−1 (𝑢𝑖 ) and hence that 𝐵 𝑚/2 𝜏𝑝3/2 (𝛾) 𝑚/2 (𝐵 𝑏 (𝑢𝑖 ) ∪ 𝐵 𝑏 (𝑢𝑖+1 )) ≥ 𝜏𝑝3/2 (𝐵tz(𝑚/2)−1 ) ≥ (log 𝑛) −1 𝐵 for all 𝑛 larger than some universal constant by (3.6.4). Let 𝑛0 = 𝑛0 (𝑑, 𝐷) ≥ 𝑁 be the maximum of these two constants and 𝑁. Let 𝑟 be the minimum positive integer such that P𝑞 (Piv[1, 𝑟 ℎ]) < ℎ for every 𝑞 ∈ [ 𝑝 3/2 , Spr( 𝑝 3/2 ; 𝛿/4)]. The previous paragraph shows that if 𝑛 ≥ 𝑛0 then the hypotheses of Proposition 3.4.1 are met. Thus, since 𝑝 7/4 ≥ Spr( 𝑝 3/2 ; 𝛿/4)], we obtain from that proposition that for a universal constant 𝑐 > 0,   𝐵 𝑚/2+2𝑟 (𝛾) 1/2 𝑐 𝐵𝑚/2 (𝛾) 𝐵𝑚/2 (𝛾) P 𝑝7/4 𝑢 ←−−−−−−→ 𝑣 ≥ 𝑐𝜏𝑝3/2 (𝐵 𝑏 (𝑢)) · 𝜏𝑝3/2 (𝐵 𝑏 (𝑣)) ≥ 𝑒 −2(log log 𝑛) . 4 105 Since 𝛾 was arbitrary, this implies that if 𝑛 ≥ 𝑛0 then  𝑚  𝑐 1/2 𝑚 𝜅 𝑝7/4 tz [log 𝑛] 𝑐3 𝐾 , + 2𝑟 ≥ 𝑒 −2(log log 𝑛) . 2 2 4 All that remains is to verify that 2𝑟 ≤ 𝑚/2 when 𝑛 is sufficiently large. Let 𝑞 ∈ [ 𝑝 3/2 , Spr( 𝑝 3/2 ; 𝛿/4)] be arbitrary. Since 2/5 < 1/2, Proposition 3.2.1 yields that there exists a constant 𝐶𝑑 such that   2/5  2/5  log Gr((𝑚/4)ℎ) 4 log Gr(𝑚) P𝑞 (Piv[1, (𝑚/4)ℎ]) ≤ 𝐶𝑑 ≤ 𝐶𝑑 . (𝑚/4)ℎ 𝑚ℎ Since ℎ ≥ 𝑛−1/15 , 𝑛1/3 ≤ 𝑚 ≤ 𝑛, log Gr(𝑚) ≤ (log 𝑚) 𝐷 , and (2/5) · (4/15) > 1/15, it follows that there exists a constant 𝑛2 = 𝑛2 (𝑑, 𝐷) such that if 𝑛 ≥ 𝑛2 then   2/5 4(log 𝑛) 𝐷 ≤ 𝑛−1/15 ≤ ℎ, P𝑞 (Piv[1, (𝑚/4)ℎ]) ≤ 𝐶𝑑 1/3 −1/15 𝑛 ·𝑛 and hence that 𝑟 ≤ 𝑚/4 as claimed. This completes the proof. □ We now apply Lemma 3.6.4 together with the polylog-plentiful tubes condition to prove Lemma 3.6.2. Proof of Lemma 3.6.2. Let 𝑛0 = 𝑛0 (𝑑, 𝐷) and the universal constant 𝑐 4 be as in Lemma 3.6.4. Suppose that 𝑛 ≥ 𝑛0 , 𝑏 ≤ 81 𝑛1/3 , and 𝑝 1 ∈ [1/𝑑, 1] are such that A 𝐷,𝐾 (𝑏, 𝑛, 𝑝 1 ) holds, and write 𝛿 = 𝛿(𝑏, 𝑛). We have by Lemma 3.6.4 that   𝑚 1/2 𝜅 𝑝7/4 tz [log 𝑛] 𝑐3 𝐾 , 𝑚 ≥ 𝑐 4 𝑒 −2(log log 𝑛) for every 𝑛1/3 ≤ 𝑚 ≤ 𝑛 2 and in particular that if B𝑑,𝐷,𝜆,𝐾 (𝑛, 𝑝) holds then there exists 𝑚 ∈ S (𝑛) = S𝑑,𝐷,𝜆 (𝑛) such that
𝜅 𝑝7/4 𝑚(log 𝑛) 3𝜆/𝑐1 , 𝑚 ≥ (log 𝑛) −1 (3.6.6) whenever 𝑛0 is larger than a suitable universal constant. (Note that we have not yet put any restrictions on the parameter 𝜆.) Suppose that 𝑚 ∈ S (𝑛) satisfies (3.6.6) and let 𝑟 = 𝑚 [log 𝑛] 3𝜆/(2𝑐1 ) , so that if 𝑛 ≥ 2 we have the inclusion of intervals h9 i h i 𝑟 (log 𝑟) −𝜆/𝑐1 , 𝑟 (log 𝑟) 𝜆/𝑐1 ⊆ 𝑚, 𝑚(log 𝑛) 3𝜆/𝑐1 . (3.6.7) 10 9 Since 10 𝑟 ∈ [𝑚 1 , 𝑚 2 ] (where 𝑚 1 and 𝑚 2 are as in the definition of S (𝑛) above), 𝐺 has (𝑐 1 , 𝜆)9 polylog-plentiful radial tubes at scale 10 𝑟. Let Γ be a family of paths witnessing this fact (which 106 cross the annulus from 𝑆0.9𝑟 to 𝑆4·(0.9𝑟) = 𝑆3.6𝑟 ) and let (𝑇𝛾 ) 𝛾∈Γ be the associated family of tubes 9 9 𝑟 [log( 10 𝑟)] −𝜆/𝑐1 ). Since each tube 𝑇𝛾 has thickness at least 𝑚 and length given by 𝑇𝛾 := 𝐵(𝛾, 10 len 𝛾 ≤ 𝑟 (log 𝑟) 𝜆/𝑐1 ≤ 𝑚(log 𝑛) 3𝜆/𝑐1 , it follows from (3.6.6) that 𝑇𝛾 P 𝑝7/4 (𝑆0.9𝑟 ←→ 𝑆3.6𝑟 ) ≥ (log 𝑛) −1 for every 𝛾 ∈ Γ and hence that there exists a constant 𝑛1 = 𝑛1 (𝑑, 𝐷, 𝜆) ≥ 𝑛0 such that if 𝑐 1𝜆 > 4 and 𝑛 ≥ 𝑛1 then Ö 𝑇𝑖 1/3 𝑐1 𝜆 P 𝑝7/4 (𝑆0.9𝑟 ↔ 𝑆3.6𝑟 ) ≥ 1 − P 𝑝7/4 (𝑆0.9𝑟 ← → 𝑆3.6𝑟 ) ≥ 1 − (1 − (log 𝑛) −1 ) (log(𝑛 )) 𝛾∈Γ
≥ 1 − exp 3−𝑐1 𝜆 (log 𝑛) 𝑐1 𝜆−1 ≥ 1 − exp(−(log 𝑛) 3𝑐1 𝜆/4 ), (3.6.8) where we used that 1 − 𝑥 ≤ 𝑒 −𝑥 in the second line. We next use the fact that 𝐺 also has (𝑐 1 , 𝜆)-polylog-plentiful annular tubes at scale 𝑟. (We will no longer use the paths and tubes from the radial case that we defined in the previous paragraph, so it is not a problem to reuse the same notation for the annular tubes we consider in the rest of the proof.) We will work with the standard monotone coupling (𝜔 𝑝 ) 𝑝∈[0,1] of percolation at different parameters. To lighten notation, we write 𝜔1 = 𝜔 𝑝1 , 𝜔3/2 = 𝜔 𝑝3/2 , 𝜔7/4 = 𝜔 𝑝7/4 , and 𝜔2 = 𝜔 𝑝2 . Let 𝑢, 𝑣 ∈ 𝑆𝑟 and consider the event 𝐴𝑢,𝑣 = {𝑢 ↔ 𝑆3𝑟 and 𝑣 ↔ 𝑆3𝑟 in the configuration 𝜔7/4 }. Define 𝐶𝑢 := 𝐾𝑢 (𝜔7/4 ) and 𝐶𝑣 := 𝐾𝑣 (𝜔7/4 ), so that the event 𝐴𝑢,𝑣 is entirely determined by the pair (𝐶𝑢 , 𝐶𝑣 ). Whenever 𝐴𝑢,𝑣 holds, let Γ be a family of paths from 𝐶𝑢 to 𝐶𝑣 that is guaranteed to exist by the fact that 𝐺 has (𝑐 1 , 𝜆)-polylog-plentiful annular tubes at scale 𝑟 (choosing these paths as a function of (𝐶𝑢 , 𝐶𝑣 )), and let (𝑇𝛾 ) 𝛾∈Γ be the associated family of tubes, noting that Ð 𝜆/𝑐 1 ). Define the configurations 𝛾∈Γ 𝑇𝛾 ⊆ 𝐵(2𝑟 [log 𝑟] 𝛼 := 𝜔2 ∩ (𝜕𝐸 𝐶𝑢 ∪ 𝜕𝐸 𝐶𝑣 ) ∩ 𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 ) and 𝛽 := (𝜔7/4 \𝐶𝑢 ∪ 𝐶𝑣 ) ∩ 𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 ), where we recall that 𝐶𝑢 ∪ 𝐶𝑣 denotes the set of edges with at least one endpoint in 𝐶𝑢 ∪ 𝐶𝑣 . In order for 𝐶𝑢 to be connected to 𝐶𝑣 in the configuration (𝛼 ∪ 𝛽) ∩ 𝑇𝛾 , it suffices that in at least one of the tubes 𝑇𝛾 , there is a 𝛽-path from an endpoints of an 𝛼-open edge in 𝜕𝐸 𝐶𝑢 to an endpoint of an 𝛼-open edge in 𝜕𝐸 𝐶𝑣 . The estimate (3.6.6) together with the interval inclusion (3.6.7) therefore 107 yields that " 𝛼∪𝛽 P(𝐶𝑢 ←−−→ 𝐶𝑣 | 𝐶𝑢 , 𝐶𝑣 ) ≥ 1( 𝐴𝑢,𝑣 ) · 1 − Ö # 𝑇𝑖 ∩(𝛼∪𝛽) P(𝐶𝑢 ←−−−−−→ 𝐶𝑣 | 𝐶𝑢 , 𝐶𝑣 ) 𝛾∈Γ ! [log(𝑛1/3 )] 𝑐1 𝜆    2   𝑐 𝛿 −1  ≥ 1( 𝐴𝑢,𝑣 ) · 1 − 1 − [log 𝑛] −1  4   h   i −𝑐 1 𝜆 𝑐 1 𝜆−2 ≥ 1( 𝐴𝑢,𝑣 ) · 1 − exp −3 (log 𝑛) , where we used that 𝑐 −1 𝛿 ≥ 4(log 𝑛) −1/2 (which holds by definition of 𝛿 and 𝐾0 ) in the final inequality. As such, there exist constants 𝜆 0 = 𝜆 0 (𝑑, 𝐷) and 𝑛2 = 𝑛2 (𝑑, 𝐷, 𝜆) ≥ 𝑛1 such that if 𝜆 ≥ 𝜆 0 , 𝑛 ≥ 𝑛2 , and 𝑐 1𝜆 ≥ 4 then h i 𝛼∪𝛽 −[log 𝑛] 3𝑐1 𝜆/4 P(𝐶𝑢 ←−−→ 𝐶𝑣 | 𝐶𝑢 , 𝐶𝑣 ) ≥ 1( 𝐴𝑢,𝑣 ) · 1 − 𝑒 . Since 𝛼 ∪ 𝛽 ⊆ 𝜔2 ∩ 𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 ), it follows that 𝜔2 ∩𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 ) P(𝐶𝑢 ←−−−−−−−−−−−−−−→ 𝐶𝑣 | 𝐴𝑢,𝑣 ) ≥ 1 − 𝑒 −[log 𝑛] 3𝑐1 𝜆/4 under the same conditions. Letting U be the event that all 𝜔7/4 -clusters that intersect both 𝑆𝑟 and 𝑆3𝑟 are contained in a single 𝜔2 ∩ 𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 )-cluster, it follows by a union bound that there exists 𝜆 1 = 𝜆 1 (𝑑, 𝐷) ≥ 𝜆0 such that if 𝜆 ≥ 𝜆 1 and 𝑛 ≥ 𝑛2 then P(U) ≥ 1 − ∑︁   𝜔2 ∩𝐵(2𝑟 [log 𝑟] 𝜆/𝑐1 ) P 𝐴𝑢,𝑣 ∩ {𝐶𝑢 ←−−−−−−−−−−−−−→ 𝐶𝑣 } 𝑢,𝑣∈𝑆𝑟 ≥ 1 − Gr(𝑟) 2 𝑒 −[log 𝑛] 3𝑐1 𝜆/4 ≥ 1 − 𝑒 [log 𝑛] 𝑒 −[log 𝑛] 2𝐷 3𝑐1 𝜆/4 ≥ 1 − 𝑒 −[log 𝑛] 2𝑐1 𝜆/3 . (3.6.9) For each vertex 𝑥, let E𝑥 be the event that 𝑆0.9𝑟 (𝑥) is 𝜔7/4 -connected to 𝑆3.6𝑟 (𝑥) and let U𝑥 be the event that all 𝜔7/4 -clusters that intersect both 𝑆𝑟 (𝑥) and 𝑆3𝑟 (𝑥) are contained in a single 𝜔2 ∩ 𝐵(𝑥, 2𝑟 [log 𝑟] 𝜆/𝑐1 )-cluster. Observe that if 𝜁 is a path in 𝐺 then we have the inclusion of events len Ù𝜁 𝜔7/4 𝜔7/4 𝜔2 ∩𝐵 𝑛 (𝜁) {𝑢 ←−−−−−→ 𝑣} ⊇ {𝑢 ←−→ 𝑆𝑛 (𝑢)} ∩ {𝑣 ←−→ 𝑆𝑛 (𝑣)} ∩ (U𝜁𝑡 ∩ E 𝜁𝑡 ). 𝑡=0 Thus, it follows by the Harris-FKG inequality and a union bound that there exists a constant 108 𝑛3 = 𝑛3 (𝑑, 𝐷, 𝜆) ≥ 𝑛2 such that if 𝜆 ≥ 𝜆 1 , 𝑛 ≥ 𝑛3 , and the path 𝜁 satisfies len 𝜁 ≤ 𝑒 [log 𝑛] 1 𝑐 𝜆/4 𝐵 𝑛 (𝜁) P 𝑝2 (𝑢 ←−−→ 𝑣) ≥ P 𝑝7/4 (𝑢 ↔ 𝑆𝑛 (𝑢)) · P 𝑝7/4 (𝑣 ↔ 𝑆𝑛 (𝑣)) · len Ö𝜁 𝑡=1  ≥ 𝑒 −[log log 𝑛]  h 1/2 2 1 − len(𝜁)𝑒 −[log 𝑛] i 3𝑐 𝜆/4 1 P(E 𝜁𝑡 ) − len ∑︁𝜁 then P(U𝜁𝑐𝑡 ) 𝑡=1 − len(𝜁)𝑒 −[log 𝑛] 3𝑐1 𝜆/4 ≥ 𝑒 −3[log log 𝑛] . 1/2 The claimed lower bound on the corridor function follows since 𝜁 was an arbitrary path of length 𝑐 𝜆/4 at most 𝑒 [log 𝑛] 1 . □ Concluding from a negative assumption The next lemma explains how to use the negative information encapsulated in an upper bound on the two-point zone tz(𝑚) to find a set of vertices for which point-to-point connection probabilities within a ball are uniformly small. This lemma plays an analogous role to that of Section 7.2 in [CMT22], but our proof is completely different and relies on the machinery developed in Section 3.4. Note that this lemma does not use the plentiful tubes condition. Lemma 3.6.5. There exists a constant 𝑛0 = 𝑛0 (𝑑, 𝐷) such that if 𝑛 ≥ 𝑛0 and 𝑏 ≤ 18 𝑛1/3 satisfy A 𝐷,𝐾 (𝑏, 𝑛) then for every 𝑛1/3 ≤ 𝑚 ≤ 𝑛, there exists a subset 𝑈 ⊆ 𝐵tz(𝑚) with |𝑈| ≥ (log 𝑛) 𝑐3 𝐾 such that 𝐵 𝑚/2 P 𝑝1 (𝑢 ←−→ 𝑣) ≤ (log 𝑛) −𝑐3 𝐾 for all distinct 𝑢, 𝑣 ∈ 𝑈. (Reminder: The implicit assumption 𝐾 ≥ 𝐾0 remains in force.) Proof of Lemma 3.6.5. Fix 𝑛 and 𝑏 ≤ 81 𝑛1/3 such that A 𝐷,𝐾 (𝑏, 𝑛) holds. We will assume that the claim is false for some particular 𝑛1/3 ≤ 𝑚 ≤ 𝑛, and show that this implies a contradiction when 𝑛 is sufficiently large. By definition of the two-point zone tz(𝑚) = tz(𝑚, 𝑛), there exist vertices 𝐵𝑚 𝑢, 𝑣 ∈ 𝐵tz(𝑚) such that P 𝑝3/2 (𝑢 ←−→ 𝑣) < (log 𝑛) −1 . Let 𝛾 be a geodesic in 𝐵tz(𝑚) from 𝑢 to 𝑣. Recursively pick a sequence of indices 0 = 𝑖 0 < 𝑖1 < . . . < 𝑖 𝑘 = len 𝛾 starting with 𝑖0 := 0 and for each 𝑗 ≥ 0 with 𝑖 𝑗 < len 𝛾 setting n n oo 𝐵𝑚/2 𝑖 𝑗+1 = max len 𝛾, 1 + max 𝑖 ≥ 𝑖 𝑗 : P 𝑝1 (𝑢𝑖 𝑗 ←−→ 𝑢𝑖 ) ≥ (log 𝑛) −𝑐3 𝐾 . To lighten notation, define 𝑣 𝑗 := 𝑢𝑖 𝑗 for every 0 ≤ 𝑗 ≤ 𝑘. This sequence has the property that P 𝑝1 (𝑣 𝑗 ↔ 𝑣 𝑗+1 ) ≥ 𝑝 1 (log 𝑛) −𝑐3 𝐾 for every 0 ≤ 𝑗 < 𝑘 and P 𝑝1 (𝑣 𝑗 ↔ 𝑣 ℓ ) ≤ (log 𝑛) −𝑐3 𝐾 for every 109 distinct 0 ≤ 𝑗, ℓ < 𝑘. (The last connection probability, from 𝑣 𝑘−1 to 𝑣 𝑘 = 𝑣, may be larger.) As such, our assumption guarantees that 𝑘 ≤ (log 𝑛) 𝑐3 𝐾 . To conclude the proof, it suffices to show that P 𝑝3/2 (𝑢 ↔ 𝑣) ≥ (log 𝑛) −1 when 𝑛 is sufficiently large. To this end, we would like to apply Proposition 3.4.1 where the sets ‘𝐴𝑖 ’ are the balls 𝐵 𝑏 (𝑣 𝑗 ), the superset ‘Λ’ is the bigger ball 𝐵𝑚/2 , the ghost field intensity ‘ℎ’ is ℎ := min{log 𝑛, Gr(𝑏)}−1 , and the sprinkling amount ‘𝛿’ is 𝛿/2. To do this, we need to verify that ℎ𝑐2 (𝛿/2) ≤ 𝑐 2 𝑘 −1 3 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑣 𝑗 ) ∪ 𝐵 𝑏 (𝑣 𝑗+1 )) ≥ 4ℎ−𝑐2 (𝛿/2) 𝐵 and 4 (3.6.10) for every 0 ≤ 𝑗 ≤ 𝑘 − 1 when 𝑛 is sufficiently large. We have by definition of 𝛿 (and the assumption 𝐾 ≥ 𝐾0 ) that if 𝑛 is larger than some universal constant then   3 4 ℎ𝑐2 (𝛿/2) ≤ 4ℎ𝑐2 (𝛿/2) ≤ 4(log 𝑛) −2𝑐3 𝐾 ≤ 𝑐 2 (log 𝑛) −𝑐3 𝐾 + 1 . (3.6.11) Since 𝑘 ≤ (log 𝑛) 𝑐3 𝐾 , this is easily seen to imply that the first inequality of (3.6.10) holds. Moreover, we have by the same calculation performed in (3.6.4) that 𝐵 𝜏𝑝1𝑚/2 (𝐵 𝑏 ) ≥ 1 −(log log 𝑛) 1/2 𝑒 ≥ (log 𝑛) −1 2 (3.6.12) for all 𝑛 larger than some universal constant, and it follows by the Harris-FKG inequality that if 𝑛 is larger than some constant depending only on 𝑑 then 𝐵 𝐵 𝑚/2 𝐵 𝐵 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑣 𝑗 ) ∪ 𝐵 𝑏 (𝑣 𝑗+1 )) ≥ 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑣 𝑗 )) · P 𝑝1 (𝑣 𝑗 ←−→ 𝑣 𝑗+1 ) · 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑣 𝑗+1 )) i h 1 ≥ (log 𝑛) −1 · (log 𝑛) −𝑐3 𝐾 · (log 𝑛) −1 ≥ (log 𝑛) −𝑐3 𝐾−3 2𝑑 (3.6.13) for every 0 ≤ 𝑗 ≤ 𝑘 − 1. The estimates (3.6.11) and (3.6.13) together yield that there exists a constant 𝑛0 = 𝑛0 (𝑑) ≥ 𝑁 such that the required estimates (3.6.10) hold whenever 𝑛 ≥ 𝑛0 . Thus, Proposition 3.4.1 and (3.6.12) yield that there exists a constant 𝑛1 = 𝑛1 (𝑑) ≥ 𝑛0 such that if 𝐾 ≥ 𝐾0 and 𝑛 ≥ 𝑛1 and we define 𝑟 to be the minimum positive integer such that P𝑞 (Piv[1, 𝑟 ℎ]) < ℎ for every 𝑞 ∈ [ 𝑝 1 , 𝑝 3/2 ] then (since 𝑝 3/2 ≥ Spr( 𝑝 1 ; 𝛿/2))   𝐵𝑚/2+2𝑟 1/2 𝑐2 𝐵 𝐵 P 𝑝3/2 𝑢 ←−−−−→ 𝑣 ≥ 𝑐 2 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑢)) · 𝜏𝑝1𝑚/2 (𝐵 𝑏 (𝑣)) ≥ 𝑒 −2[log log 𝑛] . 4 Finally, the same argument as in the proof of Lemma 3.6.4 yields that 2𝑟 ≤ 𝑚/2 when 𝑛 is larger than some constant depending on 𝑑 and 𝐷, and it follows that there exists a constant 𝑛2 = 𝑛2 (𝑑, 𝐷) ≥ 𝑛1 𝐵𝑚 such that if 𝑛 ≥ 𝑛2 then P 𝑝3/2 (𝑢 ←→ 𝑣) ≥ (log 𝑛) −1 — a contradiction. □ 110 We now use the existence of this set of poorly connected vertices (negative information) to prove that 𝑆tz(𝑚) is very likely to be connected to the boundary of 𝑆 𝑚/2 (positive information). This only works because we are working under the positive hypothesis of a two-point lower bound at scale 𝑛. This step is essentially the same as Section 7 in [CMT22], with our two-point lower bound at scale 𝑛 playing the role of ‘being in the supercritical regime’ in their setting. Lemma 3.6.6. There exists a constant 𝑛0 = 𝑛0 (𝑑, 𝐷) such that if 𝑛 ≥ 𝑛0 and 𝑏 ≤ 18 𝑛1/3 satisfy A 𝐷,𝐾 (𝑏, 𝑛) then 𝑐3 𝐾 −1 P 𝑝3/2 (𝑆tz(𝑚) ↔ 𝑆 𝑚/2 ) ≥ 1 − 𝑒 −(log 𝑛) for every 𝑚 ∈ S (𝑛). Note that we are not actually assuming a negative information assumption (such as ¬B (𝑛, 𝑝)) in the hypotheses of this lemma. The lemma holds without any such assumption, but is stronger when tz(𝑚) is small. The proof of Lemma 3.6.6 will apply the following proposition. Proposition 3.6.7. Let 𝐺 be a finite connected graph, let 𝑝 ∈ [0, 1], and let 𝐴, 𝐵 ⊆ 𝑉 (𝐺). If 𝜃 ∈ (0, 1) is such that min P 𝑝 (𝑥 ↔ 𝐵) ≥ 𝜃 ≥ 2 | 𝐴| max P 𝑝 (𝑥 ↔ 𝑦), 𝑥∈𝐴 𝑥,𝑦∈𝐴 𝑥≠𝑦 then P𝑞 ( 𝐴 ↔ 𝐵) ≥ 1 − 𝑒 −𝛿( 𝑝,𝑞)𝜃| 𝐴| for every 𝑞 ∈ ( 𝑝, 1). Proof of Proposition 3.6.7. This proposition is essentially the same as [CMT22, Proposition 7.2], except that it is stated in terms of our sprinkling coordinates introduced in Section 3.3 (which are natural from the perspective of Talagrand’s inequality) and we get a factor 𝛿 rather than 2𝛿 in the exponential in the conclusion. Both versions of the proposition are elementary consequences of the differential inequality   1 𝑑 (− log P 𝑝 ( 𝐴)) ≥ E 𝑝 Hamming distance from 𝜔 to 𝐴 , 𝑑𝑝 𝑝(1 − 𝑝) which holds for every finite graph and every decreasing event 𝐴 [Gri06, Theorem 2.53]. In our coordinates, this inequality reads   𝑑 log 1/(1 − Spr( 𝑝; 𝑡)) (− log PSpr( 𝑝;𝑡) ( 𝐴)) ≥ ESpr( 𝑝;𝑡) Hamming distance from 𝜔 to 𝐴 . 𝑑𝑡 Spr( 𝑝; 𝑡) In our case the prefactor −(log(1 − Spr( 𝑝; 𝑡)))/Spr( 𝑝; 𝑡) is at least 1 whereas in the original inequality the prefactor 1/( 𝑝(1 − 𝑝)) is at least 2, leading to the difference between our two conclusions. □ 111 Proof of Lemma 3.6.6. Let 𝑛0 = 𝑛0 (𝑑, 𝐷) be as in Lemma 3.6.5, and suppose that 𝑛 ≥ 𝑛0 and 𝑏 ≤ 81 𝑛1/3 are such that A (𝑏, 𝑛) holds. Let 𝑈 ⊆ 𝐵tz(𝑚) be the set of vertices guaranteed to exist by Lemma 3.6.5, and let 𝐴 be a subset of 𝑈 with | 𝐴| = ⌈[log 𝑛] 𝑐3 𝐾−1/2 ⌉. Since A (𝑏, 𝑛) holds, we have that 1/2 min P 𝑝1 (𝑥 ↔ 𝑆 𝑚/2 ) ≥ 𝜅 𝑝1 (𝑛, ∞) ≥ 𝑒 −[log log 𝑛] , 𝑥∈𝐴 while our choice of 𝐴 guarantees that 𝐵 𝑚/2 2 | 𝐴| max P 𝑝1 (𝑥 ←−→ 𝑦) ≤ 2⌈(log 𝑛) 𝑐3 𝐾−1/2 ⌉ (log 𝑛) −𝑐3 𝐾 ≤ 𝑒 −(log log 𝑛) 1/2 𝑥,𝑦∈𝐴 𝑥≠𝑦 whenever 𝑛 is larger than some constant 𝑛1 = 𝑛1 (𝑑, 𝐷) ≥ 𝑛0 . (This constant does not depend on 𝐾 1/2 since 𝑐 3 𝐾 ≥ 𝑐 3 𝐾0 ≥ 2 > 1/2.) Thus, applying Proposition 3.6.7 with 𝜃 = 𝑒 −[log log 𝑛] yields that (since 𝑝 3/2 ≥ Spr( 𝑝 1 , 𝛿/2))   𝛿 −[log log 𝑛] 1/2 | 𝐴| . P 𝑝3/2 (𝐵tz(𝑚) ↔ 𝑆 𝑚/2 ) ≥ P 𝑝3/2 ( 𝐴 ↔ 𝑆 𝑚/2 ) ≥ 1 − exp − 𝑒 (3.6.14) 2 On the other hand, the definition of 𝛿 ensures that 𝛿 ≥ [log 𝑛] −1/4 and hence that there exists a constant 𝑛2 = 𝑛2 (𝑑, 𝐷) ≥ 𝑛1 such that 1/2 1 𝛿 −[log log 𝑛] 1/2 | 𝐴| ≥ 𝑒 −[log log 𝑛] (log 𝑛) 𝑐3 𝐾−1/2−1/4 ≥ (log 𝑛) 𝑐3 𝐾−1 𝑒 2 2 whenever 𝑛 ≥ 𝑛2 , which implies the claim in conjunction with (3.6.14). □ The next lemma is completely elementary. It tells us that we can find two nearby reals 𝑚 and 𝑚′ where tz(𝑚) is close to tz(𝑚′). Lemma 3.6.8. Let 𝑅 ≥ 1. There exists a constant 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝑅) such that if 𝑛 ≥ 𝑛0 ∨ 𝑁 then there exists 𝑚 ∈ S (𝑛) such that 𝑚(log 𝑛) −𝑅 ∈ S (𝑛) and tz(𝑚) ≤ (log 𝑛) 8𝑅/𝑐1 . tz(𝑚(log 𝑛) −𝑅 ) Proof of Lemma 3.6.8. Let 𝑛 ≥ 𝑁 so that S (𝑛) is defined. Let 𝑠 and 𝑡 denote the left and right endpoints of S (𝑛), and define       𝑐 1 log(𝑚 1 ) 𝑐 1 log 𝑛 𝑘 := log [log 𝑛] 𝑅 (𝑡/𝑠) = ≥ 2𝑅 log log 𝑛 6𝑅 log log 𝑛 112 where 𝑛1/3 ≤ 𝑚 1 ≤ 𝑛 is as in the definition of S (𝑛). If a suitable 𝑚 ∈ S (𝑛) does not exist then, using the trivial inequalities tz(𝑡) ≤ 𝑡 ≤ 𝑛 and tz(𝑠) ≥ 1, we must have that 𝑛 ≥ tz(𝑡) ≥ tz(𝑠) 𝑘 Ö tz(𝑠[log 𝑛] 𝑖𝑅 ) tz(𝑠[log 𝑛] (𝑖−1)𝑅 ) 𝑖=1    8𝑅 𝑐 1 log 𝑛 ≥ tz(𝑠) [log 𝑛] ≥ exp · log log 𝑛 . 𝑐 1 6𝑅 log log 𝑛 Since 8 > 6, this yields a contradiction when 𝑛 is larger than some constant 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝑅) (allowing us to approximately remove the effect of rounding down). □  8𝑅/𝑐 1  𝑘 We will now combine our lemmas to prove Lemma 3.6.3. The idea here is inspired by the uniqueness via sprinkling argument from [CMT22, Section 8], which itself used ideas from [BT17]. Our approach is different because we do not know that exposed spheres are well-connected. Instead, we have the polylog-plentiful tubes condition. This is a much weaker geometric control because the tubes are not constrained to lie within narrow annuli. The main step is to use the strong connectivity bound from Lemma 3.6.6 to deduce that with high probability, every 𝜔7/4 -cluster crossing a thick annulus is contained in a single 𝜔2 -cluster. (As before we abbreviate 𝜔2 = 𝜔 𝑝2 and so on.) In [CMT22], the analogous step was carried out by dividing the thick annulus into thinner annuli before showing that if two clusters cross multiple annuli, then, after sprinkling in those annuli, the clusters will merge with high probability. This works because in every thin annulus, there is some good probability that the clusters will merge after sprinkling in the annulus, thanks to the connectivity of exposed spheres. In our case, we also track how many clusters survive un-merged as they cross through multiple annuli. The difference is that we will have to sprinkle everywhere each time we cross a thin annulus. Nevertheless, we will sprinkle so little at each stage that the net effect is to sprinkle by less than 𝛿/4, as required. −1 Proof of Lemma 3.6.3. Let 𝐾1 = 𝐾1 (𝑑, 𝐷, 𝜆) = max{𝐾0 , 40𝜆(𝑐 1 ∨ 1) −2 𝑐−1 3 , 4𝑐 3 (𝐷 ∨ 1)} and suppose that 𝐾 ≥ 𝐾1 and 𝑛 ≥ 𝑁. Fix 𝑅 := 5𝜆/𝑐 1 and let 𝑛0 = 𝑛0 (𝑑, 𝐷, 𝑅) be the constant from Lemma 3.6.8, which by our choice of 𝑅 depends only on 𝑑, 𝐷, and 𝜆. We also let 𝑛1 = 𝑛1 (𝑑, 𝐷) and 𝑐 4 be the constants from Lemma 3.6.4. 𝑅 Suppose that 𝑛 ≥ 𝑛2 = 𝑛2 (𝑑, 𝐷, 𝜆) = 𝑛0 ∨ 𝑛1 ∨ 𝑁 ∨ 𝑒 2 and 𝑏 ≤ 81 𝑛1/3 are such that A (𝑏, 𝑛, 𝑝) holds, and let 𝑚 ∈ S (𝑛) be the element guaranteed to exist by Lemma 3.6.8 applied with this 𝑅 value of 𝑅. Taking 𝑛2 ≥ 𝑒 2 guarantees that 𝑚/2 ≥ 𝑚(log 𝑛) −𝑅 and hence that 𝑚/2 ∈ S (𝑛) when 𝑛 ≥ 𝑛2 . Since 𝑛 ≥ 𝑛1 , we have by Lemma 3.6.4 that   1/2 −𝑅 𝑐3 𝐾 −𝑅 𝜅 𝑝7/4 tz(𝑚(log 𝑛) )(log 𝑛) , 2𝑚(log 𝑛) ≥ 𝑐 4 𝑒 −2[log log 𝑛] . 113 On the other hand, our choice of 𝑅 and 𝑚 ensure that there exists a constant 𝑛3 = 𝑛3 (𝑑, 𝐷, 𝜆) ≥ 𝑛2 such that if 𝑛 ≥ 𝑛3 then we have the inclusion of intervals    𝑚
[log 𝑚] −4𝜆/𝑐1 , 2 tz(𝑚/2) + 2 , 2𝑚 [log 𝑛] −𝑅 , tz 𝑚 [log 𝑛] −𝑅 [log 𝑛] 𝑐3 𝐾 ⊇ 3 so that   1/2 𝑚 𝜅 𝑝7/4 2 tz(𝑚/2) + 2, [log 𝑚] −4𝜆/𝑐1 ≥ 𝑐 4 𝑒 −2[log log 𝑛] . (3.6.15) 3 Note that this estimate holds as a consequence of A (𝑏, 𝑛, 𝑝) alone: we have not yet made use of the negative information ¬B (𝑛, 𝑝). −1 Now suppose that ¬B (𝑛, 𝑝) holds. Since 𝐾 ≥ 𝐾1 ≥ 12𝑐−1 3 𝑐 1 𝜆, there exists a constant 𝑛3 = 𝑛3 (𝑑, 𝐷, 𝜆) ≥ 𝑛3 such that if 𝑛 ≥ 𝑛3 then tz(𝑚/2) ≤ 𝑚 [log 𝑛] 3𝜆/𝑐1 −𝑐3 𝐾 ≤ 𝑚(log 𝑛) −3𝑐3 𝐾/4 ≤ 𝑚 (log 𝑚) −𝑐3 𝐾/2 17 (3.6.16) for every 𝑚 ∈ S (𝑛). Our next goal is to prove a good upper bound on the probability of the non-uniqueness event Piv 𝑝7/4 ,𝑝2 [𝑚/16, 𝑚/8], where Piv 𝑝,𝑞 [𝑚, 𝑛] denotes the event that there are at least two distinct 𝜔 𝑝 -clusters that each intersect both 𝐵𝑚 and 𝑆𝑛 but that are not connected to each other by any path in 𝐵𝑛 ∩ 𝜔 𝑞 . We will do this using a variation on the “orange peeling” argument of [CMT22], where we iteratively sprinkle and zoom in closer to 𝑚/16 over a number of steps. Let 𝑘 := 2⌊(log 𝑛) 𝐷 ⌋, 𝜀 := (log 𝑛) −(𝐷+1) , and for each 𝑖 ∈ {0, ..., 𝑘 } set 𝑟𝑖 := 𝑚 𝑖𝑚 − [log 𝑛] −𝐷 8 40 and 𝑞𝑖 := Spr( 𝑝 7/4 ; 𝑖𝜀). Note that 𝑟𝑖 ∈ [𝑚/16, 𝑚/8] and 𝑞𝑖 ∈ [ 𝑝 7/4 , 𝑝 2 ] for every 0 ≤ 𝑖 ≤ 𝑘. We work with the standard monotone coupling (𝜔 𝑞 ) 𝑞∈[0,1] , and write 𝜔 (𝑖) = 𝜔 𝑞𝑖 for each 𝑖 ∈ {0, . . . , 𝑘 }. (Be careful not to confuse this with our previous notational shorthand 𝜔1 = 𝜔 𝑝1 , 𝜔2 = 𝜔 𝑝2 .) Given the family of configurations (𝜔 𝑞 ) 𝑞∈[0,1] , recursively define a set of 𝐵𝑚/8 ∩ 𝜔 (𝑖) -clusters C𝑖 for each 𝑖 ∈ {0, ..., 𝑘 } as follows: 1. Let C0 be the set of all 𝐵𝑚/8 ∩ 𝜔 (0) -clusters that contain a vertex in 𝑆𝑟0 . 2. Given C𝑖 for some 𝑖 < 𝑘, let C𝑖+1 be the set of 𝐵𝑚/8 ∩ 𝜔 (𝑖+1) -clusters 𝐶 such that there exists 𝐶 ′ ∈ C𝑖 with 𝐶 ′ ⊆ 𝐶 and 𝐶 ′ ∩ 𝑆𝑟𝑖+1 ≠ ∅. 114 Figure 3.4: Schematic illustration of the construction of C𝑖+1 from C𝑖 : The black, purple, and green circles represent the spheres 𝑆𝑟0 , 𝑆𝑟𝑖 , and 𝑆𝑟𝑖+1 respectively. The purple regions in the left figure represent the clusters making up C𝑖 . By construction each of these is a 𝐵𝑟0 ∩ 𝜔 (𝑖) -cluster that connects 𝑆𝑟0 to 𝑆𝑟𝑖 . The green regions in the middle figure represent the 𝐵𝑟0 ∩ 𝜔 (𝑖+1) clusters that contain some cluster in C𝑖 . Finally, the green regions remaining in the third figure represent the subset of these clusters that happen to intersect 𝑆𝑟𝑖+1 ; these make up C𝑖+1 . See fig. 3.4 for an illustration. This definition ensures that the cardinality |C𝑖 | is a decreasing function of 𝑖 and that we have the inclusion of events  Piv 𝑝7/4 ,𝑝2 [𝑚/16, 𝑚/8] ⊆ C 𝑘 ∈ {0, 1} . (3.6.17) Roughly speaking, our goal is to show that, for each 𝑖, if C𝑖 is not a singleton then the cardinality |C𝑖+1 | is smaller than |C𝑖 | by a factor of roughly 1/2 with high probability under P. Ideally, we would like to show that, with high probability on the event {|C𝑖 | > 1}, every cluster in C𝑖 that intersects 𝑆𝑟𝑖+1 is 𝐵𝑚/8 ∩ 𝜔 (𝑖+1) -connected to a distinct cluster in C𝑖 . This is what we will prove, except that we will allow one distinguished cluster to go un-merged. For every non-empty set of clusters F , permanently fix a choice of element min(F ) ∈ F such that dist(𝑜, min(F )) = Ð dist(𝑜, F ). For each 0 ≤ 𝑖 ≤ 𝑘 − 1, consider the event n 𝐵𝑚/8 ∩ 𝜔 (𝑖+1) Ø o E𝑖 = {C𝑖 = ∅}∪ 𝐶 ←−−−−−−−−→ (C𝑖 \{𝐶}) for every 𝐶 ∈ C𝑖 \{min(C𝑖 )} with dist(𝑜, 𝐶) ≤ 𝑟𝑖+1 . We will prove the following lemma at the end of the section after explaining how it may be used to conclude the proof of Lemma 3.6.3 (and hence of Proposition 3.6.1). Lemma 3.6.9 (Merging clusters). There exists a constant 𝑛11 = 𝑛11 (𝑑, 𝐷, 𝜆) ≥ 𝑛4 such that if 𝑛 ≥ 𝑛11 then 𝑐1 𝜆/2 P(E𝑖 ) ≥ 1 − 𝑒 −[log 𝑛] , (3.6.18) for every 0 ≤ 𝑖 ≤ 𝑘 − 1. 115 (We name this constant 𝑛11 to leave room for the constants 𝑛6 through 𝑛10 that will appear in the proof of this claim.) Let us now conclude the proof of Lemma 3.6.3 given Lemma 3.6.9. On the event E𝑖 we have that   |C𝑖 | − 1 |C𝑖+1 | ≤ + 1. 2 Since |C0 | ≤ 𝑆𝑟0 ≤ 𝑒 (log 𝑛) and 𝑘 := 2⌊[log 𝑛] 𝐷 ⌋, it follows that |C 𝑘 | ∈ {0, 1} on the event Ñ 𝑘−1 𝑖=0 E𝑖 . It follows by a union bound that there exists a constant 𝑛6 = 𝑛6 (𝑑, 𝐷, 𝜆) ≥ 𝑛5 such that if 𝑛 ≥ 𝑛6 then 𝐷   P Piv 𝑝7/4 ,𝑝2 [𝑚/16, 𝑚/8] ≤ 𝑘−1 ∑︁ P 𝑝 (E𝑖𝑐 ) ≤ 2(log 𝑛) 𝐷 𝑒 −(log 𝑛) 𝑐1 𝜆/2 ≤ 𝑒 −(log 𝑛) 𝑐1 𝜆/3 . (3.6.19) 𝑖=0 On the other hand, (3.6.16) ensures that if 𝑛 ≥ 𝑛6 then tz(𝑚/2) ≤ 𝑚/17, and it follows from Lemma 3.6.6 applied to 𝑚/2 that
𝑐1 𝜆/3 P 𝑝7/4 𝑆 𝑚/17 ↔ 𝑆 𝑚/4 ≥ 1 − 𝑒 −(log 𝑛) if 𝑛 ≥ 𝑛6 . It follows in particular that there exists 𝑛7 = 𝑛7 (𝑑, 𝐷, 𝜆) ≥ 𝑛6 such that
𝑒 [log 𝑛] 𝑐1 𝜆/4 1 (3.6.20) 2 if 𝑛 ≥ 𝑛7 ; this will be used to form a chain of connected annuli using the Harris-FKG inequality. P 𝑝7/4 𝑆 𝑚/17 ↔ 𝑆 𝑚/4 ≥ We now apply (3.6.19) and (3.6.20) to bound the corridor function. Let 𝛾 be a path of length at 𝑐 𝜆/4 most 𝑒 [log 𝑛] 1 , starting at some vertex 𝑢 and ending at some vertex 𝑣, and observe that we have the inclusion of events   𝜔7/4  𝜔7/4 𝐵 𝑛 (𝛾)∩𝜔2 𝑢 ←−−−−−−→ 𝑣 ⊆ 𝑢 ←−→ 𝑆𝑛 (𝑢) ∩ 𝑣 ←−→ 𝑆𝑛 (𝑣)} ∩ len Ù𝛾   𝜔7/4  𝑆 𝑚/17 (𝛾𝑡 ) ←−→ 𝑆 𝑚/4 (𝛾𝑡 ) ∩ Piv 𝑝7/4 ,𝑝2 [𝑚/16, 𝑚/8] (𝛾𝑡 ) . 𝑡=1 Applying the Harris-FKG inequality and a union bound, we deduce that there exists a constant 𝑛8 = 𝑛8 (𝑑, 𝐷, 𝜆) such that if 𝑛 ≥ 𝑛8 then
len(𝛾) 𝐵 𝑛 (𝛾) P 𝑝2 (𝑢 ←−−→ 𝑣) ≥ P 𝑝7/4 (𝑜 ↔ 𝑆𝑛 ) 2 · P 𝑝7/4 𝑆 𝑚/17 ↔ 𝑆 𝑚/4   − len(𝛾) · (1 − P Piv 𝑝7/4 ,𝑝2 [𝑚/16, 𝑚/8]) 𝑐1 𝜆/4 −(log 𝑛) 𝑐1 𝜆/3 1 −2(log log 𝑛) 1/2 𝑒 − 𝑒 (log 𝑛) 𝑒 2 1/2 ≥ 𝑒 −3[log log 𝑛] , ≥ 116 where we used the assumption that A (𝑏, 𝑛, 𝑝) holds to bound P 𝑝7/4 (𝑜 ↔ 𝑆𝑛 ) ≥ 𝜅 𝑝1 (𝑛, ∞) ≥ 1/2 𝑒 −(log log 𝑛) . The claimed lower bound on the corridor function follows since 𝛾 was an arbitrary 𝑐 𝜆/4 path of length at most 𝑒 [log 𝑛] 1 . □ It remains only to prove Lemma 3.6.9. Proof of Lemma 3.6.9. We continue to use the notation from the proof of Lemma 3.6.3, and in particular will use the constants 𝐾1 = 𝐾1 (𝑑, 𝐷, 𝜆) and 𝑛4 = 𝑛4 (𝑑, 𝐷, 𝜆) defined in that proof. Consider the event Ω defined by Ù 𝜔3/2 Ω := {𝑆tz(𝑚/2) (𝑢) ←−→ 𝑆 𝑚/8 }. 𝑢∈𝐵 𝑚/8 It follows by Lemma 3.6.6 (applied to 𝑚/2) and a union bound that there exists a constant 𝑛5 = 𝑛5 (𝑑, 𝐷, 𝜆) ≥ 𝑛4 such that if 𝑛 ≥ 𝑛5 then P(Ω) ≥ 1 − Gr(𝑚) sup P 𝑝3/2 (𝑆tz(𝑚/2) (𝑢) ↔ 𝑆 𝑚/8 ) 𝑢∈𝐵 𝑚/8 ≥ 1 − Gr(𝑚) sup P 𝑝3/2 (𝑆tz(𝑚/2) (𝑢) ↔ 𝑆 𝑚/4 (𝑢)) 𝑢∈𝐵 𝑚/8 ≥ 1−𝑒 (log 𝑚) 𝐷 𝑒 −(log 𝑛) 𝑐3 𝐾 −1 ≥ 1 − 𝑒 −(log 𝑛) 𝑐3 𝐾/2 , (3.6.21) where we used that 𝐾 ≥ 𝐾1 ≥ 4𝑐−1 3 (𝐷 ∨ 1) in the final inequality. For each 0 ≤ 𝑖 ≤ 𝑘 − 1, let F𝑖 be the set F𝑖 = {F : P(C𝑖 = F | Ω) > 0}. (Note that F𝑖 is a set of sets of sets of vertices.) It follows from the definitions that  Ø  𝑑 𝑢, 𝐶 ≤ tz(𝑚/2) for every 𝑢 ∈ 𝐵𝑚/8 and F ∈ F𝑖 . (3.6.22) 𝐶∈F By (3.6.21) and a union bound, it suffices to prove that there exists 𝑛11 = 𝑛11 (𝑑, 𝐷, 𝜆) ≥ 𝑛5 such that 𝑐1 𝜆/2 1 (3.6.23) P(E𝑖 | C𝑖 = F ) ≥ 1 − 𝑒 −[log 𝑛] 2 for every 0 ≤ 𝑖 ≤ 𝑘 − 1 and every F ∈ F𝑖 . Before doing this, we will need to prove a purely geometric preliminary claim. Let 0 ≤ 𝑖 ≤ 𝑘 − 1, let F ∈ F𝑖 , and let 𝐶 ∈ F \{min(F )}. We claim that there exists a constant 𝑛8 = 𝑛8 (𝑑, 𝐷, 𝜆) ≥ 𝑛5 such that if 𝑛 ≥ 𝑛8 and dist(𝑜, 𝐶) ≤ 𝑟𝑖+1 then there exists a set of vertices 𝑈 ⊆ 𝐵(𝑟𝑖+1 + 𝑚(log 𝑚) −𝜆/𝑐1 ) with |𝑈| ≥ 12 (log 𝑚) 𝑐1 𝜆 such that the following hold: 117 (i) 𝑈 is 𝑚(log 𝑚) −4𝜆/𝑐1 -separated. That is, pairwise distances between distinct points in 𝑈 are at least 𝑚(log 𝑚) −4𝜆/𝑐1 . (ii) For each 𝑢 ∈ 𝑈, the ball 𝐵tz(𝑚/2)+1 (𝑢) intersects both 𝐶 and Ð (F \{𝐶}). (As before, we name this constant 𝑛8 to leave room for the constants 𝑛6 and 𝑛7 that will appear in the proof of this claim.) We let 𝑟 := 𝑚(log 𝑚) −2𝜆/𝑐1 and split the proof of this claim into two cases according to whether dist(𝑜, 𝐶) is smaller or larger than 𝑟. Ð Case 1: (dist(𝑜, 𝐶) ≤ 𝑟.) Since 𝐶 ≠ min(F ), we have that dist(𝑜, (F \{𝐶})) ≤ dist(𝑜, 𝐶) ≤ 𝑟 Ð also. Let Γ be the family of paths from 𝐵3𝑟 ∩ 𝐶 to 𝐵3𝑟 ∩ (F \{𝐶}) that is guaranteed to exist by the fact that 𝐺 has (𝑐 1 , 𝜆)-polylog-plentiful annular tubes at scale 𝑟. We now observe that for each 𝛾 ∈ Γ, there exists a vertex 𝑢 𝛾 on the path 𝛾 satisfying the ball-intersection condition (ii): • If max𝑡 dist(𝐶, 𝛾𝑡 ) ≤ tz(𝑚/2) then we may take 𝑢 𝛾 to be the final vertex of 𝛾. • Otherwise, if max𝑡 dist(𝐶, 𝛾𝑡 ) > tz(𝑚/2), we may take 𝑢 𝛾 = 𝛾𝑡 𝛾 where 𝑡 𝛾 is the maximum index such that dist(𝐶, 𝛾𝑡 ) ≤ tz(𝑚/2). To see that this choice of 𝑢 𝛾 satisfies (ii), note that Ð dist(𝛾𝑡 𝛾 +1 , 𝐶) > tz(𝑚/2) and hence by (3.6.22) that dist(𝛾𝑡 𝛾 +1 , (F \{𝐶})) ≤ tz(𝑚/2). Now define 𝑈 := {𝑢 𝛾 : 𝛾 ∈ Γ}. Since the family Γ is 2𝑟 (log 𝑟) −𝜆/𝑐1 -separated (and 𝑟 was defined to be 𝑟 = 𝑚(log 𝑚) −2𝜆/𝑐1 ), it follows that there exists a constant 𝑛6 = 𝑛6 (𝑑, 𝐷, 𝜆) ≥ 𝑛5 such that if 𝑛 ≥ 𝑛6 then 𝑈 is 𝑚(log 𝑚) −4𝜆/𝑐1 -separated and |𝑈| = |Γ| ≥ (log 𝑟) 𝑐1 𝜆 ≥ 1 (log 𝑚) 𝑐1 𝜆 . 2 Moreover, since every path in Γ was contained in 𝐵(3𝑟 + 𝑟 [log 𝑟] 𝜆/𝑐1 ), there exists a constant 𝑛7 = 𝑛7 (𝑑, 𝐷, 𝜆) ≥ 𝑛6 such that if 𝑛 ≥ 𝑛7 then 3𝑟 + 𝑟 [log 𝑟] 𝜆/𝑐1 ≤ 𝑚/16 ≤ 𝑟𝑖+1 + 𝑚(log 𝑚) −𝜆/𝑐1 and hence 𝑈 ⊆ 𝐵(𝑟𝑖+1 + 𝑚(log 𝑚) −𝜆/𝑐1 ). Case 2: (dist(𝑜, 𝐶) > 𝑟.) Let 𝑣 ∈ 𝐶 be such that dist(𝑜, 𝑣) = dist(𝑜, 𝐶), let 𝛾 𝑎 be a path in 𝐶 from 𝑣 to 𝑆𝑟 (𝑣), and let 𝛾 𝑏 be the portion of a geodesic from 𝑣 to 𝑜 starting at a neighbour 𝑢 of 𝑣 with dist(𝑜, 𝑢) < dist(𝑜, 𝑣) and ending at the first intersection with 𝑆𝑟 (𝑣). These path are both finite, start in 𝑆1 (𝑣) and end in 𝑆𝑟 (𝑣), and are contained in 𝐶 and disjoint from 𝐶 respectively. Let Γ be the family of paths from 𝛾 𝑎 to 𝛾 𝑏 that is guaranteed to exist by the fact that 𝐺 has (𝑐 1 , 𝜆)-polylog-plentiful annular tubes at scale 𝑟/3. We can construct the desired set 𝑈 by picking a vertex 𝑢 𝛾 in 𝛾 satisfying the condition (ii) for each 𝛾 ∈ Γ; the fact that such a vertex exists for each 118 𝛾 follows by the same argument used in case 1 above. Moreover, it follows by the same argument used in the first case that there exists a constant 𝑛8 = 𝑛8 (𝑑, 𝐷, 𝜆) ≥ 𝑛7 such that if 𝑛 ≥ 𝑛8 then the required bounds on the cardinality and separation of the set 𝑈 = {𝑢 𝛾 : 𝛾 ∈ Γ} hold, as well as the containment 𝑈 ⊆ 𝐵(𝑟𝑖+1 + 𝑚(log 𝑚) −𝜆/𝑐1 ). This concludes the proof of the geometric claim. We now use this geometric claim to establish the estimate (3.6.23), which will complete the proof of the lemma. Let 𝑖 ∈ {0, . . . , 𝑘 − 1} and let F ∈ F𝑖 be arbitrary. Consider also an arbitrary 𝐶 ∈ F \{min(F )} with dist(𝑜, 𝐶) ≤ 𝑟𝑖+1 , and let 𝑈 be the corresponding set of vertices guaranteed to exist by the geometric claim above. For each 𝑢 ∈ 𝑈, let 𝛽𝑢 := 𝐵(𝑢; 𝑚2 (log 𝑚) −4𝜆/𝑐1 ) ∩ 𝜔 (𝑖+1) . Consider an arbitrary vertex 𝑢 ∈ 𝑈. By construction of 𝑈, there exists a path 𝛾 that starts in 𝐶, Ð ends in (F \{𝐶}), is contained in 𝐵(𝑢; tz(𝑚/2) + 1), and has length at most 2 tz(𝑚/2) + 2. The estimate eq. (3.6.16) yields the existence of a constant 𝑛9 = 𝑛9 (𝑑, 𝐷, 𝜆) ≥ 𝑛8 such that if 𝑛 ≥ 𝑛9 then 𝑚 𝑚 [tz(𝑚/2) + 1] + (log 𝑚) −4𝜆/𝑐1 ≤ (log 𝑚) −4𝜆/𝑐1 , 3 2 so that the tube 𝐵(𝛾; 𝑚3 (log 𝑚) −4𝜆/𝑐1 ) associated to this path 𝛾 is contained in the ball 𝐵(𝑢; 𝑚2 (log 𝑚) −4𝜆/𝑐1 ). Thus, if 𝑛 ≥ 𝑛9 , the estimate (3.6.15) yields that   𝛽𝑢 Ø 1/2 P 𝐶 ←→ (F \{𝐶}) ≥ 𝑐 4 𝑒 −2[log log 𝑛] . We stress that the 𝐶 and F appearing in this inequality are deterministic, and do not depend on the configuration 𝛽𝑢 . Under P, the conditional law of 𝛽𝑢 given that C𝑖 = F is simply (inhomogeneous) bond percolation on 𝐵(𝑢; 𝑚2 (log 𝑚) −4𝜆/𝑐1 ) where every edge has probability at least Ð 𝑐 −1 𝜀 of being open, and every edge that does not touch F has probability 𝑞𝑖+1 of being open. In particular, recalling that 𝜀 = 𝛿(𝑞𝑖+1 , 𝑞𝑖 ) = (log 𝑛) −(𝐷+1) , it follows that there exists a constant 𝑛10 = 𝑛10 (𝑑, 𝐷, 𝜆) ≥ 𝑛9 such that if 𝑛 ≥ 𝑛10 then   𝛽𝑢 Ø 1/2 P 𝐶 ←→ (F \{𝐶}) | C𝑖 = F ≥ 𝑐2−1 𝜀 2 𝑐 4 𝑒 −2[log log 𝑛] ≥ (log 𝑚) −2𝐷−3 . Notice that under P, the configurations (𝛽𝑢 )𝑢∈𝑈 are independent. Moreover, this still holds after conditioning on the event that C𝑖 = F . So, by independence, there exist constants 𝜆 0 = 𝜆 0 (𝑑, 𝐷) = 8𝑐−1 1 (2𝐷 + 3) and 𝑛11 = 𝑛11 (𝑑, 𝐷, 𝜆) ≥ 𝑛10 such that if 𝜆 ≥ 𝜆 0 and 𝑛 ≥ 𝑛11 then    Ö  𝛽𝑢 Ø 𝐵𝑚/8 ∩𝜔 (𝑖+1) Ø P 𝐶 ←−−−−−−−→ (F \{𝐶}) | C𝑖 = F ≥ 1 − P 𝐶← → (F \{𝐶}) | C𝑖 = F 𝑢∈𝑈 1 ≥ 1 − (1 − (log 𝑚) −2𝐷−3 ) 2 (log 𝑚) ≥ 1 − 𝑒 −2(log 𝑛) 119 𝑐1 𝜆/2 . 𝑐1 𝜆 Finally, since |F | ≤ 𝑆𝑟0 ≤ 𝑒 (log 𝑛) , we have by a union bound that if 𝜆 ≥ 𝜆 0 and 𝑛 ≥ 𝑛11 then 𝐷 P(E𝑖 | C𝑖 = F ) ≥ 1 − 𝑒 (log 𝑛) 𝑒 −2(log 𝑛) 𝐷 𝑐1 𝜆/2 𝑐1 𝜆/2 1 . ≥ 1 − 𝑒 −(log 𝑛) 2 Since F was arbitrary, this implies the claimed bound (3.6.23), completing the proof. □ Completing the proof of the main theorem: The implications (C0 ) and (C) In this section we apply Proposition 3.6.1 to complete the proof of the Proposition 3.3.1 and hence of Theorem 8.1.1. Given Proposition 3.4.5, what remains is to verify the implications (C0 ) and (C). Proof of Proposition 3.3.1. Let 𝐷 := 20, and accordingly let 𝜆 0 (𝑑, 𝐷) and 𝑐 = 𝑐(𝑑, 𝐷) be the constants from Proposition 3.6.1 with this value of 𝐷. Let 𝜆 := max{𝜆 0 , 10/𝑐}, and let 𝐾 (𝑑, 𝐷, 𝜆) and 𝑀 (𝑑, 𝐷, 𝜆) be the corresponding constants from Proposition 3.6.1 that are there called 𝐾1 and 𝑛0 . (We want to avoid reusing the label 𝑛0 .) We claim that if we define 𝛿0 using this value of 𝐾, then the implications (C0 ) and (C) (for all 𝑖 ≥ 1) hold whenever 𝑝 0 ≥ 1/𝑑, 𝛿0 ≤ 1, and 𝑛0 is sufficiently large with respect to 𝑑, which in particular guarantees that 𝑛0 ≥ max{16, 𝑀 }. The implication (C0 ) is immediate (i.e., is a direct consequence of Proposition 3.6.1 after unpacking the definitions), so we will just explain how to prove the implication (C). Fix 𝑖 ≥ 1 and assume that Full-space(𝑖) holds and that Corridor(𝑘) holds for all 1 ≤ 𝑘 ≤ 𝑖. Our goal is to establish that Corridor(𝑖 + 1) holds provided that 𝑛0 is sufficiently large with respect to 𝑑. This follows immediately from Proposition 3.6.1 if we can show that for every 𝑛 ∈ L (𝐺, 20) ∩ [𝑛𝑖−1 , 𝑛𝑖 ] there exists some 𝑏 ≤ 81 𝑛1/3 such that  𝐾 log log 𝑛 min{log 𝑛, log Gr(𝑏)}  1/4 ≤ (log log 𝑛𝑖 ) −1/2 and   P 𝑝𝑖 Piv[4𝑏, 𝑛1/3 ] ≤ (log 𝑛) −1 . (3.6.24) Consider an arbitrary 𝑛 ∈ L (𝐺, 20) ∩ [𝑛𝑖−1 , 𝑛𝑖 ] (assuming one exists; the claim is vacuous if not). We split into two cases according to whether (log 𝑛) 2/3 ∈ L (𝐺, 20). First suppose that (log 𝑛) 2/3 ∉ L (𝐺, 20), so in particular Gr((log 𝑛) 2/3 ) ≥ 𝑒 ((log 𝑛) 2/9 ) 20 . By Corollary 3.2.4, provided 𝑛0 is sufficiently large with respect to 𝑑, we know that   P 𝑝𝑖 Piv[4(log 𝑛) 2/3 , 𝑛1/3 ] ≤ (log 𝑛) −1 . So in this case both conditions in (3.6.24) are satisfied for 𝑏 = (log 𝑛) 2/3 ≤ 18 𝑛1/3 provided that 𝑛0 is sufficiently large with respect to 𝑑. 120 Now instead suppose that (log 𝑛) 2/3 ∈ L (𝐺, 20). Note that we can always find some 𝑘 ∈ {1, . . . , 𝑖} such that (log 𝑛) 2/3 ∈ [𝑛 𝑘−2 , 𝑛 𝑘−1 ]. (This is why we took 𝑛−1 = (log 𝑛0 ) 1/2 as small as we did in the statement of the proposition.) So by our hypothesis that Corridor(𝑘) holds for this particular value of 𝑘, we have that   2/3 10 1/2 𝜅 𝑝 𝑘 𝑒 [log( [log 𝑛] )] , [log 𝑛] 2/3 ≥ 𝑒 −(log log 𝑛 𝑘 ) . (3.6.25) We now claim that 𝑏 := 15 min{𝑒 (log log 𝑛) , Gr−1 (𝑒 (log 𝑛) )} satisfies both conditions from eq. (3.6.24) provided that 𝑛0 is sufficiently large with respect to 𝑑. The inequality 𝑏 ≤ 18 𝑛1/2 again holds trivially when 𝑛0 is large. The definition of 𝑏 ensures that log Gr(𝑏) ≥ 51 log Gr(5𝑏) ≥ 15 (log log 𝑛) 9 , so that the first condition also trivially holds when 𝑛0 is large. To see that the second condition holds, we apply Lemma 3.2.3 and Proposition 3.2.1 to obtain that there exists a constant 𝐶 such that 9 1/10 P 𝑝𝑖 (Piv[4𝑏, 𝑛1/3 ]) ≤ P 𝑝𝑖 (Piv[1, 𝑛1/3 /2]) · |𝑆4𝑏 | 2 Gr(5𝑏) 𝐵5𝑏 min𝑎,𝑏∈𝑆4𝑏 (𝑎 ←−→ 𝑏)  1/4  1/10 1/2 (log 𝑛) 20 𝑒 3(log 𝑛) +(log log 𝑛 𝑘 ) ≤ (log 𝑛) −1 ≤𝐶 𝑛 whenever 𝑛0 is sufficiently large with respect to 𝑑, where we used the estimate (3.6.25) and the fact 2/3 10 that 4𝑏 ≤ 𝑒 [log( [log 𝑛] )] and 5𝑏 ≥ (log 𝑛) 2/3 (when 𝑛0 is sufficiently large) to bound the term 𝐵5𝑏 min𝑎,𝑏∈𝑆4𝑏 (𝑎 ←−→ 𝑏). This completes the proof. □ 3.7 Closing discussion and open problems Joint continuity of the supercritical infinite cluster density Recall that G ∗ is the space of all infinite, connected, transitive graphs that are not one-dimenional, which we endow with the local topology, and recall that for all 𝑝 ∈ (0, 1) and 𝐺 ∈ G ∗ , the infinite cluster density is defined to be 𝜃 (𝐺, 𝑝) := P𝐺𝑝 (𝑜 ↔ ∞). Consider a sequence (𝐺 𝑛 )𝑛≥1 in G ∗ converging to some 𝐺 ∈ G ∗ . The main result of the present paper Theorem 8.1.1 states that 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺). One could ask the following more refined question: does 𝜃 𝐺 𝑛 → 𝜃 𝐺 pointwise? (One can observe from the mean-field lower bound, say, that a positive answer to this question would imply our result that 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺).) For 𝑝 < 𝑝 𝑐 (𝐺), it follows immediately from the lower semi-continuity of 𝑝 𝑐 that 𝜃 (𝐺 𝑛 , 𝑝) → 𝜃 (𝐺, 𝑝) = 0, so the only non-trivial cases are when 𝑝 = 𝑝 𝑐 (𝐺) and 𝑝 > 𝑝 𝑐 (𝐺). The case 𝑝 = 𝑝 𝑐 (𝐺) appears to be hard. Indeed, if one could prove this result in the case of toroidal slabs (Z2 × Z/𝑛Z)𝑛≥1 converging to the cubic grid Z3 , then it would follow from the main result of [DST16] that 𝜃 (Z3 , 𝑝 𝑐 (Z3 )) = 0, a notorious open question. We have nothing interesting to say about this case. However, in our other work [EH23+a] 121 we prove the following theorem, which together with Theorem 8.1.1 completely resolves the case when 𝑝 > 𝑝 𝑐 (𝐺). Theorem 3.7.1 ([EH23+a]). Let (𝐺 𝑛 )𝑛≥1 be a sequence in G ∗ that converges in the local topology to some 𝐺 ∈ G ∗ . Then 𝜃 (𝐺 𝑛 , 𝑝) → 𝜃 (𝐺, 𝑝) as 𝑛 → ∞ for every 𝑝 > lim sup𝑛→∞ 𝑝 𝑐 (𝐺 𝑛 ). Note that the main theorem of [EH23+a] is much more general than this and also establishes a form of locality for the density of the giant cluster on finite transitive graphs that may have divergent degree. Together with Theorem 8.1.1 this theorem yields the following elementary corollary. Corollary 3.7.2. 𝜃 (𝐺, 𝑝) is continuous on the open set {(𝐺, 𝑝) : 𝐺 ∈ G ∗ , 𝑝 > 𝑝 𝑐 (𝐺)}. Moreover, if (𝐺 𝑛 )𝑛≥1 is a sequence in G ∗ that converges in the local topology to some 𝐺 ∈ G ∗ then 𝜃 ( 𝑝, 𝐺 𝑛 ) → 𝜃 ( 𝑝, 𝐺) as 𝑛 → ∞ for each 𝑝 ∈ [0, 1]\{𝑝 𝑐 (𝐺)}. Let us roughly indicate how the tools built in Section 3.4 could be used to give an alternative proof of Theorem 3.7.1. This alternative proof is less general and (arguably) more involved and than the one given in [EH23+a], but the result is quantitatively stronger: it can be used to prove that 𝜃 (𝐺, 𝑝) is not just continuous but even locally Hölder continuous on the supercritical set (with the power in the definition of Hölder continuity possibly degenerating near the boundary of the set). Let G𝑑∗ denote the set of infinite transitive graphs with vertex degree exactly 𝑑 that are not onedimensional. As explained in detail in [EH23+a], it suffices to prove a tail estimate on the size of finite clusters in supercritical percolation that is uniform over G𝑑∗ for each 𝑑, i.e. it suffices to prove that (3.7.1) lim sup sup P𝐺𝑝 (𝑚 ≤ |𝐾𝑜 | < ∞) = 0 for all 𝜀 > 0 and 𝑑 ≥ 1, 𝑚→∞ 𝐺∈G ∗ 𝑝≥ 𝑝 (𝐺)+𝜀 𝑑 𝑐 where we recall that 𝐾𝑜 denotes the cluster of the root vertex 𝑜. We will focus on proving an estimate of this form for the set of unimodular graphs U𝑑∗ instead of G𝑑∗ ; in the nonunimodular case much stronger results (with optimal dependence on 𝑝 − 𝑝 𝑐 and 𝑚) can be proven by invoking the results of [Hut20e; Hut22] as explained in detail in [EH23+a]. Our proof will yield quantitatively that for every 𝜀 > 0 and 𝑑 ≥ 1 there exist constants 𝐶 and 𝑐 such that sup sup 𝐺∈U𝑑∗ 𝑝≥ 𝑝 𝑐 (𝐺)+𝜀 P𝐺𝑝 (𝑚 ≤ |𝐾𝑜 | < ∞) ≤ 𝐶𝑚 −𝑐 ; running the proof of continuity with this quantitative estimate yields the aforementioned local Hölder continuity of 𝜃 (𝐺, 𝑝). This bound is quantitatively much better than the bound coming from the proof in [EH23+a]. On the other hand, it is also much worse than the conjectured 122 optimal bounds, which are stretched exponential in 𝑚 (see Section 5.3 of [HH21c]). Having any superpolynomial tail bound would imply that 𝜃 (𝐺, 𝑝) is a smooth function of 𝑝 ∈ ( 𝑝 𝑐 (𝐺), 1] for each fixed 𝐺; for Z𝑑 and for nonamenable graphs it is known that the density is not just smooth but real analytic on this set [HH21c; GP23]. Let 𝜀 > 0, 𝑑 ≥ 1, 𝐺 ∈ U𝑑∗ , 𝑝 ≥ 𝑝 𝑐 (𝐺) + 𝜀, 𝑚 ≥ 1, and 𝜂 > 0 be arbitrary, and suppose that P𝐺𝑝 (𝑚 ≤ |𝐾𝑜 | < ∞) ≥ 𝜂. It suffices to prove that 𝑚 is necessarily bounded above by some constant 𝑀 (𝜀, 𝑑, 𝜂) < ∞. Let P denote the canonical monotone coupling (𝜔 𝑞 ) 𝑞∈[0,1] of the percolation measures (P𝐺 𝑞 ) 𝑞∈[0,1] . By the mean-field lower bound and transitivity, one can find vertices 𝑢, 𝑣 ∈ 𝑉 (𝐺) such that   𝜂𝜀 , P 𝐾𝑢 (𝜔 𝑝−𝜀/2 ) ≥ 𝑚 and 𝐾𝑣 (𝜔 𝑝 ) ≥ 𝑚 but 𝑢 ↮ 𝑣 ≥ 2 where 𝐾𝑢 (𝜔 𝑝−𝜀/2 ) denotes the cluster of 𝑢 in 𝜔 𝑝−𝜀/2 . In particular, writing G1/𝑚 for the law of a ghost-field G of intensity 1/𝑚 on the whole vertex set 𝑉 (𝐺),  1 G 1 ⊗ P(𝑢 ←−−−→ G ← → 𝑣 but 𝑢 ← → 𝑣) ≥ 1 − 𝑚 𝑒 𝜔 𝑝− 𝜀/2 𝜔𝑝 𝜔𝑝 2 𝜂𝜀 𝜂𝜀 ≥ . 2 8 (3.7.2) Assume for now that 𝐺 is amenable so that there is at most one infinite cluster P𝐺 𝑞 -almost surely for every 𝑞 ∈ [ 𝑝 − 𝜀/2, 𝑝]. Then by Lemma 3.4.4 with (𝑋, 𝐴, 𝑌 ) := ({𝑢}, 𝑉 (𝐺), {𝑣}), one can deduce that for some constants 𝑐 3 (𝜀, 𝑑) > 0 and 𝐶 (𝜀, 𝑑) < ∞, 𝜔 𝑝− 𝜀/2 𝜔𝑝 𝜔𝑝 G 1 ⊗ P(𝑢 ←−−−→ G ← → 𝑣 but 𝑢 ← → 𝑣) ≤ 𝐶𝑚 −𝑐3 . 𝑚 (3.7.3) By combining (3.7.2) and (3.7.3), we deduce that 𝑚 ≤ 𝑀 (𝜀, 𝑑, 𝜂) := (8𝐶/(𝜂𝜀)) 1/𝑐3 , as required. Finally, to handle the case when 𝐺 is nonamenable (but still unimodular), one can still run essentially the same argument as above but with some technical modifications to handle the possible existence of multiple infinite clusters. The key difference is that instead of tracking connections 𝑢 ↔ 𝑣 as wired usual, we instead track wired connections 𝑢 ←−−→ 𝑣 where wired {𝑢 ←−−→ 𝑣} := {𝑢 ↔ 𝑣} ∪ {𝑢 ↔ ∞ and 𝑣 ↔ ∞}. One can verify that the proof of Lemma 3.4.4 works just as well with this alternative notion of connnectivity, without requiring the hypothesis about the uniqueness of the infinite cluster. The rest of the argument explained above can then be adapted to work with this wired notion of connectivity also. 123 As explained in [EH23+a], the estimate (6.1.2) has the following nice interpretation. Consider the space (0, 1) × G ∗ → (0, 1) with the product topology, and consider the function 𝜃 : (0, 1) × G ∗ → (0, 1) mapping ( 𝑝, 𝐺) ↦→ 𝜃 𝐺 ( 𝑝). It is natural to ask whether 𝜃 is continuous as a function of two variables. One can show that a priori, 𝜃 is continuous if and only if the conclusion of Theorem 8.1.1 holds (locality of 𝑝 𝑐 ), the estimate (6.1.2) holds (i.e., there is a uniform tail bound on supercritical finite clusters), and 𝜃 (𝐺, 𝑝 𝑐 (𝐺)) = 0 for every 𝐺 ∈ G ∗ (continuity of the phase transition), the last statement being one of the most important open conjectures in the general study of percolation in G ∗ . (In this decomposition, (6.1.2) handles the interior of the supercritical region S := {( 𝑝, 𝐺) : 𝑝 > 𝑝 𝑐 (𝐺)}, Theorem 8.1.1 implies that this region is open, and the continuity conjecture handles the boundary values.) Finite graphs. As mentioned above, in [EH23+a] we prove versions of Theorem 3.7.1 and eq. (6.1.2) that also apply to families of bounded-degree finite transitive graphs. The above sketches work just as well in this context too; we have stated things in terms of infinite graphs purely for simplicity. Moreover, the above sketch can be used to give an alternative proof that for supercritical percolation on bounded-degree finite transitive graphs, the giant cluster is unique and has concentrated density, recovering the results of our two papers [EH21a; EH23+a] in this case. Note however that all of the tools from Section 3.4 break down rather badly when working with families of finite graphs that have large vertex degrees (e.g. vertex degrees that grow at least as a power of the total number of vertices), partly because for such graphs the emergence of a giant cluster can occur around values of 𝑝 close to 0. To handle this more general setting of arbitrary finite transitive graphs, we know of no alternative proofs of the uniqueness or concentration of the supercritical giant cluster to those we give in [EH21a; EH23+a]. The 𝑝 𝑐 gap and its witnesses Since G𝑑∗ is compact for each 𝑑 ≥ 1, it is a consequence of Theorem 8.1.1 that 𝑝 𝑐 attains its maximum on G𝑑∗ for each 𝑑 ≥ 1. In [PS23b], Panagiotis and Severo improved upon the results of [DGRSY20; HT21a] to establish that there exists a universal 𝜀 > 0 (independent of the degree) such that every Cayley graph with 𝑝 𝑐 < 1 has 𝑝 𝑐 ≤ 1 − 𝜀; Lyons, Mann, Tessera, and Tointon [LMTT23] give the explicit bound 𝜀 ≥ exp(− exp(17 exp(100 · 8100 ))). Presumably a similar result holds for transitive graphs that are not Cayley. The following natural conjecture would strengthen this result, and would also imply by Theorem 8.1.1 that 𝑝 𝑐 attains its global maximum on G ∗ . Conjecture 3.7.3. There exists a universal constant 𝐶 such that if 𝐺 is an infinite, connected, transitive, simple graph of vertex degree 𝑑 that is not one-dimensional, then 𝑝 𝑐 (𝐺) ≤ 𝐶/𝑑. 124 Figure 3.5: The transitive graph formed by laying out copies of 𝐾4𝑛 in an infinite square grid as above has 𝑝 𝑐 ≥ (4 log 2 − 𝑜(1))/deg as 𝑛 → ∞ as can be seen by coupling with bond percolation on Z2 and using a Poisson approximation. Since 4 log 2 ≈ 1.2 > 1, this shows that the asymptotic estimate 𝑝 𝑐 ∼ 1/deg can fail for high-degree vertex-transitive graphs even when these graphs are not one-dimensional in any sense. An exact asymptotic estimate 𝑝 𝑐 ∼ 𝐶/deg can be proven with a little further work. (Indeed, the constant 𝐶 ≈ 3.095 is the unique solution to the equation 𝐶 (1 + 𝐶 −1𝑊 [−𝑒 −𝐶 𝐶]) 2 = 4 log 2 where 𝑊 is the Lambert 𝑊 function.) For many natural families of high-degree graphs we have the stronger statement that 𝑝 𝑐 ∼ 1/deg as the degree diverges. For example, this holds for Z𝑑 as 𝑑 → ∞ by a theorem of Kesten [Kes90] (see also [ABS04a]). Moreover, this is not just a high-dimensional phenomenon: Penrose [Pen93] proved that a similar-estimate holds for the “spread-out” 𝑑-dimensional lattice, in which 𝑥, 𝑦 ∈ Z𝑑 are connected by an edge whenever ∥𝑥 − 𝑦∥ ≤ 𝑅, when 𝑑 is fixed and 𝑅 → ∞. On the other hand, the example illustrated in fig. 3.5 shows that high-degree transitive graphs do not always have 𝑝 𝑐 ∼ 1/deg even when they are not one-dimensional in any sense. Once one knows that 𝑝 𝑐 attains a maximum (either globally on G ∗ or on G𝑑∗ ), it becomes interesting to understand which graphs attain this maximum. Martineau and Severo [MS19] proved that 𝑝 𝑐 is strictly increasing under quotients, so that any maximal graph must have no non-trivial quotients in G ∗ . It seems reasonable to believe that the maximal graph would be a lattice of low degree and in low dimension. Consulting tables of numerical values of 𝑝 𝑐 for these lattices (as can be found on https://en.wikipedia.org/wiki/Percolation_threshold) leads to the highly speculative conjecture that 𝑝 𝑐 is maximized by the so-called super-kagome lattice (a.k.a. 3-12 125 Figure 3.6: The 3-12 (a.k.a. super-kagome) lattice is the current best candidate for the transitive graph with the highest non-trivial value of 𝑝 𝑐 for bond percolation. Its critical value 𝑝 𝑐 ≈ 0.7404 . . . has been estimated to great precision numerically in [SJ20]. The transitive graph with the next highest value of 𝑝 𝑐 to have been investigated numerically is the truncated trihexagonal lattice, which has 𝑝 𝑐 ≈ 0.6937. lattice); see fig. 3.6 for an illustration. Problem 3.7.4. Investigate the transitive graphs in G𝑑∗ that maximize 𝑝 𝑐 for each degree 𝑑, as well as the global maximum in G ∗ if this maximum exists. Are these maxima uniquely attained? Does 𝑝 𝑐 : G ∗ → [0, 1] attain its maximum uniquely at the 3-12 lattice (a.k.a. super-kagome lattice), which has 𝑝 𝑐 ≈ 0.7404207? When restricted to edge-transitive graphs, does 𝑝 𝑐 attain its unique maximum at the hexagonal lattice, which has 𝑝 𝑐 = 0.65270 . . . = 1 − 2 sin(𝜋/18)? One may wish to restrict attention to simple graphs. Similar questions have been investigated for self-avoiding walk by Grimmett and Li [GL20]. Acknowledgments This work was supported by NSF grant DMS-2246494. TH thanks Ariel Yadin for discussions on the history of the “cool inequality” and thanks Vincent Tassion for discussions on the plausibility of Conjecture 3.7.3. Both authors thank Geoffrey Grimmett, Russ Lyons, and Sébastien Martineau for helpful comments on an earlier version of the manuscript. 126 Chapter 4 127 UNIFORM FINITE PRESENTATION FOR GROUPS OF POLYNOMIAL GROWTH Joint with Tom Hutchcroft Abstract We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let 𝐺 be a group with finite generating set 𝑆 and let Gr(𝑟) be the volume of the ball of radius 𝑟 in the associated Cayley graph. For each 𝑘 ≥ 0, let 𝑅 𝑘 be the set of words of length at most 2 𝑘 in the free group 𝐹𝑆 that are equal to the identity in 𝐺, and let ⟨⟨𝑅 𝑘 ⟩⟩ be the normal subgroup of 𝐹𝑆 generated by 𝑅 𝑘 , so that the quotient map 𝐹𝑆 /⟨⟨𝑅 𝑘 ⟩⟩ → 𝐺 induces a covering map of the associated Cayley graphs that has injectivity radius at least 2 𝑘−1 − 1. Given a non-negative integer 𝑘, we say that (𝐺, 𝑆) has a new relation on scale k if ⟨⟨𝑅 𝑘+1 ⟩⟩ ≠ ⟨⟨𝑅 𝑘 ⟩⟩. We prove that for each 𝐾 < ∞ there exist constants 𝑛0 and 𝐶 depending only on 𝐾 and |𝑆| such that if Gr(3𝑛) ≤ 𝐾 Gr(𝑛) for some 𝑛 ≥ 𝑛0 , then there exist at most 𝐶 scales 𝑘 ≥ log2 (𝑛) on which 𝐺 has a new relation. We apply this result in another paper as part of our proof of Schramm’s locality conjecture in percolation theory. 4.1 Introduction It is a seminal theorem of Gromov [Gro81c] (see also [Kle10; Oza18]) that a finitely generated group has polynomial volume growth if and only if it is virtually nilpotent. This theorem and its extension to transitive graphs due to Trofimov [Tro85] are of foundational importance in the study of geometry and probability on transitive graphs, implying in particular that every transitive graph of polynomial growth has a well-defined volume growth dimension and that this dimension is an integer. In probability, these theorems are often used together with the isoperimetric inequality of Coulhon and Saloff-Coste [CS93] to prove results for general transitive graphs via a “structure vs. expansion” dichotomy: that is, proceeding by a case analysis according to whether the graph is virtually nilpotent or satisfies a 𝑑-dimensional isoperimetric inequality for every 𝑑 < ∞. Important results in probability employing the structure theory of transitive graphs in this way include Varopoulos’s theorem [Var86] that an infinite transitive graph is recurrent for simple random walks if and only if it has linear or quadratic volume growth, and Duminil-Copin, Goswami, Severo, Raoufi, and Yadin’s proof that transitive graphs admit a percolation phase transition if and only if they have superlinear growth [Dum+20c]. Over the last twenty years, an extensive literature in approximate group theory has been developed establishing finitary versions of Gromov’s theorem and Trofimov’s theorem, highlights of which include [BGT12b; ST10a; Hru12; TT21b; BGT11]. See [Bre14] for a detailed overview, [Toi20a] for a textbook introduction, and [Toi20b] for a concise survey. For groups, this theory culminated in the celebrated work of Breuillard, Green, and Tao [BGT12b], a special case of whose results can be stated1 as follows. Given a group 𝐺 and a finite generating set 𝑆, we write Gr(𝑟) = Gr𝐺,𝑆 (𝑟) for the cardinality of the ball of radius 𝑟 in the Cayley graph Cay(𝐺, 𝑆). Theorem 4.1.1 (Breuillard, Green, and Tao 2012). For each 𝐾 ≥ 1 there exist constants 𝑟 0 = 𝑟 0 (𝐾) and 𝐶 = 𝐶 (𝐾) such that the following holds. Let 𝐺 be a group with finite generating set 𝑆, and suppose that there exists 𝑟 ≥ 𝑟 0 such that Gr(3𝑟) ≤ 𝐾 Gr(𝑟). Then Gr(𝑚𝑟) ≤ 𝑚 𝐶 Gr(𝑟) for every 𝑚 ≥ 3 and there exists a finite normal subgroup 𝑄 ⊳ 𝐺 such that: 1. Every fibre of the projection 𝜋 : 𝐺 → 𝐺/𝑄 has diameter at most 𝐶𝑟. 2. 𝐺/𝑄 has a nilpotent normal subgroup 𝑁 of rank, step and index at most 𝐶. 3. The projection 𝑥 ↦→ 𝜋(𝑥) is a (1, 𝐶𝑟)-quasi-isometry from Cay(𝐺, 𝑆) to Cay(𝐺/𝑄, 𝜋(𝑆)). Here, we recall that a function 𝜙 : 𝑉1 → 𝑉2 between the vertex sets of two graphs 𝐺 1 = (𝑉1 , 𝐸 1 ) and 𝐺 2 = (𝑉2 , 𝐸 2 ) is said to be an (𝛼, 𝛽)-quasi-isometry (a.k.a. rough isometry) if 𝛼−1 𝑑 (𝑥, 𝑦) − 𝛽 ≤ 𝑑 (𝜙(𝑥), 𝜙(𝑦)) ≤ 𝛼𝑑 (𝑥, 𝑦) + 𝛽 for every 𝑥, 𝑦 ∈ 𝑉1 , and every vertex 𝑧 ∈ 𝑉2 is within distance at most 𝛽 of 𝜙(𝑉1 ). (The second property holds automatically if 𝜙 is surjective.) Informally, the Breuillard-Green-Tao theorem states that polynomial growth at one sufficiently large scale forces the group to have polynomial growth at every subsequent scale, and moreover to be metrically “well-modelled” by a nilpotent group at all larger scales. Similar theorems for 1 We state their theorem in a ‘metric’ form that is convenient for our applications, and which is adapted from Tessera and Tointon’s structure theorem for vertex-transitive graphs of polynomial growth [TT21b, Theorem 2.3]. Indeed, the statement given below is equivalent to the special case of their theorem in which the graph Γ is the Cayley graph of 𝐺, together with the growth bound of [BGT12b, Corollary 11.9]. 128 vertex-transitive graphs that are not necessarily Cayley graphs have recently been established in the work of Tessera and Tointon [TT21b; TT18]. These results have recently found many probabilistic applications, particularly for problems concerning families of transitive graphs (such as sequences of finite transitive graphs converging to an infinite graph); such problems often require estimates that are “uniform in the graph”, so that structure theoretic results invoked in their solutions must typically be finitary. Results proven using finitary structure theory include a finite-graph version of Varopoulos’s theorem [TT20b], universality theorems for cover time fluctuations [BHT22], locality of the critical probability for graphs of polynomial growth [CMT23b], non-triviality of the supercritical phase for percolation on finite transitive graphs [HT21b], and “gap at 1” theorems for the critical probability on infinite vertex transitive graphs [HT21b; Lyo+23b; PS23b]. Several of these works exploit finitary versions of the “structure vs. expansion” dichotomy provided by the finitary structure theory of [BGT12b; TT21b], with key technical difficulties arising from the fact that the same graph may exhibit different sides of this dichotomy at different scales. Uniform finite presentation Since virtually nilpotent groups are finitely presented, it is a consequence of Gromov’s theorem that every group of polynomial volume growth is finitely presented. The purpose of this paper is to prove a uniform version of this fact, stating roughly that every group of polynomial growth has a bounded number of scales witnessing a new relation after the first scale that polynomial growth is witnessed. This result is used in our work [EH23c] as part of our proof of Schramm’s locality conjecture for Bernoulli bond percolation [BNP11a], where it plays an important part in our “uniformization” of the methods of Contreras, Martineau, and Tassion [CMT21]. A comparison of our results with the previous literature is given at the end of this section. Let us now state our result formally. Let 𝐺 be a group with finite generating set 𝑆, so that 𝐺  𝐹𝑆 /𝑅 for some normal subgroup 𝑅 of 𝐹𝑆 . For each 𝑛 ≥ 0, let 𝑅𝑛 be the set of words of length at most 2𝑛 in the free group 𝐹𝑆 that are equal to the identity in 𝐺, and let ⟨⟨𝑅𝑛 ⟩⟩ be the normal subgroup of 𝐹𝑆 generated by 𝑅𝑛 , so that the quotient map 𝐹𝑆 /⟨⟨𝑅𝑛 ⟩⟩ → 𝐺 induces a covering map of the associated Cayley graphs that has injectivity radius at least 2𝑛−1 − 1 (see Lemma 4.5.6). We say that (𝐺, 𝑆) has a new relation on scale n if ⟨⟨𝑅𝑛+1 ⟩⟩ ≠ ⟨⟨𝑅𝑛 ⟩⟩. A finitely generated group 𝐺 is finitely presented if and only if it has a new relation on at most finitely many scales, so that the following theorem can indeed be thought of as stating that groups of polynomial growth are “uniformly finitely presented". 129 Theorem 4.1.2. For each 𝐾, 𝑘 < ∞ there exist constants 𝑟 0 = 𝑟 0 (𝐾) and 𝐶 = 𝐶 (𝐾, 𝑘) such that if 𝐺 is a group and 𝑆 is a finite generating set for 𝐺 with |𝑆| ≤ 𝑘 whose growth function Gr satisfies Gr(3𝑟) ≤ 𝐾 Gr(𝑟) for some integer 𝑟 ≥ 𝑟 0 then n o # 𝑛 ∈ N : 𝑛 ≥ log2 (𝑟) and (𝐺, 𝑆) has a new relation on scale 𝑛 ≤ 𝐶. Î𝑘 Remark 4.1.1. Considering the abelian group 𝑖=1 (Z/𝑛𝑖 Z) with its standard generating set, where 𝑛1 , . . . , 𝑛 𝑘 are arbitrary, we see that it is not possible to control on which scales we find a new relation; we only claim that the total number of scales on which we find a new relation is bounded. We will prove the following theorem about the number of times we find an “unexpected element” during a breadth-first exploration of a (not necessarily normal) subgroup of a group of polynomial growth; we will see in Section 4.5 that this theorem easily implies Theorem 8.1.1. Theorem 4.1.3 (Breadth-first exploration of subgroups). For each 𝐾 and 𝑘 there exist constants 𝑟 0 = 𝑟 0 (𝐾) and 𝐶 = 𝐶 (𝐾, 𝑘) such that the following holds. Let 𝐺 be a group with finite generating set 𝑆 satisfying |𝑆| ≤ 𝑘, let 𝐻 be a subgroup of 𝐺, and for each 𝑛 ≥ 1 let 𝐻𝑛 be the subgroup of 𝐻 generated by elements that have word length at most 2𝑛 in (𝐺, 𝑆). If 𝑟 ≥ 𝑟 0 is such that Gr(3𝑟) ≤ 𝐾 Gr(𝑟) then #{𝑛 ≥ log2 𝑟 : 𝐻𝑛+1 ≠ 𝐻𝑛 } ≤ 𝐶. Remark 4.1.2. We believe that it should be possible to take the constants in Theorems 4.1.3 and 8.1.1 to be independent of the size of the generating set. We do not pursue this here. Other previous results. A more classical way to quantify the sense in which a presentation is finite is through Dehn functions, filling length functions, and so-called isoperimetric functions (which do not refer to the same kind of isoperimetry mentioned in our above discussion of the structure vs. expansion dichotomy); see [Bri02] for an overview and [GHR03] for results on nilpotent groups. As far as we can tell, however, the literature on these notions focusses on asymptotic properties of a fixed group and is not suitable for the kind of uniform-in-the-group results we wish to prove. Besides this, the notion of uniform finite presentation we consider is also rather different from these notions in terms of Dehn functions etc. There is a striking resemblance between our theorem and the following theorem of Tao [Tao17b] (see also [TT17, Appendix A]), which also relies on the structure theory of Breuillard, Green, and Tao. 130 Theorem 4.1.4 ([Tao17b], Theorem 1.9). For each non-negative integer 𝑑 there exist constants 𝑚 0 and 𝐶 depending only on 𝑑 such that if 𝐺 is a group, 𝑆 is a finite, symmetric generating set for 𝐺 containing the identity and satisfying |𝑆 𝑛𝑚 | ≤ 𝑚 𝑑 |𝑆 𝑛 | for some integers 𝑛 ≥ 1 and 𝑚 ≥ 𝑚 0 then there exists a continuous, piecewise-linear, non-decreasing function 𝑓 : [0, ∞) → [0, ∞) with 𝑓 (0) = 0 that has at most 𝐶 pieces, each of which has slope equal to an integer bounded by 𝐶, such that |𝑆 𝑘𝑛𝑚 | log 𝑛𝑚 − 𝑓 (log 𝑘) ≤ 𝐶 |𝑆 | for every integer 𝑘 ≥ 1. Informally, this theorem states that, once we witness polynomial growth on a sufficiently large scale, the log-log plot of the growth function is well-approximated by a continuous, piecewiselinear function with bounded, integer valued slopes and a bounded number of “kinks” connecting the different pieces. Naively, one might hope that our bounded number of scales on which a new relation occurs are in correspondence with Tao’s bounded number of scales on which the growth function has a “kink” in its log-log plot. Unfortunately this is not the case, at least when one allows generating sets of unbounded size: one can have a new relation without having a kink, and can have a kink without having a new relation. Indeed, as explained in [Tao17b, Example 1.11], taking 1 Z Z © ª 𝐺 = ­­0 1 Z®® «0 0 1 ¬ and 1 [−𝑁, 𝑁] [−𝑁 3 , 𝑁 3 ] © ª 𝑆 = ­­0 1 [−𝑁, 𝑁] ®® 0 1 «0 ¬ for a large integer 𝑁 yields   3 log 𝑛 ± 𝑂 (1)  |𝑆 𝑛 |  = log  |𝑆|  4 log 𝑛 − log 𝑁 ± 𝑂 (1)  1≤𝑛≤𝑁 𝑛 > 𝑁, so that this example’s growth function has a kink at scale log2 𝑁. On the other hand the pair (𝐺, 𝑆) does not have a new relation at any at 𝑘 ≥ 3, and in particular does not have a new relation on the scale where it has a kink when 𝑁 is large. Indeed, the relations of (𝐺, 𝑆) are generated by the usual relations for the Heisenberg group       110 100 101 010 , 011 = 010 , 001  110 010 001  101 010 001   = 101 010 001  110 010 001 001  001  , and 131 100 011 001  101 010 001   = 101 010 001  100 011 001  together with the following three sets of relations relating the extra generators in 𝑆 to the standard generators:       1 𝑎±1 𝑐 0 1 𝑏 0 0 1   1𝑎 𝑐 0 1 𝑏±1 0 0 1 1 𝑎 𝑐±1 0 1 𝑏 0 0 1   =   = 1 ±1 0 0 1 0 0 0 1 1𝑎 𝑐 0 1 𝑏 0 0 1 = 10 0 0 1 ±1 00 1 1𝑎 𝑐 0 1 𝑏 0 0 1   1𝑎 𝑐 0 1 𝑏 0 0 1 1 0 ±1 01 0 00 1 : 𝑎, 𝑏 ∈ [−𝑁, 𝑁], 𝑐 ∈ [−𝑁 3 , 𝑁 3 ] ,   3 3 3 3 : 𝑎, 𝑏 ∈ [−𝑁, 𝑁], 𝑐 ∈ [−𝑁 , 𝑁 ] ,   : 𝑎, 𝑏 ∈ [−𝑁, 𝑁], 𝑐 ∈ [−𝑁 , 𝑁 ] . These relations all have word length at most five in (𝐺, 𝑆), so that the example has the desired properties. Conversely, taking the direct product 𝐺 × Z/𝑁 with generating set 𝑆 × {−1, 0, 1} yields an example where there is a new relation at scale log2 𝑁 but where the growth function does not have any kinks. About the proof. It is natural to describe the proof of Theorems 4.1.3 and 8.1.1 “backwards”, as a sequence of reductions, although we have written it “forwards” as a sequence of extensions and generalizations. In this backwards description, the “first” step (which is the last part of the paper) is to reduce from groups of polynomial growth to nilpotent groups of bounded step using the Breuillard-Green-Tao theorem. Next, this statement about nilpotent groups is in turn reduced to an analogous statement about breadth-first exploration of discrete subgroups in a Carnot group, a simply connected nilpotent Lie group carrying the additional structure of a stratification and homogeneous left-invariant metric. Nilpotent groups are related to Carnot groups for example by Pansu’s theorem [Pan83; BL13], which states roughly that the large-scale geometry of a finitely generated nilpotent group is well-modelled by an appropriate Carnot group equipped with a leftinvariant homogeneous metric. Finally, this statement about Carnot groups is reduced to a statement about vector spaces using the close connection between the discrete subgroups of a simply connected nilpotent Lie group and the additive bracket-closed subgroups of its associated Lie algebra. This step of the reduction is the most involved part of the paper, with the connection between additive and multiplicative lattices being developed at length in Section 4.4. This ends the chain of reductions, and leaves us with a problem we must actually solve directly: Bounding the number of times we find an “unexpected element” of a discrete subgroup of R𝑑 as we explore the subgroup with an increasing family of convex, symmetric sets. This is done in Section 4.3 as an application of Minkowski’s second theorem, a classical result in the geometry of numbers. Let us stress again that we have described the argument here in the opposite order to the way we carry it out, so that the result about subgroups of R𝑑 is the first thing we prove. 132 Disclaimer: Neither author is an expert in approximate groups, Lie theory, or the geometry of numbers. As such, it is likely that we have included a larger amount of detail in the proofs than would be considered necessary by experts, or have re-derived known results from scratch. While we have attempted to provide appropriate attribution to the intermediate results of the paper as much as possible, we would be happy to receive comments and corrections from experts. Remark 4.1.3. Since the present paper first appeared, two relevant papers by Tessera and Tointon have since appeared. The first paper [TT24] establishes a stronger version of Theorem 8.1.1 as a corollary, with optimal bounds on the number of scales at which new relations appear that do not depend on the degree. The proof is very different than that given here. The second paper [TT23] allows us to replace the “small tripling” hypothesis Gr(3𝑟) ≤ 𝐾 Gr(𝑟) in Theorems 4.1.3 and 8.1.1 with the more natural hypothesis of “small doubling” Gr(2𝑟) ≤ 𝐾 Gr(𝑟) because, as the title of the paper says: Small doubling implies small tripling on large scales. 4.2 Background on nilpotent groups and Lie groups In this section we review the relevant background material and establish some notational conventions. We have included a rather thorough account of the basic theory with the hope that our paper can be easily understood by probabilists. Given a group 𝐺, the commutator of two elements 𝑥, 𝑦 ∈ 𝐺 is defined by [𝑥, 𝑦] = 𝑥𝑦𝑥 −1 𝑦 −1 . The lower central series of 𝐺 is defined recursively by 𝐺 1 = 𝐺 and 𝐺 𝑖+1 = [𝐺 𝑖 , 𝐺] for each 𝑖 ≥ 1, where we write [ 𝐴, 𝐵] := {[𝑎, 𝑏] : 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} for subsets 𝐴 and 𝐵 of 𝐺. The group 𝐺 is said to be nilpotent if 𝐺 𝑖+1 = {id} for all sufficiently large 𝑖, with the minimal such 𝑖 denoted by 𝑠 and known as the step of 𝐺. A group is said to be virtually nilpotent if it has a nilpotent subgroup of finite index. Given 𝑠 ≥ 1 and a set 𝑆, the free step 𝑠 nilpotent group 𝑁 𝑠,𝑆 is defined to be the quotient of the free group 𝐹𝑆 by the step-𝑠 nilpotency relations, which state that all iterated commutators of length at least 𝑠 + 1 are equal to the identity. The free step 𝑠 nilpotent group 𝑁 𝑠,𝑆 can also be defined up to unique 𝑆-preserving isomorphism by the universal property that it is nilpotent of step at most 𝑠, contains 𝑆, and every function from 𝑆 to a nilpotent group of step at most 𝑠 can be uniquely extended to a homomorphism from 𝑁 𝑠,𝑆 to that group. (Nilpotent) Lie groups and the Baker-Campbell-Hausdorff formula Recall that a (real) Lie group is a group that is also a finite-dimensional real smooth manifold, in such a way that the group operations of multiplication and inversion are smooth maps 𝐺 × 𝐺 → 𝐺 and 𝐺 → 𝐺. By Gleason, Montgomery, and Zippin’s solution to Hilbert’s fifth problem [MZ52; Gle52], one can equivalently define a Lie group as a group that is also a finite-dimensional 133 topological manifold with continuous multiplication and inversion operations; such a group carries a unique smooth structure compatible with its algebraic structure. More generally, Yamabe [Yam50] proved that every locally compact, connected topological group is a projective limit of Lie groups (possibly of divergent dimension). These facts underlie the ubiquity of Lie groups in the scaling limit theory of discrete groups, and in particular are used directly in Gromov’s original proof of his polynomial growth theorem [Gro81c]. Further background on these topics can be found in [Tao14]. For our purposes, Lie groups become relevant primarily via a theorem of Pansu, which allows us to approximate the balls in the Cayley graph of a nilpotent group in terms of the balls in a certain left-invariant homogeneous metric on a Carnot group; this is explained in Section 4.2. Lie algebras. A Lie algebra 𝔤 is a vector space equipped with a binary operation, the Lie bracket [·, ·] : 𝔤 × 𝔤 → 𝔤, that is bilinear, antisymmetric ([𝑋, 𝑌 ] = −[𝑌 , 𝑋] for all 𝑋, 𝑌 ∈ 𝔤), and satisfies the Jacobi identity ([𝑋, [𝑌 , 𝑍]] + [𝑌 , [𝑍, 𝑋]] + [𝑍, [𝑋, 𝑌 ]] = 0 for all 𝑋, 𝑌 , 𝑍 ∈ 𝔤). A subset 𝐴 of a Lie algebra is said to be bracket-closed if [𝑋, 𝑌 ] ∈ 𝐴 for every 𝑋, 𝑌 ∈ 𝐴. For ease of reading, we will loosely follow the convention that points in a Lie algebra are denoted using upper-case letters, while points in a Lie group are denoted using lower-case letters. We will also assume without further comment that all Lie algebras are finite-dimensional. To each Lie group 𝐺, we can associate a Lie algebra 𝔤 arising from the tangent space at the identity; the details of this construction are not important to us and can be found in any textbook on the subject. In the concrete case that 𝐺 is a Lie subgroup of a general linear group GL𝑛 for some 𝑛 ≥ 1, the affine space 𝐼 + 𝔤 is precisely the tangent space at the identity to 𝐺 in the space of all 𝑛 × 𝑛 matrices, so that 𝔤 is a Lie subalgebra (i.e. a bracket-closed linear subspace) of the Lie algebra 𝔤𝔩 𝑛 of all 𝑛 × 𝑛 matrices with Lie bracket defined by the commutator [𝑋, 𝑌 ] = 𝑋𝑌 − 𝑌 𝑋. (In particular, 𝔤𝔩 𝑛 is the Lie algebra associated to the Lie group GL𝑛 .) In fact this case is not particularly special: Ado’s theorem states that every Lie algebra is isomorphic to a Lie subalgebra of 𝔤𝔩 𝑛 for some 𝑛 ≥ 1 [Tao14, Chapter 2.3]. (Ado’s theorem does not imply that every Lie group is isomorphic to a Lie subgroup of a general linear group, although it does imply a “local” version of the same claim.) The fundamental theorems of Lie (see e.g. [Tao14, Chapter 2.5.1]) state in particular that there is a one-to-one correspondence between (isomorphism classes of) Lie algebras and simply connected Lie groups. On the other hand, Lie groups that are connected but not simply connected have universal covers which are simply connected Lie groups with the same Lie algebra. The lower central series of the Lie algebra 𝔤 is defined recursively by 𝔤1 = 𝔤 and 𝔤𝑖+1 = [𝔤𝑖 , 𝔤] for each 𝑖 ≥ 1, where if 𝐴 and 𝐵 are two subsets of 𝔤 then we write [ 𝐴, 𝐵] = {[𝑎, 𝑏] : 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵}. 134 It is a simple consequence of the Jacobi identity that [𝔤𝑖 , 𝔤 𝑗 ] ⊆ 𝔤𝑖+ 𝑗 for every 𝑖, 𝑗 ≥ 1 [Wan23, Lemma 1.4.3]. The Lie algebra 𝔤 is said to be nilpotent if 𝔤𝑖+1 = {0} for all sufficiently large 𝑖; the minimal such 𝑖 is called the step of 𝔤 and is usually denoted 𝑠. Nilpotence of a Lie group is defined as for any other group (meaning that the lower central series terminates at the identity subgroup); a connected Lie group is nilpotent if and only if its corresponding Lie algebra is nilpotent. Lie polynomials. A Lie monomial of degree 𝑑 in the terms 𝑋1 , 𝑋2 , . . . , 𝑋𝑛 ∈ 𝔤 is an expression obtained by taking iterated Lie brackets of these terms in some way, so that the sum over 𝑖 of the total number of times 𝑋𝑖 appears is 𝑑. In other words, a Lie monomial of degree 1 in the terms 𝑋1 , . . . , 𝑋𝑛 is an expression of the form 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) = 𝑋𝑖 for some 1 ≤ 𝑖 ≤ 𝑛, while each Lie monomial of degree 𝑑 can be written 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) = [𝐿 1 (𝑋1 , . . . , 𝑋𝑛 ), 𝐿 2 (𝑋1 , . . . , 𝑋𝑛 )] for some Lie monomials 𝐿 1 and 𝐿 2 whose degrees sum to 𝑑. A Lie polynomial 𝑃(𝑋1 , 𝑋2 , . . . , 𝑋𝑛 ) in elements 𝑋1 , 𝑋2 , . . . , 𝑋𝑛 ∈ 𝔤 is a linear combination of Lie monomials; it is said to be homogeneous of degree 𝑑 if every Lie monomial in the linear combination has degree 𝑑. Thus, homogeneous Lie polynomials of degree 𝑑 obey the scaling transformation 𝑃(𝜆𝑋1 , . . . , 𝜆𝑋𝑛 ) = 𝜆 𝑑 𝑃(𝑋1 , . . . , 𝑋𝑛 ). The exponential map. Given a Lie group 𝐺 and associated Lie algebra 𝔤, there is a canonically defined exponential map exp : 𝔤 → 𝐺, which for Lie subgroups of GL𝑛 coincides with ordinary matrix exponentiation. (We will omit the general definition of the exponential map; everything we need to know about it will be captured by the Baker-Campbell-Hausdorff formula.) The exponential map is smooth, and is a diffeomorphism in a neighbourhood of the identity, but might not be injective or surjective. The Baker-Campbell-Hausdorff (BCH) formula [Tao14, Chapter 1.2.5] states that if 𝐺 is a Lie group with Lie algebra 𝔤 then there exists an open neighbourhood 𝑈 of the origin such that ∞ ∑︁ 1 1 1 𝐿 𝑖 (𝑋, 𝑌 ) log [exp 𝑋 exp 𝑌 ] = 𝑋 + 𝑌 + [𝑋, 𝑌 ] + [𝑋, [𝑋, 𝑌 ]] − [𝑌 , [𝑋, 𝑌 ]] + · · · = 2 12 12 𝑖=1 for all 𝑋, 𝑌 ∈ 𝑈, where each 𝐿 𝑖 is a homogeneous Lie polynomial of degree 𝑖 with rational coefficients and where we write log for the inverse of the exponential map on exp(𝑈). (The Lie polynomials 𝐿 𝑖 appearing in the BCH formula are universal and do not depend on the choice of Lie group 𝐺.) When 𝐺 is a simply connected nilpotent Lie group, all terms with 𝑖 > 𝑠 are identically zero, the exponential function exp : 𝔤 → 𝐺 is defined globally, and the BCH formula holds for all 𝑋, 𝑌 ∈ 𝔤. This lets us define the BCH product on the Lie algebra 𝔤 as   1 1 1 𝑋 ⋄ 𝑌 := log 𝑒 𝑋 𝑒𝑌 = 𝑋 + 𝑌 + [𝑋, 𝑌 ] + [𝑋, [𝑋, 𝑌 ]] − [𝑌 , [𝑋, 𝑌 ]] + · · · , 2 12 12 135 so that (𝔤, ⋄) is isomorphic to 𝐺 as a Lie group. Note that (𝑎𝑋) ⋄ (𝑏𝑋) = (𝑎 + 𝑏) 𝑋 for every 𝑋 ∈ 𝔤 and 𝑎, 𝑏 ∈ R, so that 0 is both the additive and BCH identity and that (−𝑋) is both the additive and BCH inverse of 𝑋. Example 4.2.1. The Heisenberg group 𝐻 and its Lie algebra 𝔥 are given by 1 R R © ª 𝐻 = ­­0 1 R®® «0 0 1 ¬ and 0 R R © ª 𝔥 = ­­0 0 R®® , «0 0 0 ¬ and 1 𝑎 𝑐 0 𝑎 𝑐 − 𝑎𝑏 2 ª © ª © log ­­0 1 𝑏 ®® = ­­0 0 𝑏 ®® . 0 ¬ «0 0 1 ¬ «0 0 with exponential map and logarithm 0 𝑎 𝑐 1 𝑎 𝑐 + 𝑎𝑏 2 ª © ª © exp ­­0 0 𝑏 ®® = ­­0 1 𝑏 ®® 1 ¬ «0 0 0 ¬ «0 0 Note that the exponential map here is just the usual matrix exponential, which for 𝑋 ∈ 𝔥 satisfies 𝑒 𝑋 = 𝐼 + 𝑋 + 21 𝑋 2 . The Lie bracket on 𝔥 is given by  0 𝑎 𝑐 0 𝑥 𝑧  0 0 𝑎𝑦 − 𝑥𝑏 © ª © ª © ª ­ ® ­ ® ­ 0 ®® . ­0 0 𝑏 ® , ­0 0 𝑦 ® = ­0 0    0 0 0  0 0 0 0 ¬ « ¬ « ¬   «0 0 Since 𝐻 is step-2 nilpotent, the BCH multiplication on 𝔥 is given by  0 𝑎 𝑐 0 𝑥 𝑧  0 𝑎 𝑐 0 𝑥 𝑧 0 𝑎 𝑐 0 𝑥 𝑧 ª © ª © ª © ª © ª © ª 1 © ­0 0 𝑏 ® ⋄ ­0 0 𝑦 ® = ­0 0 𝑏 ® + ­0 0 𝑦 ® + ­0 0 𝑏 ® , ­0 0 𝑦 ® ­ ® ­ ® ­ ® ­ ® 2 ­ ® ­ ®   «0 0 0 ¬ «0 0 0¬ «0 0 0 ¬ «0 0 0¬  «0 0 0 ¬ «0 0 0 ¬ 0 𝑎 + 𝑥 𝑐 + 𝑧 + 𝑎𝑦−𝑏𝑥 2 ª © ­ ®. = ­0 0 𝑏+𝑦 ® 0 0 0 ¬ « We will see in Example 4.4.1 that the non-integer rational coefficient 1/2 appearing in this expression leads to complications when comparing lattices in 𝐻 and 𝔥. Remark 4.2.1. It follows from the BCH formula that 1 1 log[𝑒 𝑋 𝑒𝑌 𝑒 −𝑋 𝑒 −𝑌 ] = [𝑋, 𝑌 ] + [𝑋, [𝑋, 𝑌 ]] + [𝑌 , [𝑋, 𝑌 ]] + · · · 2 2 for 𝑋 and 𝑌 in a neighbourhood of the origin in 𝔤 (all of 𝔤 when 𝐺 is nilpotent and simply connected). As such, the BCH commutator and the Lie bracket agree to first order as 𝑋, 𝑌 → 0, but are not exactly equal unless 𝐺 is nilpotent of step at most 2. 136 We will also make use of the Zassenhaus formula [CMN12], a dual form of the BCH formula which states in particular that if 𝐺 is 𝑠-step nilpotent then 1 −1 1 𝑒 𝑋+𝑌 = 𝑒 𝑋 𝑒𝑌 𝑒 − 2 [𝑋,𝑌 ] 𝑒 6 (2[𝑌 ,[𝑋,𝑌 ]]+[𝑋,[𝑋,𝑌 ]]) 𝑒 24 ( [[[𝑋,𝑌 ],𝑋],𝑋]+3[[[𝑋,𝑌 ],𝑋],𝑌 ]+3[[[𝑋,𝑌 ],𝑌 ],𝑌 ]) · · · = 𝑒 𝑋 𝑒𝑌 𝑒 𝐿 2 (𝑋,𝑌 ) 𝑒 𝐿 3 (𝑋,𝑌 ) · · · 𝑒 𝐿 𝑠 (𝑋,𝑌 ) ˜ ˜ ˜ (4.2.1) for every 𝑋, 𝑌 ∈ 𝔤, where 𝐿˜ 𝑖 is a homogeneous Lie polynomial of degree 𝑖 with rational coefficients for each 𝑖 ≥ 2. Subgroups and lattices. Let 𝐺 be a simply connected nilpotent Lie group with Lie algebra 𝔤. For each set 𝐴 ⊆ 𝔤, we define L ( 𝐴) to be the smallest Lie subalgebra of 𝔤 containing 𝐴. The exponential map identifies closed connected subgroups of 𝐺 with Lie subalgebras of 𝔤, so that every closed connected subgroup of 𝐺 is itself a simply connected nilpotent Lie group. (For general Lie groups, the image under the exponential map of a Lie subalgebra might not be closed, but for simply connected nilpotent groups it is always closed since the exponential map is a diffeomorphism.) As such, for each subset 𝐴 of 𝐺, the intersection of all closed connected subgroups of 𝐺 containing 𝐴 is a closed connected subgroup of 𝐺 that is equal to exp(L (log 𝐴)). We write C ( 𝐴) for this minimal closed connected subgroup of 𝐺 containing 𝐴. Theorem 4.2.2 (Mal’cev). If 𝐺 is a simply connected nilpotent Lie group and 𝐻 is a closed subgroup of 𝐺 then 𝐺/𝐻 is compact if and only if 𝐻 is not contained in any proper closed connected subgroup of 𝐺. In particular, C (𝐻)/𝐻 is compact. (The quotients 𝐺/𝐻 and C (𝐻)/𝐻 appearing here are topological spaces, and do not carry group structures in general.) We call a subgroup Γ of a simply connected nilpotent Lie group 𝐺 a lattice in 𝐺 if it is discrete with compact quotient 𝐺/Γ. (Note that discrete subgroups of Hausdorff topological groups are automatically closed.) Remark 4.2.2. For a general Lie group, a lattice is defined to be a discrete subgroup for which the quotient admits a finite left-invariant measure (a.k.a. Haar measure); for simply connected nilpotent Lie groups this is equivalent to 𝐺/Γ being compact by Mal’cev’s theorem. This theorem also gives several further characterisations of a discrete subgroup being a lattice that we omit since we do not use them. A further theorem of Mal’cev [Rag72, Theorem 2.12] states that a simply connected nilpotent Lie group 𝐺 admits a lattice if and only if its Lie algebra 𝔤 admits a basis 𝑒 1 , . . . , 𝑒 𝑑 for which the Í structure constants (𝑇𝑖,𝑘𝑗 )𝑖, 𝑗,𝑘 , defined by [𝑒𝑖 , 𝑒 𝑗 ] = 𝑘 𝑇𝑖,𝑘𝑗 𝑒 𝑘 , are rational. It is a consequence of 137 this theorem [Wan23, Proposition 2.3.7] that if Γ is a lattice in a simply connected nilpotent Lie group then there exist (additive) lattices Λ− and Λ+ in 𝔤 such that Λ− ⊆ log Γ ⊆ Λ+ . We will prove significantly stronger versions of this fact in Section 4.4. Carnot groups and Pansu’s theorem A Carnot group is a simply connected nilpotent Lie group 𝐺 of some step 𝑠 whose Lie algebra 𝔤 is equipped with a decomposition 𝔤 = 𝑉1 ⊕ 𝑉2 · · · ⊕ 𝑉𝑠 for some non-trivial linear subspaces 𝑉1 , . . . , 𝑉𝑠 such that [𝑉1 , 𝑉𝑖 ] = 𝑉𝑖+1 for every 1 ≤ 𝑖 < 𝑠 and [𝑉1 , 𝑉𝑠 ] = {0}. It can be shown that for such a decomposition we have moreover that [𝑉𝑖 , 𝑉 𝑗 ] ⊆ 𝑉𝑖+ 𝑗 for every 𝑖, 𝑗 ≥ 1, where 𝑉𝑖+ 𝑗 := {0} for 𝑖 + 𝑗 > 𝑠. A decomposition 𝔤 = 𝑉1 ⊕ · · · ⊕𝑉𝑠 satisfying these conditions is known as a stratification of 𝔤; the subspace 𝑉1 , which generates 𝔤 as a Lie algebra, is known as the horizontal subspace. Not every nilpotent lie algebra admits a stratification. While a Carnot group consists of both a nilpotent Lie group and a choice of stratification of its Lie algebra, we will nevertheless write e.g. “let 𝐺 be a Carnot group” when this does not cause confusion. Given a Carnot group 𝐺 and a real number 𝜆 > 0 the dilation maps 𝛿𝜆 : 𝔤 → 𝔤 and 𝐷 𝜆 : 𝐺 → 𝐺 are defined by 𝛿𝜆 (𝑥 1 + 𝑥2 + · · · + 𝑥 𝑠 ) = 𝜆𝑥 1 + 𝜆2 𝑥 2 + · · · 𝜆 𝑠 𝑥 𝑠 and 𝐷 𝜆 (𝑥) = exp(𝛿𝜆 (log(𝑥))), where we write 𝑥 = 𝑥 1 + 𝑥 2 + · · · + 𝑥 𝑠 for the decomposition of 𝑥 ∈ 𝔤 associated to the stratification 𝔤 = 𝑉1 ⊕𝑉2 ⊕ · · · ⊕𝑉𝑠 . These dilation maps satisfy the semigroup properties 𝛿𝜆𝜇 = 𝛿𝜆 𝛿 𝜇 = 𝛿 𝜇 𝛿𝜆 and 𝐷 𝜆𝜇 = 𝐷 𝜆 𝐷 𝜇 = 𝐷 𝜇 𝐷 𝜆 for every 𝜆, 𝜇 > 0. It is a consequence of the Baker-Campbell-Hausdorff formula and the fact that [𝑉𝑖 , 𝑉 𝑗 ] ⊆ 𝑉𝑖+ 𝑗 that 𝐷 𝜆 is a Lie group automorphism of 𝐺 for every 𝜆 > 0. A metric 𝑑 : 𝐺 × 𝐺 → [0, ∞) on a Carnot group is said to be left-invariant and homogeneous if 𝑑 (𝑧𝑥, 𝑧𝑦) = 𝑑 (𝑥, 𝑦) and 𝑑 (𝐷 𝜆 𝑥, 𝐷 𝜆 𝑦) = 𝜆𝑑 (𝑥, 𝑦) for every 𝑥, 𝑦, 𝑧 ∈ 𝐺 and 𝜆 > 0. Note that the abelian group R𝑑 is a Carnot group with 𝑉1 = R𝑑 , and the left-invariant homogeneous metrics on R𝑑 seen as a Carnot group are equivalent to norms on R𝑑 . In general, left-invariant homogeneous metrics are closely analogous to norms, and also have the property that they are determined by the unit ball 𝐵 around the origin: 𝑑 (𝑥, 𝑦) = inf{𝜆 : 𝑥 −1 𝑦 ∈ 𝐷 𝜆 (𝐵)}. In particular, the ball {𝑥 : 𝑑 (0, 𝑥) ≤ 𝜆} is equal to 𝐷 𝜆 (𝐵). 138 We now introduce the free step-𝑠 nilpotent Lie algebra and the free step-𝑠 Carnot group, referring the reader to [BLU07, Chapter 14.1] for proofs the objects we discuss here are well-defined. Let 𝑆 be a finite set. The free step-𝑠 nilpotent Lie algebra 𝔣𝑠,𝑆 is defined to be the unique-up-to-𝑆preserving-isomorphism nilpotent Lie algebra of step 𝑠 that is generated by 𝑆 and is such that if 𝔤 is any nilpotent Lie algebra of step at most 𝑠 and 𝜙 : 𝑆 → 𝔤 is any function, then there exists a unique Lie algebra homomorphism 𝔣𝑠,𝑆 → 𝔤 extending 𝜙. The Lie algebra 𝔣𝑠,𝑆 may be equipped with a canonical stratification defined in terms of Hall bases, making its associated BCH-multiplication Lie group 𝐺 𝑠,𝑆 = (𝔣𝑠,𝑆 , ⋄) into a Carnot group known as the free step-𝑠 Carnot group or free step-𝑠 nilpotent Lie group over 𝑆; see [BLU07, Chapters 14.1 and 14.2] for details. (It is convenient to define the free step-𝑠 nilpotent Lie group over 𝑆 via BCH multiplication so that it contains the set 𝑆.) Moreover, this stratification 𝔣𝑠,𝑆 = 𝑉1 ⊕ · · · ⊕ 𝑉𝑠 has the property that 𝑉1 is equal to the linear span of 𝑆. Finally, if we define Γ𝑠,𝑆 to be the subgroup of 𝐺 𝑠,𝑆 generated by 𝑆, then Γ𝑠,𝑆 is a lattice in 𝐺 𝑠,𝑆 that is isomorphic (via an 𝑆-preserving isomorphism) to the discrete free step-𝑠 nilpotent group 𝑁 𝑠,𝑆 . Indeed, the fact that Γ𝑠,𝑆 is discrete in 𝐺 𝑠,𝑆 can be proven using Mal’cev’s theorem on rational structure constants, since the structure constants in the Hall basis are all equal to 1, while Γ is a lattice since 𝑆 generates 𝐺 𝑠,𝑆 as a Lie group. (The fact that Γ𝑠,𝑆 is discrete can also be proven using the techniques of Section 4.4.) Finally, the fact that Γ𝑠,𝑆 is isomorphic to 𝑁 𝑠,𝑆 can be deduced straightforwardly from the relevant universal properties since every torsion-free finitely generated nilpotent group can be embedded as a lattice in a nilpotent Lie group, sometimes known as the Mal’cev completion of the group [Rag72, Theorem 2.18]. In his thesis [Pan83], Pansu proved that Cayley graphs of finitely generated nilpotent groups converge under rescaling to Carnot groups equipped with certain left-invariant homogeneous metrics known as sub-Finsler metrics. (We will not need the definition of sub-Finsler metrics in this paper.) Note that the Carnot group arising in this limit might not be isomorphic to the Mal’cev completion of the relevant nilpotent group, and indeed the Mal’cev completion might not admit a stratification. However, if 𝑁 is a torsion-free nilpotent group with finite generating set 𝑆 and the lie algebra 𝔤 of the Mal’cev completion 𝐺 happens to be simply connected and admit a stratification 𝔤 = 𝑉1 ⊕ · · · ⊕ 𝑉𝑠 with log 𝑆 ⊆ 𝑉1 , then there exists a left-invariant homogeneous metric 𝑑𝐺 on 𝐺 such that 𝑑 𝑆 (𝑥, 𝑦) = (1 ± 𝑜(1))𝑑𝐺 (𝑥, 𝑦) as 𝑑 𝑆 (𝑥, 𝑦) → ∞, (4.2.2) where 𝑑 𝑆 denotes the word metric on 𝑁. It follows in particular that (𝑁, 𝑛1 𝑑 𝑆 ) converges to (𝐺, 𝑑𝐺 ) in the Gromov-Hausdorff sense as 𝑛 → ∞. See [BL13; Tas22] for details and quantitative refinements of this theorem. 139 4.3 Exploring abelian lattices with convex sets In this section we prove the following theorem, which will eventually be used to prove our main theorems by reduction to the abelian case. Theorem 4.3.1. Let 𝑑 ≥ 1, let Λ be a discrete subgroup of R𝑑 , and let 𝐾1 ⊆ 𝐾2 ⊆ · · · be an increasing sequence of non-empty, symmetric, convex sets in R𝑑 . For each 𝑛 ≥ 1 let Λ𝑛 = spanZ (Λ ∩ 𝐾𝑛 ) so that Λ1 ⊆ Λ2 ⊆ · · · is an increasing sequence of subgroups of Λ. Then #{𝑛 : Λ𝑛+1 ≠ Λ𝑛 } ≤ 𝑑 + 1 + 𝑑 ∑︁ ⌊log2 ℓ!⌋. ℓ=1 Remark 4.3.1. By Stirling’s formula, the upper bound appearing here is asymptotic to 21 𝑑 2 log2 𝑑 as 𝑑 → ∞. We have not investigated the optimality of this bound. We will deduce this theorem as a consequence of Minkowski’s second theorem [Cas97, p. 203], which states that if Λ is a lattice in R𝑑 with 𝑑 ≥ 1 and 𝐾 is a non-empty, symmetric convex subset of R𝑑 then vol(R𝑑 /Λ) 1≤ ≤ 𝑑!, (4.3.1) Î𝑑 2𝑑 vol(𝐾) 𝑖=1 𝜆𝑖 (Λ, 𝐾) where  𝜆𝑖 (Λ, 𝐾) = inf 𝜆 > 0 : 𝜆𝐾 ∩ Λ contains at least 𝑖 linearly independent vectors for each 1 ≤ 𝑖 ≤ 𝑑. For our purposes, the most important feature of (4.3.1) is that the expression Î𝑑 2𝑑 vol(𝐾) 𝑖=1 𝜆𝑖 (Λ, 𝐾) is determined by 𝐾 and Λ ∩ 𝐾 whenever Λ ∩ 𝐾 has real span equal to R𝑑 . Indeed, if Λ is a lattice in R𝑑 then every fundamental domain for Λ has volume vol(R𝑑 /Λ) and if Λ1 ⊆ Λ2 are two lattices in R𝑑 then Λ1 is a finite-index subgroup of Λ2 with index [Λ2 : Λ1 ] = vol(R𝑑 /Λ1 ) . vol(R𝑑 /Λ2 ) Thus, if Λ1 ⊆ Λ2 are two lattices and 𝐾 is a symmetric convex set such that Λ1 ∩ 𝐾 and Λ2 ∩ 𝐾 are equal and both have real linear span equal to R𝑑 then Minkowski’s second theorem implies that [Λ2 : Λ1 ] ≤ 𝑑!. Proof of Theorem 4.3.1. It suffices to prove that if 𝐾1 ⊆ 𝐾2 ⊆ · · · ⊆ 𝐾𝑛 is an increasing sequence of non-empty, symmetric convex subsets of R𝑑 and Λ is a discrete subgroup of R𝑑 such that the Í𝑑 subgroups Λ𝑖 := spanZ (Λ∩ 𝐾𝑖 ) satisfy Λ𝑖+1 ≠ Λ𝑖 for every 1 ≤ 𝑖 < 𝑛 then 𝑛 ≤ 𝑑 +1+ ℓ=0 ⌊log2 ℓ!⌋. 𝑑 We may also assume without loss of generality that Λ = Λ𝑛 is a lattice in R , replacing Λ with Λ𝑛 140 and R𝑑 with the subspace spanned by Λ𝑛 otherwise. Fix such a pair Λ and (𝐾1 , . . . , 𝐾𝑛 ) and for each 0 ≤ ℓ ≤ 𝑑 let 𝑖ℓ be minimal such that the real span of Λ𝑖ℓ has dimension at least ℓ, setting 𝑖 𝑑+1 = 𝑛 + 1 for notational convenience. Since Λ2 ≠ Λ1 , we must have that 𝑖0 = 1 and 𝑖 1 ∈ {1, 2}. For each 1 ≤ ℓ ≤ 𝑑 define 𝑉ℓ to be the real span of Λ𝑖ℓ , so that Λ 𝑗 is a lattice in 𝑉ℓ for every 1 ≤ ℓ ≤ 𝑑 and 𝑖ℓ ≤ 𝑗 < 𝑖ℓ+1 . (Note that 𝑉ℓ might have dimension strictly larger than ℓ, in which case 𝑖ℓ+1 = 𝑖ℓ .) Suppose that 0 ≤ ℓ ≤ 𝑑 is such that 𝑖ℓ+1 ≥ 𝑖ℓ + 2. In this case, the subgroups Λ𝑖ℓ and Λ𝑖ℓ+1 −1 are both lattices in 𝑉ℓ with Λ𝑖ℓ ∩ 𝐾𝑖ℓ = Λ𝑖ℓ+1 −1 ∩ 𝐾𝑖ℓ and with Λ𝑖ℓ ∩ 𝐾𝑖ℓ having real span equal to 𝑉ℓ . As such, it follows by Minkowski’s second theorem that vol(𝑉ℓ /Λ𝑖ℓ ) ≤ ℓ! vol(𝑉ℓ /Λ𝑖ℓ+1 −1 ) [Λ𝑖ℓ+1 −1 : Λ𝑖ℓ ] = for each 1 ≤ ℓ ≤ 𝑑 such that 𝑖ℓ+1 ≥ 𝑖ℓ + 2. Now, using that [Λ𝑖ℓ+1 −1 : Λ𝑖ℓ ] = 𝑖ℓ+1 −2 Ö [Λ 𝑗+1 : Λ 𝑗 ] ≥ 2𝑖ℓ+1 −1−𝑖ℓ 𝑗=𝑖ℓ it follows that 𝑖ℓ+1 − 𝑖ℓ ≤ 1 + ⌊log2 ℓ!⌋ for every 1 ≤ ℓ ≤ 𝑑 such that 𝑖ℓ+1 ≥ 𝑖ℓ + 2. Since the same inequality also holds trivially when 𝑖ℓ+1 < 𝑖ℓ + 2, it follows that 𝑛 = 𝑖 𝑑+1 − 𝑖0 = 𝑑 ∑︁ (𝑖ℓ+1 − 𝑖ℓ ) ≤ 𝑑 + 1 + ℓ=0 𝑑 ∑︁ ⌊log2 ℓ!⌋ ℓ=1 as claimed. 4.4 □ Additive and multiplicative subgroups of nilpotent Lie algebras Our goal in this section is to clarify the relationship between lattices in a simply connected nilpotent Lie group and its associated Lie algebra. As mentioned above, closed, connected subgroups of a simply connected nilpotent Lie group 𝐺 are in bijection with Lie subalgebras of the Lie algebra 𝔤, which are precisely the closed, connected, bracket-closed additive subgroups of 𝔤. Without the assumption of connectivity, the exponential map need not interact this nicely with the subgroup structure of 𝐺: it is possible to have subgroups 𝐻 of 𝐺 for which log 𝐻 is not an additive subgroup of 𝔤 and to have bracket-closed additive subgroups of 𝔤 whose image under the exponential is not a subgroup of 𝐺. Example 4.4.1. Let the Heisenberg group 𝐻 and its Lie algebra 𝔥 be as in Example 4.2.1. The set of all elements of 𝐻 whose matrix entries are integers is a lattice in 𝐻 whose logarithm is given by     0 𝑎 𝑐 1 Z Z    ©  © ª  ª 1 ­ ® ­ ® log ­0 1 Z® = ­0 0 𝑏 ® : 𝑎, 𝑏 ∈ Z, 𝑐 ∈ Z, 2𝑐 = 𝑎𝑏 mod 2 .   2    0 0 0  0 0 1 « ¬ « ¬  141 This is not an additive subgroup of 𝔥 since 0 1 0 0 0 0 0 1 0 1 Z Z © ª © ª © ª © ª ­0 0 0® + ­0 0 1® = ­0 0 1® ∉ log ­0 1 Z® . ­ ® ­ ® ­ ® ­ ® 0 0 0 0 0 0 0 0 0 0 0 1 « ¬ « ¬ « ¬ « ¬ Similarly, the set of all elements of 𝔥 whose matrix entries are integers is a bracket-closed additive subgroup of 𝔥 whose exponential is not a subgroup of 𝐻. We call a subgroup 𝐻 of 𝐺 harmonious if log 𝐻 is an additive subgroup of 𝔤 that is bracket-closed in the sense that [log 𝑥, log 𝑦] ∈ log 𝐻 for every 𝑥, 𝑦 ∈ 𝐻. (This terminology is not standard.) Thus, harmonious subgroups of 𝐺 are those that most closely mimic the behaviour of the connected subgroups of 𝐺 under the exponential map. We call a lattice in 𝐺 that is also a harmonious subgroup of 𝐺 a harmonious lattice in 𝐺, noting that if Γ is a harmonious lattice in 𝐺 then Λ = log Γ is a bracket-closed lattice in 𝔤. The remainder of this section is devoted to proving the following theorem, which states intuitively that subgroups of 𝐺 and bracket-closed additive subgroups of 𝔤 are, in some sense, “equivalent up to bounded index”. This result allows us to deduce various statements about lattices in simply connected nilpotent Lie groups from analogous statements for vector spaces (i.e., statements in the geometry of numbers), which are classical. Note that the theorem does not require Γ to be discrete. Recall that if 𝐴 ⊆ 𝔤 and 𝜆 ∈ R then we define 𝜆 · 𝐴 = {𝜆𝑎 : 𝑎 ∈ 𝐴}, so that if Λ is an additive subgroup of 𝔤 then (𝑚𝜆) · Λ ⊆ 𝜆 · Λ for every 𝜆 ∈ R and 𝑚 ∈ Z. We also write B( 𝐴) for the smallest bracket-closed set containing 𝐴. Theorem 4.4.2. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤. There exist positive integers 𝐶1 and 𝐶2 depending only on 𝑠 such that if Γ is a subgroup of 𝐺 then the sets      1 · spanZ (log Γ) H− (Γ) := exp 𝐶1 · spanZ (log Γ) and H+ (Γ) := exp 𝐶1 · B 𝐶1 are harmonious subgroups of 𝐺 such that 𝐶1 · log Γ ⊆ log H− (Γ) ⊆ log Γ ⊆ log H+ (Γ) ⊆ 1 · log Γ. 𝐶2 In corollary 4.4.13 we show moreover that if Γ is discrete then the harmonious subgroups H− (Γ) and H+ (Γ) have index bounded by a constant depending only on the step and dimension of 𝐺. 142 Example 4.4.3. We continue to analyze the Heisenberg group as studied in Examples 4.2.1 and 4.4.1. Although the subgroup Γ of 𝐻 consisting of those elements of 𝐻 with integer matrix entries is not harmonious, we can write 1 2Z Z 1 Z Z 1 Z © ª © ª © ­0 1 2Z® ⊆ Γ = ­0 1 Z® ⊆ ­0 1 ­ ® ­ ® ­ 0 0 1 0 0 1 « ¬ « ¬ «0 0 1 2 Zª Z ®® 1¬ with the two outer two lattices being harmonious subgroups of 𝐻. Moreover, these subgroups arise naturally from the original subgroup Γ as 1 2Z Z © ª
­0 1 2Z® = exp span (2 · log Γ) and Z ­ ® 0 0 1 « ¬ 1 Z © ­0 1 ­ «0 0 1 2 Zª
Z ®® = exp spanZ (log Γ) . 1¬ Similar remarks apply to the bracket-closed additive lattice in 𝔥 consisting of those elements of 𝔥 with integer matrix entries. The methods used to prove Theorem 4.4.2 are based on those of Breuillard and Green [BG11]. In particular, we will make use of the following lemma of Tessera and Tointon [TT18] that is also proved using the methods of [BG11]. Lemma 4.4.4. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤. There exists an integer constant 𝐶 = 𝐶 (𝑠) such that if Λ is a bracket-closed additive subgroup of the Lie algebra 𝔤 then exp(𝐶 · Λ) is a harmonious subgroup of 𝐺. Proof. This is essentially [TT18, Lemma 4.3]. It is not stated that 𝐶 · Λ is bracket-closed, but this is obvious since [𝐶 · Λ, 𝐶 · Λ] = 𝐶 2 · [Λ, Λ] ⊆ 𝐶 · Λ. □ Remark 4.4.1. The constant 𝐶1 appearing in Theorem 4.4.2 will be taken to be a multiple of the constant 𝐶 appearing in lemma 4.4.4. This will be important in the proof of proposition 4.5.1. We begin by stating the following lemma of Lazard [Laz54] as presented in [BG11, Lemmas 5.2 and 5.3]. This lemma was first applied to the structure theory of approximate groups in the work of Fisher, Katz, and Peng [FKP09]. Given a simply connected nilpotent Lie group 𝐺, we define 𝑥 𝛼 = exp(𝛼 log 𝑥) for every 𝑥 ∈ 𝐺 and 𝛼 ∈ R. Lemma 4.4.5 (Lazard). Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤. There exists ℓ ≥ 1 and sequences of rational numbers 𝛼1 , . . . , 𝛼ℓ , 𝛽1 , . . . , 𝛽ℓ , 𝛾1 , . . . , 𝛾ℓ , and 143 𝛿1 , . . . , 𝛿ℓ depending only on 𝑠 such that exp(log 𝑥 + log 𝑦) = 𝑥 𝛼1 𝑦 𝛽2 · · · 𝑥 𝛼ℓ 𝑦 𝛽ℓ and exp([log 𝑥, log 𝑦]) = 𝑥 𝛾1 𝑦 𝛿1 · · · 𝑥 𝛾ℓ 𝑦 𝛿ℓ for every 𝑥, 𝑦 ∈ 𝐺. Corollary 4.4.6 (Expansion of sums). Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤. For each 𝑛 ≥ 2 there exists ℓ ≥ 1, a sequence of rational numbers 𝛼1 , . . . , 𝛼ℓ , and a sequence of indices 𝑖1 , . . . , 𝑖ℓ ∈ {1, . . . , 𝑛}, all depending only on 𝑠 and 𝑛, such that exp(log 𝑥1 + log 𝑥 2 + · · · + log 𝑥 𝑛 ) = 𝑥𝑖𝛼11 𝑥𝑖𝛼22 · · · 𝑥𝑖𝛼ℓℓ for every 𝑥 1 , . . . , 𝑥 𝑛 ∈ 𝐺. Corollary 4.4.7. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤, let Γ be a subgroup of 𝐺 and let Λ = log Γ. For each 𝑛 ≥ 1, there exists a natural number 𝐶 = 𝐶 (𝑠, 𝑛) such that if 𝑋1 , . . . , 𝑋𝑛 ∈ 𝐶 · Λ then 𝑋1 + · · · + 𝑋𝑛 ∈ Λ. Proof. Let 𝐶 = 𝐶 (𝑠, 𝑛) be the least common multiple of the denominators of the numbers 𝛼1 , . . . , 𝛼ℓ appearing in corollary 4.4.6 when written in reduced form. □ Lemma 4.4.8. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤, let Γ be a subgroup of 𝐺 and let Λ = log Γ. There exists a natural number 𝐶 = 𝐶 (𝑠) such that n o 𝐿(𝑋1 , . . . , 𝑋𝑛 ) : 𝑋1 , . . . , 𝑋𝑛 ∈ 𝐶 𝑛−1 · Λ ⊆ Λ, for every multilinear Lie monomial 𝐿. (A Lie monomial 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) is multilinear when each variable appears at most once.) Proof. Let 𝐶 = 𝐶 (𝑠) be the least common multiple of the denominators of the numbers 𝛾1 , . . . , 𝛾ℓ and 𝛿1 , . . . , 𝛿ℓ appearing in Lazard’s lemma when written in reduced form. We will prove the claim by induction on 𝑛, the case 𝑛 = 1 being vacuous. If 𝑛 > 1 then 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) = [𝐿 1 (𝑋𝜋(1) , . . . , 𝑋𝜋( 𝑗) ), 𝐿 2 (𝑋𝜋( 𝑗+1) , . . . , 𝑋𝜋(𝑛) )] 144 for some 1 ≤ 𝑗 < 𝑛, some multilinear Lie monomials 𝐿 1 , 𝐿 2 and some permutation 𝜋 : {1, . . . , 𝑛} → {1, . . . , 𝑛}. We may assume without loss of generality that 𝜋 is the identity permutation. By the induction hypothesis, if 𝑋1 , . . . , 𝑋𝑛 ∈ 𝐶 𝑛−2 ·Λ then 𝐿 1 (𝑋1 , . . . , 𝑋 𝑗 ), 𝐿 2 (𝑋 𝑗+1 , . . . , 𝑋𝑛 ) ∈ Λ. As such, if 𝑋1 , . . . , 𝑋𝑛 ∈ 𝐶 𝑛−1 · Λ then we have by multlilinearity that 𝐿 1 (𝑋1 , . . . , 𝑋 𝑗 ) = 𝐶 𝑗 𝐿 1 (𝐶 −1 𝑋1 , . . . , 𝐶 −1 𝑋 𝑗 ) ∈ 𝐶 𝑗 · Λ ⊆ 𝐶 · Λ and 𝐿 2 (𝑋 𝑗+1 , . . . , 𝑋𝑛 ) = 𝐶 𝑛− 𝑗 𝐿 2 (𝐶 −1 𝑋 𝑗+1 , . . . , 𝐶 −1 𝑋𝑛 ) ∈ 𝐶 𝑛− 𝑗 · Λ ⊆ 𝐶 · Λ. On the other hand, if 𝑋, 𝑌 ∈ 𝐶 · Λ then 𝑒 𝛾 𝑗 𝑋 , 𝑒 𝛿 𝑗 𝑌 ∈ Γ for every 1 ≤ 𝑗 ≤ ℓ, so that exp([𝑋, 𝑌 ]) = 𝑒 𝛾1 𝑋 𝑒 𝛿1𝑌 · · · 𝑒 𝛾ℓ 𝑋 𝑒 𝛿ℓ 𝑌 ∈ Γ and hence that [𝑋, 𝑌 ] ∈ Λ. It follows that if 𝑋1 , . . . , 𝑋𝑛 ∈ 𝐶 𝑛−1 · Λ then 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) ∈ Λ as claimed. □ Corollary 4.4.9. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤, let Γ be a subgroup of 𝐺 and let Λ = log Γ. There exists a natural number 𝐶 = 𝐶 (𝑠) such that n o 𝑑 (𝑑−1) 𝐿(𝑋1 , . . . , 𝑋𝑛 ) : 𝑋1 , . . . , 𝑋𝑛 ∈ Λ, 𝑋𝑖 ∈ 𝐶 · Λ for some 1 ≤ 𝑖 ≤ 𝑛 ⊆ Λ, for every Lie monomial 𝐿 of degree 𝑑 that depends on every variable. Proof. It suffices without loss of generality to consider the case that 𝐿 is multilinear. Let 𝐶 be the constant from lemma 4.4.8. Let 𝑋1 , . . . , 𝑋𝑛 ∈ Λ, and let 𝐿 (𝑋1 , . . . , 𝑋𝑛 ) be a multlilinear Lie monomial depending on every variable. Such a Lie monomial necessarily has degree 𝑑 = 𝑛. For each 1 ≤ 𝑖 ≤ 𝑛 we can write 𝐿(𝑋1 , . . . , 𝑋𝑛 ) = 𝐿(𝐶 (𝑛−1) 𝑋1 , . . . , 𝐶 (𝑛−1) 𝑋𝑖−1 , 𝐶 −(𝑛−1) 𝑋𝑖 , 𝐶 (𝑛−1) 𝑋𝑖+1 , . . . , 𝐶 (𝑛−1) 𝑋𝑛 ). 2 If 𝑋𝑖 ∈ 𝐶 𝑛(𝑛−1) · Λ = 𝐶 (𝑛−1) +(𝑛−1) · Λ then 𝐶 −(𝑛−1) 𝑋𝑖 ∈ 𝐶 (𝑛−1) · Λ, so that the claim follows from lemma 4.4.8. □ 2 2 Lemma 4.4.10. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 with Lie algebra 𝔤. There exists a constant 𝐶 depending only on 𝑠 such that if Γ is a subgroup of 𝐺 and we write Λ = log Γ then 𝐶 · Λ is bracket-closed and 𝑋 + 𝑌 ∈ Λ for every 𝑋 ∈ 𝐶 · Λ and 𝑌 ∈ Λ. 145 Proof. We recall the Zassenhaus formula, which states for simply connected nilpotent Lie groups that exp(𝑋 + 𝑌 ) = 𝑒 𝑋 𝑒𝑌 𝑒 𝐿 2 (𝑋,𝑌 ) 𝑒 𝐿 3 (𝑋,𝑌 ) · · · 𝑒 𝐿 𝑠 (𝑋,𝑌 ) for every 𝑋, 𝑌 ∈ 𝔤, where, for each 𝑖 ≥ 1, 𝐿 𝑖 is a homogeneous Lie polynomial of degree 𝑖 of the form 𝑟𝑖 ∑︁ 𝐿 𝑖 (𝑋, 𝑌 ) = 𝑎𝑖, 𝑗 𝐿 𝑖, 𝑗 (𝑋, 𝑌 ), 𝑗=1 where 𝑎𝑖, 𝑗 are rational numbers and 𝐿 𝑖, 𝑗 are Lie monomials depending on both variables. Let 𝐶1 be the least common multiple of the denominators of the rational numbers 𝑎𝑖, 𝑗 and let 𝐶2 be the least common multiple of the constants 𝐶 (𝑠, 2), 𝐶 (𝑠, 3), . . . , 𝐶 (2, max𝑖 𝑟𝑖 ) appearing in corollary 4.4.7. By corollary 4.4.9, there exists a constant 𝐶3 = 𝐶3 (𝑠) such that if 𝑋 ∈ 𝐶3 · Λ and 𝑌 ∈ Λ then 𝐿 𝑖, 𝑗 (𝑋, 𝑌 ) ∈ Λ for every 1 ≤ 𝑖 ≤ 𝑠 and 1 ≤ 𝑗 ≤ 𝑟𝑖 . Let 𝐶 = 𝐶1𝐶2𝐶3 . Since 𝐿 𝑖, 𝑗 depends on 𝑋, it follows that if 𝑋 ∈ 𝐶 · Λ then 𝐿 𝑖, 𝑗 (𝑋, 𝑌 ) = (𝐶1𝐶2 ) 𝑑𝑖, 𝑗 𝐿 𝑖, 𝑗 ((𝐶1𝐶2 ) −1 𝑋, 𝑌 ) ∈ (𝐶1𝐶2 ) · Λ for every 𝑦 ∈ Λ, where 𝑑𝑖, 𝑗 is the degree of 𝑋 in 𝐿 𝑖, 𝑗 . It follows in particular that if 𝑋 ∈ 𝐶 · Λ then 𝑎𝑖, 𝑗 𝐿 𝑖, 𝑗 (𝑋, 𝑌 ) ∈ 𝐶2 · Λ for every 1 ≤ 𝑖 ≤ 𝑠 and 1 ≤ 𝑗 ≤ 𝑟𝑖 and hence by corollary 4.4.7 that 𝐿 𝑖 (𝑋, 𝑌 ) ∈ Λ for every 𝑋 ∈ 𝐶 · Λ and 𝑌 ∈ Λ. Since Γ is a subgroup of 𝐺, it follows by the Zassenhaus formula that 𝑋 + 𝑌 ∈ Λ for every 𝑋 ∈ 𝐶 · Λ and 𝑌 ∈ Λ as claimed. Moreover, if 𝑋, 𝑌 ∈ 𝐶 · Λ then [𝑋, 𝑌 ] = 𝐶 [𝑋, 𝐶 −1𝑌 ] ∈ 𝐶 · Λ since [𝑋 ′, 𝑌 ′] ∈ Λ for every 𝑋 ′ ∈ 𝐶 · Λ and 𝑌 ′ ∈ Λ, so that 𝐶 · Λ is bracket-closed as claimed. □ We are now ready to prove Theorem 4.4.2. Proof of Theorem 4.4.2. We begin by proving the claim concerning H− (Γ). Let 𝐶−1 = 𝐶−1 (𝑠) be the constant from lemma 4.4.10, let 𝐶0 = 𝐶0 (𝑠) be the constant from lemma 4.4.4, and let 𝐶1 = 𝐶−1𝐶0 . Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠, let 𝔤 be the Lie algebra of 𝐺, let Γ be a subgroup of 𝐺 and let Λ = log Γ. lemma 4.4.10 implies that 𝐶1 · Λ is bracket-closed and that 𝐶−1 · Λ + Λ ⊆ Λ, and it follows by induction on the number of terms in a linear combination that spanZ (𝐶−1 · Λ) = 𝐶−1 · spanZ (Λ) is contained in Λ. Since the Z-span of a bracket-closed set is bracket-closed, it follows that spanZ (𝐶−1 · Λ) is a bracket-closed additive subgroup of 𝔤 and hence by lemma 4.4.4 that spanZ (𝐶1 · Λ) = 𝐶0 · spanZ (𝐶−1 · Λ) is a bracket-closed additive subgroup of 𝔤 whose exponential H− (Γ) is a harmonious subgroup of 𝐺 satisfying the required set inclusion 𝐶1 · log Γ ⊆ log H− (Γ) ⊆ log Γ. Now consider the set H+ (Γ) defined by       1 1 · spanZ (Λ) = exp 𝐶1 · spanZ B ·Λ . H+ (Γ) := exp 𝐶1 · B 𝐶1 𝐶1 146   Since spanZ B 𝐶11 · Λ is a bracket-closed additive subgroup of 𝔤 and 𝐶0 divides 𝐶1 , lemma 4.4.4 implies that H+ (Γ) is a harmonious subgroup of 𝐺. Moreover, log H+ (Γ) trivially contains spanZ Λ. As such, it remains only to prove that there exists a constant 𝐶2 = 𝐶2 (𝑠) such that log H+ (Γ) ⊆ 𝐶12 · Λ. Since 𝐶1 · spanZ (Λ) is bracket-closed and [𝔤𝑖 , 𝔤 𝑗 ] ⊆ 𝔤𝑖+ 𝑗 we have that  !     1 1 1 · spanZ (Λ) ∩ 𝔤𝑖 , · spanZ (Λ) ∩ 𝔤 𝑗 ⊆ Λ ∩ 𝔤𝑖+ 𝑗 𝑛 𝑚 𝐶12 𝑛𝑚 for every pair of integers 𝑛, 𝑚 ≥ 0 and 1 ≤ 𝑖, 𝑗 ≤ 𝑠. Thus, it follows by induction on 𝑖 that   1 1 · spanZ (Λ) ∩ 𝔤𝑖 ⊆ 3𝑖−2 · (Λ ∩ 𝔤𝑖 ) B 𝐶1 𝐶1 for every 𝑖 ≥ 1 and hence that   1 1 B · spanZ (Λ) ⊆ 3𝑠−2 · Λ. 𝐶1 𝐶1 This implies that the claim holds with 𝐶2 = 𝐶13(𝑠−1) . □ Comparing additive and multiplicative indices In this section we prove bounds on the index [H+ (Γ) : H− (Γ)]. We will deduce these bounds from the following proposition, which lets us compare additive and multiplicative indices in the Lie algebra of a simply connected nilpotent Lie group. Proposition 4.4.11 (Index sandwich). Let Γ1 ⊆ Γ2 be lattices in a simply connected nilpotent Lie group 𝐺 with Lie algebra 𝔤, and suppose that Λ1 ⊆ Λ2 are additive lattices in 𝔤. 1. If Λ1 ⊆ log Γ1 ⊆ log Γ2 ⊆ Λ2 then [Γ2 : Γ1 ] ≤ [Λ2 : Λ1 ]. 2. If log Γ1 ⊆ Λ1 ⊆ Λ2 ⊆ log Γ2 then [Γ2 : Γ1 ] ≥ [Λ2 : Λ1 ]. In particular, if Γ1 and Γ2 are harmonious in 𝐺 then [Γ2 : Γ1 ] = [log Γ2 : log Γ1 ]. The proof of this proposition will require the following classical fact. Proposition 4.4.12 (Compatibility of Haar measures). Let 𝐺 be a simply connected nilpotent Lie group and let 𝔤 be the Lie algebra of 𝐺. If 𝜇 is a translation-invariant, locally finite measure on 𝔤 (i.e., a Lebesgue measure) then the pushforward of 𝜇 by the exponential map is a locally finite measure on 𝐺 that is both left and right invariant (i.e., a bi-invariant Haar measure). 147 Proof of proposition 4.4.12. It suffices to prove that for every 𝑋 ∈ 𝔤, the maps ℓ𝑋 : 𝑌 ↦→ 𝑋 ⋄ 𝑌 and 𝑟 𝑋 : 𝑌 ↦→ 𝑌 ⋄ 𝑋 preserve the Lebesgue measure on 𝔤. For this, it suffices to prove that the total derivatives 𝐷ℓ𝑋 and 𝐷𝑟 𝑋 have | det(𝐷ℓ𝑋 )| = | det(𝐷𝑟 𝑋 )| = 1 at every 𝑌 ∈ 𝔤. To prove this, it suffices to prove that both 𝐷ℓ𝑋 and 𝐷𝑟 𝑋 can be expressed as the sum of the identity and a nilpotent linear transformation, which can be done via an explicit computation with the BCH formula. For details see e.g. [Wan23, Proposition 2.1.1]. □ Proof of proposition 4.4.11. We start by constructing large sets in 𝔤 that have “small boundary-tovolume ratio” in both the additive and multiplicative senses. (That is, the sets we construct will yield a Følner sequence for both addition and BCH multiplication on 𝔤.) If 𝐺 were assumed to be a Carnot group we could use the logarithms of balls in a homogeneous left-invariant metric; we will perform a similar construction for a general Lie group. Let (𝔤𝑖 )𝑖≥0 be the lower central series of 𝔤, and for each 𝑖 ≥ 0 let 𝑉𝑖 be such that 𝔤𝑖 = 𝑉𝑖 ⊕ 𝔤𝑖+1 , so that we can write 𝔤 = 𝑉1 ⊕ 𝑉2 ⊕ · · · ⊕ 𝑉𝑠 . For notational convenience we also write 𝑉𝑖 = {0} for 𝑖 > 𝑠. In contrast to the Carnot case, it is not necessarily the case that 𝑉1 generates 𝔤 as a Lie algebra or that [𝑉𝑖 , 𝑉 𝑗 ] ⊆ 𝑉𝑖+ 𝑗 , but we do have that É𝑠 [𝑉𝑖 , 𝑉 𝑗 ] ⊆ [𝔤𝑖 , 𝔤 𝑗 ] ⊆ 𝔤𝑖+ 𝑗 = 𝑘=𝑖+ 𝑗 𝑉𝑘 for every 𝑖, 𝑗 ≥ 1. Fix an isomorphism of vector spaces 𝑑 𝔤  R for some 𝑑 ≥ 1 and let ∥ · ∥ ∞ be the associated ∞-norm on 𝔤. For each 𝜆 > 0 let 𝐹𝜆 := {𝑋 ∈ 𝔤 : max 𝜆−𝑖 ∥ 𝑋𝑖 ∥ ∞ ≤ 1} = {𝑋 ∈ 𝔤 : max ∥ 𝑋𝑖 ∥ 1/𝑖 ∞ ≤ 𝜆}, 𝑖 𝑖 Í where we write 𝑋 = 𝑖 𝑋𝑖 for the decomposition of 𝑋 induced by the decomposition 𝔤 = 𝑉1 ⊕ 𝑉2 ⊕ · · · ⊕ 𝑉𝑠 . The volume of 𝐹𝜆 satisfies vol(𝐹𝜆 ) = 𝜆 𝑞 Íℓ for an integer 𝑞 ≥ 1, which can be expressed as 𝑞 = 𝑖=1 𝑖 dim(𝑉𝑖 ) (this is equal to the homogeneous dimension of 𝐺). For each 𝑋 ∈ 𝔤 we define ((𝑋)) = inf{𝜆 > 0 : 𝑋 ∈ 𝐹𝜆 } = max ∥ 𝑋𝑖 ∥ 1/𝑖 , 𝑖 so that ((𝑋)) ≤ ∥ 𝑋 ∥ 1/𝑖 ∞ for every 1 ≤ 𝑖 ≤ 𝑠 and 𝑋 ∈ 𝔤𝑖 with equality if 𝑋 ∈ 𝑉𝑖 . Note that this is not a norm on 𝔤 since it does not satisfy ((𝜆𝑋)) = 𝜆((𝑋)) for all 𝑋 ∈ 𝔤 and 𝜆 > 0. (Rather, it scales under a certain graded dilation map as in the Carnot case.) Moreover, unlike the metrics we considered on Carnot groups, ((·)) will not be left-invariant in general. Nevertheless, it does trivially satisfy the triangle inequality in the form 1/𝑖 ((𝑋 + 𝑌 )) = max ∥ 𝑋𝑖 + 𝑌𝑖 ∥ 1/𝑖 ∞ ≤ max (∥ 𝑋𝑖 ∥ ∞ + ∥𝑌𝑖 ∥ ∞ ) 𝑖 𝑖 1/𝑖 ≤ max (∥ 𝑋𝑖 ∥ 1/𝑖 ∞ + ∥𝑌𝑖 ∥ ∞ ) ≤ ((𝑋)) + ((𝑌 )). (4.4.1) 𝑖 148 As with norms, writing ((𝑋)) = (((𝑋 + 𝑌 ) − 𝑌 )) yields the reverse inequality ((𝑋 + 𝑌 )) ≥ ((𝑋)) − ((𝑌 )), (4.4.2) so that 𝐹𝜆−((𝑌 )) ⊆ 𝐹𝜆 + 𝑌 ⊆ 𝐹𝜆+((𝑌 )) for every 𝑌 ∈ 𝔤 and 𝜆 ≥ ((𝑌 )). We will need similar inequalities for the BCH product ((𝑋 ⋄ 𝑌 )). Since the Lie bracket is bilinear and 𝔤 is finite-dimensional, there exists a constant 𝐶1 such that ∥ [𝑋, 𝑌 ] ∥ ∞ ≤ 𝐶1 ∥ 𝑋 ∥ ∞ ∥𝑌 ∥ ∞ for every 𝑋, 𝑌 ∈ 𝔤. This implies that there exists a constant 𝐶2 such that 1/(𝑖+ 𝑗) (( [𝑋𝑖 , 𝑌 𝑗 ])) ≤ ∥ [𝑋𝑖 , 𝑌 𝑗 ] ∥ ∞ 1/(𝑖+ 𝑗) ≤ 𝐶1 1/(𝑖+ 𝑗) ∥ 𝑋𝑖 ∥ ∞ 1/(𝑖+ 𝑗) ∥𝑌 𝑗 ∥ ∞ ≤ 𝐶2 ((𝑋𝑖 )) 𝑖/(𝑖+ 𝑗) ((𝑌 𝑗 )) 𝑗/(𝑖+ 𝑗) for every 𝑋𝑖 ∈ 𝑉𝑖 and 𝑌 𝑗 ∈ 𝑉 𝑗 . Together with (4.4.1) this implies that there exists a constant 𝐶3 such that   𝑠−1 1−𝜃 𝜃 1 (([𝑋, 𝑌 ])) ≤ 𝐶3 max ((𝑋)) ((𝑌 )) : ≤ 𝜃 ≤ 𝑠 𝑠 for every 𝑋, 𝑌 ∈ 𝔤. It follows by induction on 𝑘 ≥ 2 that if 𝐿 (𝑋, 𝑌 ) is any Lie monomial of degree 𝑘 ≥ 2 then, writing 𝐿 (𝑋, 𝑌 ) = [𝐿 1 (𝑋, 𝑌 ), 𝐿 2 (𝑋, 𝑌 )] for two Lie monomials of degree 𝑎, 𝑘 − 𝑎 < 𝑘,   𝑠−1 1−𝜃 𝜃 1 ((𝐿 (𝑋, 𝑌 ))) ≤ 𝐶3 max ((𝐿 1 (𝑋, 𝑌 ))) ((𝐿 2 (𝑋, 𝑌 ))) : ≤ 𝜃 ≤ 𝑠 𝑠   𝑠𝑘 − 1 1 ≤ 𝐶3𝑘−1 max ((𝑋)) 1−𝜃 ((𝑌 )) 𝜃 : 𝑘 ≤ 𝜃 ≤ 𝑠 𝑠𝑘 for every 𝑋, 𝑌 ∈ 𝔤. (The case that one of the monomials 𝐿 1 or 𝐿 2 has degree one, and so is equal to 𝑋 or 𝑌 , must be checked separately.) We deduce from this together with (4.4.1) and the definition of BCH multiplication that there exists a constant 𝐶4 such that n 𝑠𝑠 − 1 o 1 (4.4.3) ((𝑋 ⋄ 𝑌 )) ≤ ((𝑋)) + ((𝑌 )) + 𝐶4 max ((𝑋)) 1−𝜃 ((𝑌 )) 𝜃 : 𝑠 ≤ 𝜃 ≤ 𝑠 𝑠𝑠 for every 𝑋, 𝑌 ∈ 𝔤. Since −𝑌 is both the additive and BCH inverse of 𝑌 , we can write 𝑋 = 𝑋⋄𝑌 ⋄(−𝑌 ) to obtain that there exists a constant 𝐶5 such that the complementary inequality n 1 𝑠𝑠 − 1 o ((𝑋)) ≤ ((𝑋 ⋄ 𝑌 )) + ((𝑌 )) + 𝐶4 max ((𝑋 ⋄ 𝑌 )) 1−𝜃 ((𝑌 )) 𝜃 : 𝑠 ≤ 𝜃 ≤ 𝑠 𝑠𝑠 n 1 𝑠2𝑠 − 1 o ≤ ((𝑋 ⋄ 𝑌 )) + 𝐶5 ((𝑌 )) + 𝐶5 max ((𝑋)) 1−𝜃 ((𝑌 )) 𝜃 : 2𝑠 ≤ 𝜃 ≤ (4.4.4) 𝑠 𝑠2𝑠 holds for every 𝑋, 𝑌 ∈ 𝔤. We now use the sets 𝐹𝜆 to prove the claim about indices. Suppose that Λ is an additive lattice in 𝔤 and that Γ is a lattice in 𝐺. Let 𝐾Λ be a fundamental domain for Λ in 𝔤 and let 𝐾Γ be a fundamental 149 domain for Γ in 𝐺. Since 𝐾Λ ∪ log 𝐾Γ is compact, max{((𝑌 )) : 𝑌 ∈ 𝐾Λ ∪ log 𝐾Γ } is finite. As such, it follows from (4.4.1) and (4.4.2) that there exist positive constants 𝐶 and 𝜀 (depending on 𝐾Λ and 𝐾Γ ) such that ((𝑋)) − 𝐶 ≤ ((𝑋 + 𝑌 )) ≤ ((𝑋)) + 𝐶 and ((𝑋)) − 𝐶 ((𝑋)) 1−𝜀 − 𝐶 ≤ ((𝑋 ⋄ 𝑌 )) ≤ ((𝑋)) + 𝐶 ((𝑋)) 1−𝜀 + 𝐶 for every 𝑥 ∈ 𝔤 and 𝑦 ∈ 𝐾Λ ∪ log 𝐾Γ . Since the sets (𝑋 + 𝐾Λ : 𝑋 ∈ Λ) and (𝑋 ⋄ log 𝐾Γ : 𝑋 ∈ log Γ) each cover 𝔤 (with distinct sets having measure-zero intersection), it follows that Ø 𝐹𝜆−𝐶 ⊆ (𝑋 + 𝐾Λ ) ⊆ 𝐹𝜆+𝐶 𝑋∈𝐹𝜆 ∩Λ and 𝐹𝜆−𝐶𝜆1− 𝜀 −𝐶 ⊆ Ø (𝑋 ⋄ log 𝐾Γ ) ⊆ 𝐹𝜆+𝐶𝜆1− 𝜀 +𝐶 𝑋∈𝐹𝜆 ∩log Γ for every 𝜆 ≥ 1 such that 𝜆 − 𝐶𝜆1−𝜀 − 𝐶 > 0. Taking volumes and using that addition and BCH multiplication are both measure-preserving, we deduce that ! (𝜆 − 𝐶) 𝑞 ≤ |𝐹𝜆 ∩ Λ| · vol(𝐾Λ ) = vol Ø (𝑋 + 𝐾Λ ) ≤ (𝜆 + 𝐶) 𝑞 𝑋∈𝐹𝜆 ∩Λ and © Ø ª (𝜆 − 𝐶𝜆1−𝜀 − 𝐶) 𝑞 ≤ |𝐹𝜆 ∩ log Γ| · vol(log 𝐾Γ ) = vol ­ (𝑋 ⋄ log 𝐾Γ ) ® ≤ (𝜆 + 𝐶𝜆1−𝜀 + 𝐶) 𝑞 « 𝑋∈𝐹𝜆 ∩log Γ ¬ for every 𝜆 ≥ 1 such that 𝜆 − 𝐶𝜆1−𝜀 − 𝐶 > 0, so that 𝜆𝑞 𝜆→∞ |𝐹𝜆 ∩ Λ| vol(𝐾Λ ) = lim and 𝜆𝑞 . 𝜆→∞ |𝐹𝜆 ∩ log Γ| vol(log 𝐾Γ ) = lim This is easily seen to imply the claim. (4.4.5) □ Corollary 4.4.13. Let 𝐺 be a simply connected nilpotent Lie group of step 𝑠 and dimension 𝑑, and let the constants 𝐶1 = 𝐶1 (𝑠) and 𝐶2 = 𝐶2 (𝑠) be as in Theorem 4.4.2. If Γ is a discrete subgroup of 𝐺 then the harmonious subgroups H+ (Γ) and H− (Γ) satisfy the index bounds [H+ (Γ) : Γ] [Γ : H− (Γ)] = [H+ (Γ) : H− (Γ)] ≤ (𝐶2𝐶1 ) 𝑑 . Proof. We may assume without loss of generality that Γ is a lattice, replacing 𝐺 by C (Γ) otherwise. Since log H+ (Γ) ⊆ 𝐶1𝐶2 · log H− (Γ) we have that [log H+ (Γ) : log H− (Γ)] ≤ (𝐶1𝐶2 ) 𝑑 , and the claim follows from proposition 4.4.11. □ 150 4.5 Proof of the main theorems In this section we prove our main theorems, Theorems 4.1.3 and 8.1.1. We begin by proving the following proposition, which is a direct analogue of Theorem 4.3.1 for Carnot groups. Proposition 4.5.1. Let 𝐺 be a Carnot group with Lie algebra 𝔤, let 𝑑𝐺 be a left-invariant homogeneous metric on 𝐺, and for each 𝑟 > 0 let 𝐵𝑟 be the ball of radius 𝑟 around id in (𝐺, 𝑑𝐺 ). Then sup {#{⟨𝐻 ∩ 𝐵2𝑘 ⟩ : 𝑘 ∈ Z} : 𝐻 a discrete subgroup of 𝐺} < ∞. It will suffice for our applications that all relevant constants depend on the pair (𝐺, 𝑑𝐺 ) in an arbitrary fashion. Proof of proposition 4.5.1. Since 𝑑𝐺 is consistent with the usual topology of 𝐺, the (closed) ball 𝐵𝑟 is a compact subset of 𝐺 containing a neighbourhood of the identity for each 𝑟 > 0. In particular, there exists a convex, symmetric subset 𝐾 of 𝔤 and a constant 𝐶0 ≥ 1 such that 𝐾 ⊆ log 𝐵1 ⊆ 𝐶0 𝐾 ⊆ 𝛿𝐶0 (𝐾), where (𝛿𝜆 )𝜆>0 is the dilation semigroup on the stratified Lie algebra 𝔤. Thus, the balls 𝐵𝑟 are sandwiched between the exponentials of the dilates of 𝐾: 𝛿𝑟 (𝐾) ⊆ log 𝐵𝑟 ⊆ 𝛿𝐶0𝑟 (𝐾) (4.5.1) for every 𝑟 > 0. The sets 𝛿𝜆 (𝐾) are all convex and symmetric since they are linear images of the convex symmetric set 𝐾; this will allow us to apply Theorem 4.3.1 to an appropriately chosen additive subgroup of 𝔤. Let Λ = log 𝐻, and for each 𝑘 ∈ Z let 𝐻 𝑘 = ⟨𝐻 ∩ 𝐵2𝑘 ⟩ and Λ 𝑘 = log 𝐻 𝑘 . By Theorem 4.4.2, there exists an integer constant 𝐶1 such that Λ̃ := spanZ (𝐶1 · Λ) is an additive, bracket-closed lattice in 𝔤 whose exponential H− (𝐻) is a harmonious subgroup of 𝐺 that is contained in 𝐻. It follows by a direct application of Theorem 4.3.1 that there exists a constant 𝐶2 such that #{spanZ ( Λ̃ ∩ 𝛿𝜆 (𝐾)) : 𝜆 > 0} ≤ 𝐶2 . For each 𝑘 ≥ 0 we have trivially that 𝐻 𝑘 ∩ 𝐵2𝑘 = 𝐻 ∩ 𝐵2𝑘 and hence that Λ 𝑘 ∩ 𝛿2𝑘 (𝐾) = Λ ∩ 𝛿2𝑘 (𝐾). In particular, Λ̃ ∩ 𝛿2𝑘 (𝐾) ⊆ Λ̃ ∩ 𝐵2𝑘 ⊆ Λ ∩ 𝐵2𝑘 = Λ𝑘 ∩ 𝐵2𝑘 . On the other hand, letting 𝑚 = ⌈log2 𝐶0 ⌉, we also have by (4.5.1) that Λ̃ ∩ 𝛿2𝑘+𝑚 (𝐾) ⊇ Λ̃ ∩ 𝐵2𝑘 ⊇ (𝐶1 · Λ) ∩ 𝐵2𝑘 ⊇ 𝐶1 · (Λ ∩ 𝐵2𝑘 ) = 𝐶1 · (Λ 𝑘 ∩ 𝐵2𝑘 ). 151 Thus, if we define Λ̃ 𝑘 = spanZ ( Λ̃ ∩ 𝛿2𝑘 (𝐾)) for each 𝑘 > 0 then 𝐶1 · spanZ (Λ 𝑘−𝑚 ∩ 𝐵2𝑘−𝑚 ) ⊆ Λ̃𝑘 ⊆ spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) (4.5.2) for every 𝑘 ∈ Z. Consider an interval [𝑎, 𝑏] ∩ Z such that Λ̃ 𝑘 does not change as 𝑘 varies over [𝑎, 𝑏] ∩ Z. Then we have by (4.5.2) that 𝐶1 · spanZ (Λ𝑏−𝑚 ∩ 𝐵2𝑏−𝑚 ) ⊆ spanZ (Λ𝑎 ∩ 𝐵2𝑎 ) and hence that [spanZ (Λ𝑏−𝑚 ∩ 𝐵2𝑏−𝑚 ) : spanZ (Λ𝑎 ∩ 𝐵2𝑎 )] ≤ 𝐶1𝑑 . Since strict sublattices have an index of at least 2, this in turn implies that #{spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) : 𝑘 ∈ [𝑎, 𝑏]} ≤ 𝑚 + #{spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) : 𝑘 ∈ [𝑎, 𝑏 − 𝑚]} ≤ 𝑚 + 𝑑 log2 𝐶1 . Since [𝑎, 𝑏] was an arbitrary interval over which Λ̃ 𝑘 remained constant, it follows that there exist constants 𝐶3 and 𝐶4 such that #{spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) : 𝑘 ∈ Z} ≤ 𝐶3 #{Λ̃ 𝑘 : 𝑘 ∈ Z} ≤ 𝐶4 . Now, for each 𝑘, (since the constant 𝐶1 divides the constant appearing in lemma 4.4.4) the set 𝐶1 · spanZ (B( 𝐶11 (Λ 𝑘 ∩ 𝐵2𝑘 )) ⊆ log H+ (𝐻 𝑘 ) is an additive, bracket-closed subgroup of 𝔤 whose exponential is a subgroup of 𝐺 that contains a generating set for 𝐻 𝑘 , so that    1 Λ 𝑘 ⊆ 𝐶1 · spanZ B (Λ 𝑘 ∩ 𝐵2𝑘 ) ⊆ log H+ (𝐻 𝑘 ) 𝐶1 for every 𝑘. As such, if [𝑎, 𝑏] ∩ Z is an interval such that spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) does not change as 𝑘 varies over [𝑎, 𝑏] ∩ Z then we have that    1 Λ𝑎 ⊆ Λ𝑏 ⊆ 𝐶1 · spanZ B (Λ𝑎 ∩ 𝐵2𝑎 ) ⊆ log H+ (Λ𝑎 ) 𝐶1 and hence by corollary 4.4.13 that [𝐻𝑏 : 𝐻𝑎 ] ≤ 𝐶5 for some constant 𝐶5 . Arguing as above, this implies that there exist constants 𝐶6 and 𝐶7 such that #{𝐻 𝑘 : 𝑘 ∈ Z} ≤ 𝐶6 #{spanZ (Λ 𝑘 ∩ 𝐵2𝑘 ) : 𝑘 ∈ Z} ≤ 𝐶7 , completing the proof. □ 152 Our next goal is to use proposition 4.5.1 to prove the special case of Theorem 4.1.3 in which the group is nilpotent of bounded step. Proposition 4.5.2 (Exploring subgroups of nilpotent groups). For each 𝑠, 𝑘 ≥ 1 there exists a constant 𝐶 (𝑠, 𝑘) such that the following holds. Let 𝑁 be a nilpotent group of step 𝑠 generated by some set 𝑆 with |𝑆| ≤ 𝑘, let 𝐻 be a subgroup of 𝑁, and for each 𝑛 ≥ 1 let 𝐻𝑛 be the subgroup of 𝐻 generated by elements that have word length at most 2𝑛 in (𝐺, 𝑆). Then #{𝑛 : 𝐻𝑛+1 ≠ 𝐻𝑛 } ≤ 𝐶. Proof of Proposition 4.5.2. We first argue that it suffices to consider the case that 𝑁 is equal to the free step-𝑠 nilpotent group 𝑁 𝑠,𝑆 . Let 𝑁, 𝑆, and (𝐻𝑛 )𝑛≥0 be as in the statement of the theorem. Let 𝑁 𝑠,𝑆 be the free step-𝑠 nilpotent group over 𝑆 and let 𝐺 𝑠,𝑆 be the free step-𝑠 nilpotent Lie group over 𝑆, so that 𝑁 𝑠,𝑆 can be identified with the subgroup of 𝐺 𝑠,𝑆 generated by 𝑆. By the universal property of 𝑁 𝑠,𝑆 , there exists a homomorphism 𝜋 : 𝑁 𝑠,𝑆 → 𝑁 satisfying 𝜋(𝑥) = 𝑥 for every 𝑥 ∈ 𝑆, which is necessarily unique and surjective since 𝑆 generates 𝑁. Thus, 𝜋 maps the word metric 𝑟-ball in (𝑁 𝑠,𝑆 , 𝑆) to the word metric 𝑟-ball in (𝑁, 𝑆) for every 𝑟 ≥ 0. For each 𝑛 ≥ 0, let 𝐻˜ 𝑛 be the subgroup of 𝑁 𝑠,𝑆 generated by the elements of 𝜋 −1 (𝐻) that have word length at most 2𝑛 in (𝑁 𝑠,𝑆 , 𝑆). Letting 𝐾 denote the kernel of 𝜋, we observe that the subgroup 𝐾 𝐻˜ 𝑛 = {𝑘 ℎ : 𝑘 ∈ 𝐾, ℎ ∈ 𝐻˜ 𝑛 } of 𝑁 𝑠,𝑆 is equal to the preimage 𝜋 −1 (𝐻𝑛 ): On the one hand, since 𝐾 is normal in 𝐾 𝐻˜ 𝑛 , 𝜋(𝐾 𝐻˜ 𝑛 ) is a subgroup of 𝑁 that contains the set of words in 𝐻 that have word length at most 2𝑛 , and therefore contains 𝐻𝑛 . On the other hand, if 𝑥 = 𝑘 ℎ is an element of then we can write ℎ = ℎ1 ℎ2 · · · ℎℓ as a product of elements of 𝜋 −1 (𝐻) of word length at most 2𝑛 in (𝑁 𝑠,𝑆 , 𝑆), so that 𝜋(ℎ𝑖 ) is an element of 𝐻 of word length at most 2𝑛 in (𝑁, 𝑆) and 𝜋(𝑥) belongs to 𝜋 −1 (𝐻𝑛 ) as required. Since 𝜋 is surjective, we have the chain of implications ( 𝐻˜ 𝑛+1 = 𝐻˜ 𝑛 ) ⇒ (𝐾 𝐻˜ 𝑛+1 = 𝐾 𝐻˜ 𝑛 ) ⇒ (𝐻𝑛+1 = 𝐻𝑛 ). Thus, it suffices to prove that the theorem holds with 𝑁 and 𝐻 replaced by 𝑁 𝑠,𝑆 and 𝜋 −1 (𝐻). From now on we assume that 𝑁 = 𝑁 𝑠,𝑆 and let 𝐺 = 𝐺 𝑠,𝑆 . Since 𝑁, 𝐺, and the embedding 𝑁 → 𝐺 are determined up to isomorphism by 𝑠 and |𝑆|, we are now free to use constants that depend on this data in an arbitrary way (but must still be independent of the choice of subgroup 𝐻 ⊆ 𝑁). By Pansu’s theorem as formulated in (4.2.2), there exists a left-invariant Carnot metric 𝑑𝐺 on 𝐺 such that 𝑑𝐺 (id, 𝑥) →1 as 𝑥 → ∞ in 𝑁. 𝑑 𝑆 (id, 𝑥) 153 (For the argument to work we need only that the embedding of (𝑁, 𝑑 𝑆 ) into (𝐺, 𝑑𝐺 ) is a quasiisometry.) In particular, there exist positive constants 𝑐 and 𝐶 such that 𝑛 log 𝑁 ∩ 𝐵𝑐𝑛 ⊆ log(𝑆 ) ⊆ log 𝑁 ∩ 𝐵𝐶𝑛 for every 𝑛 ≥ 0, where we write 𝑆 = 𝑆 ∪ {id} ∪ 𝑆 −1 . This implies that if 𝐻 is a subgroup of 𝑁 then ⟨𝐻 ∩ 𝐵𝑐2𝑘 ⟩ ⊆ 𝐻 𝑘 ⊆ ⟨𝐻 ∩ 𝐵𝐶2𝑘 ⟩ for every 𝑘 ≥ 0, and the claim follows easily from this together with proposition 4.5.1. □ We next deduce Theorem 4.1.3 from Proposition 4.5.2 and the Breuillard-Green-Tao theorem. The proof will use the following elementary lemmas, the first of which is proven in [BT16, Lemma 4.2]. Recall that we write 𝑆 = 𝑆 ∪ {id} ∪ 𝑆 −1 . Lemma 4.5.3. Let 𝐺 be a group, and let 𝐻 be a subgroup of 𝐺 with index at most 𝑛. If 𝑆 is a finite generating set for 𝐺, then (𝑆) 2𝑛−1 ∩ 𝐻 is a generating set for 𝐻. Lemma 4.5.4. Let 𝐺 be a group, and let 𝐻1 ⊆ 𝐻2 ⊆ · · · ⊆ 𝐻𝑛 and 𝐻1′ ⊆ 𝐻2′ ⊆ · · · ⊆ 𝐻𝑛′ be two chains of subgroups such that 𝐻𝑖′ is a subgroup of 𝐻𝑖 for each 1 ≤ 𝑖 ≤ 𝑛. Then #{𝐻𝑖 : 1 ≤ 𝑖 ≤ 𝑛} ≤ (1 + ⌊log2 (max [𝐻𝑖 : 𝐻𝑖′])⌋) · #{𝐻𝑖′ : 1 ≤ 𝑖 ≤ 𝑛}. 𝑖 Proof of lemma 4.5.4. By taking a subsequence if necessary, we may assume that 𝐻𝑖+1 ≠ 𝐻𝑖 for every 1 ≤ 𝑖 < 𝑛. Let ℓ = #{𝐻𝑖′ : 1 ≤ 𝑖 ≤ 𝑛} and for each 0 ≤ 𝑘 < ℓ let 𝑖 𝑘 be the 𝑘th time 𝐻𝑖′ changes, so that 𝑖 0 = 1 and 𝑖 𝑘 = min{𝑖 > 𝑖 𝑘 : 𝐻𝑖′ ≠ 𝐻𝑖′𝑘 } for each 1 ≤ 𝑘 < ℓ. We also set 𝑖ℓ = 𝑛 + 1 for notational convenience. Since 𝐻𝑖′𝑘−1 = 𝐻𝑖′𝑘 −1 is a subgroup of 𝐻𝑖 𝑘−1 for each 1 ≤ 𝑘 ≤ ℓ, we have that [𝐻𝑖 𝑘 −1 : 𝐻𝑖 𝑘−1 ] ≤ [𝐻𝑖 𝑘 −1 : 𝐻𝑖′𝑘 −1 ] ≤ max [𝐻𝑖 : 𝐻𝑖′] 𝑖 for every 1 ≤ 𝑘 ≤ ℓ. On the other hand, we also have that [𝐻𝑖 𝑘 −1 : 𝐻𝑖 𝑘−1 ] = 2𝑖 𝑘 −1−𝑖 𝑘−1 for every 1 ≤ 𝑘 ≤ ℓ, and hence that 𝑛= ℓ ∑︁ Î𝑖 𝑘 −2 𝑖=𝑖 𝑘−1 [𝐻𝑖+1 : 𝐻𝑖 ] ≥ (𝑖 𝑘 − 𝑖 𝑘−1 ) ≤ ℓ · (1 + ⌊log2 (max [𝐻𝑖 : 𝐻𝑖′])⌋) 𝑖 𝑘=1 as claimed. □ Corollary 4.5.5. Let 𝐺 be a group, let 𝐺 ′ be a finite-index subgroup of 𝐺, and let 𝐻1 ⊆ 𝐻2 ⊆ · · · ⊆ 𝐻𝑛 be an increasing sequence of subgroups of 𝐺. Then #{𝐻𝑖 : 1 ≤ 𝑖 ≤ 𝑛} ≤ (1 + ⌊log2 [𝐺 : 𝐺 ′]⌋) · #{𝐻𝑖 ∩ 𝐺 ′ : 1 ≤ 𝑖 ≤ 𝑛}. 154 Proof. Apply lemma 4.5.4 with 𝐻𝑖′ = 𝐻𝑖 ∩ 𝐺 ′ and use that [𝐻𝑖 : 𝐻𝑖 ∩ 𝐺 ′] ≤ [𝐺 : 𝐺 ′]. □ We now have everything we need to prove Theorem 4.1.3. Proof of Theorem 4.1.3. Let 𝐾 ≥ 1 and let 𝑟 0 = 𝑟 0 (𝐾) and 𝐶1 = 𝐶1 (𝐾) be as in Theorem 4.1.1. Suppose that 𝑟 ≥ 𝑟 0 is such that Gr(3𝑟) ≤ 𝐾 Gr(𝑟), and let 𝑄⊳𝐺 and 𝑁 ⊳𝐺/𝑄 be as in Theorem 4.1.1. Since 𝑁 has index at most 𝐶1 in 𝐺/𝑄, lemma 4.5.3 implies that the set 𝑆′ := 𝜋((𝑆) 2𝐶1 −1 ) ∩ 𝑁 is a generating set for 𝑁, and the word metric associated to the pair (𝑁, 𝑆′) is bi-Lipschitz equivalent to the restriction of the word metric on (𝐺/𝑄, 𝜋(𝑆)) to 𝑁, with constants depending only on 𝐾. Let 𝐻 ′ = (𝑄𝐻)/𝑄, so that 𝐻 ′ is a subgroup of 𝐺/𝑄 and 𝐻˜ := 𝐻 ′ ∩ 𝑁 is a subgroup of 𝑁. For each ˜ respectively, that have word length 𝑛 ≥ 0 let 𝑊𝑛′ and 𝑊˜ 𝑛 be the set of the elements of 𝐻 ′ and 𝐻, at most 2𝑛 with respect to (𝐺/𝑄, 𝜋(𝑆)), and define 𝐻𝑛′ = ⟨𝑊𝑛′ ⟩ and 𝐻˜ 𝑛 = ⟨𝑊˜ 𝑛 ⟩ for every 𝑛 ≥ 0. Since [𝐻𝑛′ : 𝐻𝑛′ ∩ 𝑁] ≤ [𝐺/𝑄 : 𝑁] ≤ 𝐶1 and 𝑊𝑛′ is a finite symmetric generating set for 𝐻𝑛′ , the set (𝑊𝑛′ ) 2𝐶1 −1 ∩ 𝑁 is a generating set for 𝐻𝑛′ ∩ 𝑁, so that if we define 𝑚 = ⌈log2 (2𝐶1 − 1)⌉ then ′ 𝐻𝑛′ ∩ 𝑁 ⊆ 𝐻˜ 𝑛+𝑚 ⊆ 𝐻𝑛+𝑚 ∩𝑁 (4.5.3) for every 𝑛 ≥ 0. Using the above mentioned bi-Lipschitz equivalence between the two different word metrics on 𝑁 and the fact that the step of 𝑁 and the size of the generating set 𝑆′ are bounded by constants depending only on 𝐾 and 𝑘, it follows from Proposition 4.5.2 that there exists a constant 𝐶2 = 𝐶2 (𝐾, 𝑘) such that #{ 𝐻˜ 𝑛 : 𝑛 ≥ 0} ≤ 𝐶2 . It follows from this and (4.5.3) that there exists a constant 𝐶3 = 𝐶3 (𝐾, 𝑘) such that #{𝑁 ∩ 𝐻𝑛′ : 𝑛 ≥ 0} ≤ 𝐶3 , and hence by corollary 4.5.5 that there exists a constant 𝐶4 = 𝐶4 (𝐾, 𝑘) such that #{𝐻𝑛′ : 𝑛 ≥ 0} ≤ 𝐶4 . Now, as in the proof of Proposition 4.5.2, we have that 𝑄𝐻𝑛 = 𝜋 −1 (𝐻𝑛′ ) for every 𝑛 ≥ 0. Observe that if 𝑄𝐻𝑛+1 = 𝑄𝐻𝑛 but 𝐻𝑛+1 ≠ 𝐻𝑛 then there exist 𝑞 𝑛 , 𝑞 𝑛+1 ∈ 𝑄, ℎ𝑛 ∈ 𝐻𝑛 , and ℎ𝑛+1 ∈ 𝐻𝑛+1 \ 𝐻𝑛 −1 such that 𝑞 𝑛 ℎ𝑛 = 𝑞 𝑛+1 ℎ𝑛+1 , and hence that ℎ𝑛+1 ℎ−1 𝑛 = 𝑞 𝑛+1 𝑞 𝑛 ∈ 𝑄. Since 𝑄 has diameter at most 𝐶1𝑟, this implies that ℎ𝑛+1 ℎ−1 𝑛 has word length at most 𝐶1 𝑟, contradicting the assumption that ℎ𝑛+1 ∉ 𝐻𝑛 if 2𝑛 ≥ 𝐶1𝑟. It follows that there exists a constant 𝐶5 = 𝐶5 (𝐾) such that #{𝑛 ≥ 𝐶5 + log2 𝑟 : 𝐻𝑛+1 ≠ 𝐻𝑛 } ≤ #{𝑛 ≥ 𝐶5 + log2 𝑟 : 𝑄𝐻𝑛+1 ≠ 𝑄𝐻𝑛 } ≤ #{𝐻𝑛′ : 𝑛 ≥ 0} ≤ 𝐶4 , which easily implies the claim. □ 155 It remains only to deduce Theorem 8.1.1 from Theorem 4.1.3. We will need the following lemma about the injectivity radius of the quotient 𝐹𝑆 /⟨⟨𝑅𝑛 ⟩⟩ → 𝐺. For each 𝑛 ≥ 0 let 𝐺 𝑛 = 𝐹𝑆 /⟨⟨𝑅𝑛 ⟩⟩. Lemma 4.5.6. Let 𝐺 be a group with a finite generating set 𝑆. The quotient map 𝐺 𝑛 → 𝐺 induces a map between Cayley graphs that restricts to an isomorphism between the balls of radius 2𝑛−1 − 1 around the identity. 𝑛−1 Proof of Lemma 4.5.6. It suffices to prove that the quotient map 𝜋 : 𝐺 𝑛 → 𝐺 is injective on (𝑆) 2 . (This implies that 𝜋(𝑥𝑠) = 𝜋(𝑦) if and only if 𝑥𝑠 = 𝑦 for every 𝑠 ∈ 𝑆 and 𝑥, 𝑦 in the ball of radius 2𝑛−1 − 1 and hence that the balls of radius 2𝑛−1 − 1 are isomorphic.) Suppose for contradiction that 𝑛−1 this is false. Then there exist 𝑢, 𝑣 ∈ (𝑆) 2 ⊆ 𝐹𝑆 such that 𝑢 −1 𝑣 ∈ 𝑅 \ ⟨⟨𝑅𝑛 ⟩⟩. Since 𝑢 and 𝑣 both 𝑛−1 𝑛 belong to (𝑆) 2 , the product 𝑢 −1 𝑣 belongs to (𝑆) 2 , and since it also belongs to 𝑅 it must belong to 𝑅𝑛 by definition of 𝑅𝑛 . This contradicts the assumption that 𝑢 −1 𝑣 ∉ ⟨⟨𝑅𝑛 ⟩⟩. □ Proof of Theorem 8.1.1. Let 𝑟 0 = 𝑟 0 (𝐾) and 𝐶 = 𝐶 (𝐾, 𝑘) be the constants from Theorem 4.1.3. Let 𝑛0 = ⌈4 + log2 𝑟⌉ and let 𝐺 ′ = 𝐹𝑆 /⟨⟨𝑅𝑛0 ⟩⟩. The projection 𝐺 ′ → 𝐺 induces a surjective graph homomorphism between the Cayley graphs Cay(𝐺 ′, 𝑆) and Cay(𝐺, 𝑆) that restricts to an isomorphism between the balls of radius 2𝑛0 −1 − 1 ≥ 4𝑟. Let 𝐻 be the subgroup of 𝐺 ′ generated by Ð 𝑛≥𝑛0 (𝑅𝑛 /⟨⟨𝑅𝑛0 ⟩⟩) and, for each 𝑛 ≥ 𝑛0 , let 𝐻𝑛 be the subgroup of 𝐻 generated by 𝑅𝑛 /⟨⟨𝑅0 ⟩⟩. If 𝑟 ≥ 𝑟 0 , we may apply Theorem 4.1.3 to 𝐺 ′ and 𝐻 to obtain that #{𝐻𝑛 : 𝑛 ≥ 𝑛0 } ≤ 𝐶, and it follows that #{⟨⟨𝑅𝑛 ⟩⟩ : 𝑛 ≥ 𝑛0 } = #{⟨⟨𝐻𝑛 ⟩⟩ : 𝑛 ≥ 𝑛0 } ≤ #{𝐻𝑛 : 𝑛 ≥ 𝑛0 } ≤ 𝐶 as claimed. □ Acknowledgments TH thanks Christian Gorski and Mikolaj Fraczyk for helpful conversations and thanks Matthew Tointon and Seung-Yeon Ryoo for comments on a draft. This work was supported by NSF grant DMS-2246494. Both authors thank the anonymous referees for their comments. 156 Chapter 5 157 SUPERCRITICAL PERCOLATION ON FINITE TRANSITIVE GRAPHS I: UNIQUENESS OF THE GIANT COMPONENT Joint with Tom Hutchcroft Abstract Let (𝐺 𝑛 )𝑛≥1 = ((𝑉𝑛 , 𝐸 𝑛 ))𝑛≥1 be a sequence of finite, connected, vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters ( 𝑝 𝑛 )𝑛≥1 in [0, 1] is supercritical with respect to Bernoulli bond percolation P𝐺𝑝 if there exists 𝜀 > 0 and 𝑁 < ∞ such that 𝑛 (the largest cluster contains at least 𝜀|𝑉𝑛 | vertices) ≥ 𝜀 P𝐺(1−𝜀) 𝑝 𝑛 for every 𝑛 ≥ 𝑁 with 𝑝 𝑛 < 1. We prove that if (𝐺 𝑛 )𝑛≥1 is sparse, meaning that the degrees are sublinear in the number of vertices, then the supercritical giant cluster is unique with high probability in the sense that if ( 𝑝 𝑛 )𝑛≥1 is supercritical then lim P𝐺𝑝 𝑛𝑛 (the second largest cluster contains at least 𝑐|𝑉𝑛 | vertices) = 0 𝑛→∞ for every 𝑐 > 0. This result is new even under the stronger hypothesis that (𝐺 𝑛 )𝑛≥1 has uniformly bounded vertex degrees, in which case it verifies a conjecture of Benjamini (2001). Previous work of many authors had established the same theorem for complete graphs, tori, hypercubes, and bounded degree expander graphs, each using methods that are highly specific to the examples they treated. We also give a complete solution to the problem of supercritical uniqueness for dense vertex-transitive graphs, establishing a simple necessary and sufficient isoperimetric condition for uniqueness to hold. 5.1 Introduction Let 𝐺 = (𝑉, 𝐸) be a countable graph that is connected and vertex-transitive, meaning that for all vertices 𝑢, 𝑣 ∈ 𝑉 there is a graph automorphism 𝜙 ∈ Aut 𝐺 with 𝜙(𝑢) = 𝑣. Given 𝑝 ∈ [0, 1], Bernoulli bond percolation P𝐺𝑝 (abbreviated P 𝑝 when the choice of 𝐺 is clear from context) is the distribution of a random spanning subgraph 𝜔 formed by independently including each edge with probability 𝑝. We identify 𝜔 with an element of {0, 1} 𝐸 where 𝜔(𝑒) = 1 means that the edge 𝑒 is present in 𝜔. The edges in 𝜔 are called open and the rest are called closed. We are interested primarily in the geometry of the connected components of 𝜔, which we refer to as clusters. Much of the interest in the model stems from the existence of a phase transition: For infinite graphs, there is typically1 a critical probability 0 < 𝑝 𝑐 < 1 such that every cluster is finite almost surely when 𝑝 < 𝑝 𝑐 , while at least one infinite cluster exists almost surely when 𝑝 > 𝑝 𝑐 . For large finite graphs, one typically observes a similar phase transition in which a giant component, containing a positive proportion of all vertices, emerges as 𝑝 is varied through a small interval. The regime in which an infinite/giant cluster exists is known as the supercritical phase. It is now known that the supercritical phase is always non-degenerate for bounded degree transitive graphs that are strictly more than one-dimensional in an appropriate coarse-geometric sense: this was proven for infinite graphs by Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [Dum+20b] and for finite graphs by the second author and Tointon [HT21c]. Once one knows that the supercritical phase is non-degenerate, so that infinite/giant clusters exist for sufficiently large values of 𝑝, a central problem is to understand the number of these clusters. For infinite transitive graphs, this is the subject of a famous conjecture of Benjamini and Schramm [BS96c] stating that the infinite cluster is unique for every 𝑝 > 𝑝 𝑐 if and only if the graph is amenable. The ‘if’ direction of this conjecture follows from the classical work of Aizenman, Kesten, and Newman [AKN87a] and Burton and Keane [BK89], while the ‘only if’ direction remains open in general; see e.g. [MR3352259; Hut20f; Hut19a; LP16c; PS00] for an overview of what is known. In contrast, for finite transitive graphs, Benjamini [Ben01a, Conjecture 1.2] conjectured in 2001 that the giant cluster should always be unique in the supercritical regime, irrespective of the geometry of the graph. Several works, some of which are very classical, have established versions of this conjecture in special cases including for complete graphs [ER61; Bol84], hypercubes [AKS82a; BKŁ92], Euclidean tori of fixed dimension [HR06], and bounded degree expanders [ABS04b]. Each of these works uses methods that are very specific to the example it treats, with the analysis of the tori (Z𝑑 /𝑛Z𝑑 )𝑛≥1 in dimension 𝑑 ≥ 3 relying in particular on the important and technically challenging work of Grimmett and Marstrand [GM90b]. A related conjecture giving mild geometric conditions under which it should be impossible to have multiple giant clusters both above and at criticality was subsequently stated in the influential work of Alon, Benjamini, and Stacey [ABS04b, Conjecture 1.1]. A central difficulty in the study of this conjecture, and in the study of percolation on finite graphs 1 We keep the meaning of ‘typically’ intentionally vague. One very general conjecture [BS96c, Question 2] is that 𝑝 𝑐 < 1 for all (not necessarily transitive) infinite graphs with isoperimetric dimensional strictly greater than 1. 158 more generally, is that many of the most important qualitative tools used to study the infinite case, such as the ergodic theorem, break down completely in the finite case. For example, adapting the Burton–Keane uniqueness proof to the case of finite graphs merely shows that each vertex is unlikely to have three distinct large clusters in a small vicinity around it – a statement that need not in general be in tension with the existence of multiple giant components in a finite graph. Indeed, while the Burton–Keane proof applies to arbitrary insertion-tolerant automorphism-invariant percolation processes, it is possible to construct insertion-tolerant automorphism-invariant percolation processes on the torus (Z/𝑛Z) 2 that have multiple giant components with high probability. In fact, for the highly asymmetric torus (Z/𝑛Z) × (Z/2𝑛 Z) one even has multiple giant clusters with good probability for Bernoulli percolation at appropriate values of 𝑝, although this arises as a feature of a discontinuous phase transition rather than of the supercritical phase per se. See Section 5.5 for further discussion of both examples. In light of these difficulties, any treatment of the uniqueness problem for finite transitive graphs must involve new techniques and use finer properties of supercritical percolation than in the infinite case. In this paper we resolve Benjamini’s conjecture and hence also the supercritical case of the Alon– Benjamini–Stacey conjecture, giving a complete solution to the problem of supercritical uniqueness on large, finite, vertex-transitive graphs. In the forthcoming work in this series, we will prove moreover that the density of the giant component is concentrated, local, and equicontinuous in the supercritical regime and prove analogous theorems for the Fortuin-Kasteleyn random cluster model, Ising model, and Potts model. Definition 5.1.1. We will assume all graphs to be locally finite and to contain at least one vertex. Let F be the set of all isomorphism classes of finite, connected, simple (i.e., not containing loops or multiple edges), vertex-transitive graphs. (We will usually suppress the distinction between graphs and their isomorphism classes as much as possible when this does not cause any confusion.) Given an infinite set H ⊆ F , a function 𝜙 : H → R, and 𝛼 ∈ R we write lim𝐺∈H 𝜙(𝐺) = 𝛼 or “𝜙(𝐺) → 𝛼 as 𝐺 → ∞ in H ” to mean that for each 𝜀 > 0 there exists 𝑁 such that |𝜙(𝐺) −𝛼| ≤ 𝜀 for every 𝐺 ∈ H with at least 𝑁 vertices, or equivalently that 𝜙(𝐺 𝑛 ) → 𝛼 for some (and hence every) enumeration H = {𝐺 1 , 𝐺 2 , . . .} of H . Similar conventions apply to the definition of lim sup𝐺∈H , lim inf 𝐺∈H , and limits that may be equal to +∞ or −∞. Let 𝐺 = (𝑉, 𝐸) be a countable graph and consider a percolation configuration 𝜔 ∈ {0, 1} 𝐸 . The connected components of 𝜔 are called clusters. We write 𝐾𝑢 to denote the cluster containing the vertex 𝑢 and write 𝑢 ↔ 𝑣 for the event that 𝐾𝑢 = 𝐾𝑣 . Given a subset 𝑊 of 𝑉, the volume of 𝑊 is the number of vertices in 𝑊, denoted |𝑊 |, while if 𝐺 is finite, the density of 𝑊 is defined to be the 159 ratio ∥𝑊 ∥ := |𝑊 | /|𝑉 |. We write 𝐾1 , 𝐾2 , . . . for the clusters of 𝜔 in decreasing order of volume. (Note the slight abuse of notation: here 𝐾1 does not mean the cluster of a vertex labelled ‘1’.) Given an infinite set H ⊆ F , we say that an assignment of parameters 𝑝 𝑐 : H → [0, 1] is a percolation threshold if 1. lim𝐺∈H P𝐺(1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝑐) = 0 for every 𝜀, 𝑐 > 0, and 𝑐 2. For every 𝜀 > 0 there exists 𝛼 > 0 such that lim P𝐺(1+𝜀) 𝑝 𝑐 (∥𝐾1 ∥ ≥ 𝛼) = 1, 𝐺∈H where we set P𝐺𝑝 = P𝐺 1 for 𝑝 ≥ 1. Note that critical thresholds are not unique (when they exist), but any two percolation thresholds 𝑝 𝑐 , 𝑝˜𝑐 : H → [0, 1] must satisfy 𝑝 𝑐 (𝐺) ∼ 𝑝˜𝑐 (𝐺) as 𝐺 → ∞ in H . When a percolation threshold 𝑝 𝑐 : H → [0, 1] exists, we say that 𝑝 : H → [0, 1] is supercritical if H ′ := {𝐺 ∈ H : 𝑝(𝐺) < 1} is finite or if H ′ is infinite and satisfies lim inf ′ 𝐺∈H 𝑝(𝐺) > 1. 𝑝 𝑐 (𝐺) We generalise this definition to include the case that 𝑝 𝑐 does not exist by saying that 𝑝 is supercritical if H ′ := {𝐺 ∈ H : 𝑝(𝐺) < 1} is finite or if H ′ is infinite and there exists 𝜀 > 0 such that lim inf P𝐺(1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀. ′ 𝐺∈H Note that these two definitions of supercriticality coincide when H admits a threshold function, and in particular that the definition of supercriticality does not depend on the choice of threshold function. (Without the (1 − 𝜀) factor in P𝐺(1−𝜀) 𝑝 , these definitions would not always coincide, for example for the highly asymmetric torus discussed in Example 5.1, which has giant clusters with good probability at a percolation threshold bounded away from 1.) The reason for introducing the set H ′ is to ensure that every family has a supercritical sequence of parameters, namely the constant assignment 𝑝(𝐺) := 1 for all 𝐺. It will also be helpful to have the finitary version of this definition: Given a single finite, connected, simple, vertex-transitive graph 𝐺 = (𝑉, 𝐸) and given any 𝜀 > 0, we say that a parameter 𝑝 ∈ [0, 1] is 𝜀-supercritical for 𝐺 if P𝐺(1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀 and |𝑉 | ≥ 2𝜀 −3 . (There is some flexibility in how to choose this latter, technical condition that |𝑉 | is not too small.) 160 We say that H has the supercritical uniqueness property if lim P𝐺𝑝 (∥𝐾2 ∥ ≥ 𝜀) = 0 𝐺∈H for every supercritical 𝑝 : H → [0, 1] and every constant 𝜀 > 0. We begin by stating our main result in the simplest-to-state and most interesting case when we have an infinite set H ⊆ F that is sparse, meaning that the average vertex degree 𝑑 (𝐺) := 2|𝐸 (𝐺)|/|𝑉 (𝐺)| of 𝐺 ∈ H (which is the exact degree of every vertex because 𝐺 is regular) satisfies 𝑑 (𝐺) = 𝑜(|𝑉 (𝐺)|) as 𝐺 → ∞ in H . Note that in particular, if H has uniformly bounded vertex degrees (i.e. sup𝐺∈H 𝑑 (𝐺) < ∞) then H is sparse. Theorem 5.1.2. Let H ⊆ F be an infinite set. If H is sparse, then H has the supercritical uniqueness property. Remark 5.1.1. The restriction to simple graphs is not very important and could be replaced by e.g. the assumption that there are a bounded number of parallel edges between any two vertices. Remark 5.1.2. The Alon–Benjamini–Stacey conjecture [ABS04b, Conjecture 1.1] would follow immediately from Theorem 5.1.2 together with the plausible claim that the percolation phase transition is always continuous for bounded degree graph families satisfying the diam(𝐺) = 𝑜(|𝑉 (𝐺)|/log |𝑉 (𝐺)|) condition they consider. More formally, such a claim would state that if H ⊆ F is an infinite set with uniformly bounded vertex degrees that satisfies this condition, and 𝑝 : H → [0, 1] is any assignment of parameters such that lim inf 𝐺∈H P𝐺𝑝(𝐺) (∥𝐾1 ∥ ≥ 𝑐) > 0 for some 𝑐 > 0, then 𝑝 is supercritical in our sense. Unfortunately such a claim seems to be completely beyond the scope of present techniques and is a major open problem even for, e.g. the three-dimensional torus (Z/𝑛Z) 3 . Indeed, it appears to be an open problem to prove that there are not multiple giant components at criticality in this example. All the proofs in our paper are effective in the sense that they can in principle be used to produce explicit bounds on, say, the expected density of the second largest cluster. While we have not kept track of what these bounds are in all cases, we make note of the following simple explicit estimate implying Theorem 5.1.2 in the case that the graphs in question have bounded or subalgebraic vertex degrees. Theorem 5.1.3. There exists a universal constant 𝐶 such that if 𝐺 = (𝑉, 𝐸) is a finite, simple, connected, vertex-transitive graph with vertex degrees bounded by 𝑑, and 𝑝 ∈ [0, 1] is 𝜀-supercritical 161 for 𝐺 for some 𝜀 > 0, then √︄ P𝐺𝑝 ∥𝐾2 ∥ ≥ 𝜆𝑒 𝐶𝜀 −18 ! log 𝑑 1 ≤ log |𝑉 | 𝜆 for every 𝜆 ≥ 1. Note that this bound is only useful under the subalgebraic degree condition log 𝑑 ≪ log |𝑉 |. The constant given by our proof is fairly large, of order around 106 . This bound has not been optimized and is known to be very far from optimal in classical examples. Indeed, the second largest cluster in supercritical percolation is known to be of order Θ(log |𝑉 |) with high probability on both the complete graph [ER61; Bol84] and the hypercube [BKŁ92],
while for a Euclidean torus of fixed dimension 𝑑, it is of order Θ (log |𝑉 |) 𝑑/(𝑑−1) [HR06]. Similar results are established for a large class of dense graphs in [Bol+10a]. Remark 5.1.3. Let us now explain the relationship between our theorem and the Benjamini– Schramm 𝑝 𝑐 < 𝑝 𝑢 conjecture [BS96c]. Suppose (𝐺 𝑛 )𝑛≥1 is a bounded degree expander sequence converging locally to some infinite nonamenable transitive graph 𝐺. One may deduce either from our results or those of [ABS04b] (see also [Sar21b]) that there is always a unique giant component with high probability for supercritical percolation on 𝐺 𝑛 , a result that seems to be in tension with the conjectured existence of a non-uniqueness phase for percolation on the limit graph 𝐺. Naively, one might think that our definition of supercriticality for finite graphs should therefore be thought of more properly as an analogue of the uniqueness phase (𝑝 > 𝑝 𝑢 ) for infinite graphs. This is misleading. Indeed, it was proven in [BNP11a] that if (𝐺 𝑛 )𝑛≥1 is a sequence of transitive, bounded degree expanders converging to an infinite, transitive, nonamenable graph 𝐺, then a sequence ( 𝑝 𝑛 )𝑛≥1 is supercritical if and only if lim inf 𝑝 𝑛 > 𝑝 𝑐 (𝐺). The uniqueness/non-uniqueness transition on the limit graph 𝐺 does manifest itself in the approximating finite graphs 𝐺 𝑛 , but as a transition in the metric distortion of the giant component rather than its uniqueness: the length of the path connecting two neighbouring vertices of 𝐺 𝑛 given that both vertices belong to the giant is tight as 𝑛 → ∞ when 𝑝 > 𝑝 𝑢 (𝐺) and is not tight when 𝑝 𝑐 (𝐺) < 𝑝 < 𝑝 𝑢 (𝐺). In the second case, the open path connecting two such vertices in 𝐺 𝑛 has good probability to be very long, thus the two vertices become disconnected with positive probability in the limit. See [AB07] for related results and open problems for hypercube percolation. The dense case. We now discuss how our results extend to dense graphs, where vertices have degree proportional to the number of vertices. In contrast to Theorem 5.1.2, it is not true in general 162 that dense graph families have the supercritical uniqueness property. Suppose for example that 𝐺 𝑛 is the Cartesian product of the complete graphs 𝐾2 and 𝐾𝑛 and that 𝑝 𝑛 = 2/𝑛 for every 𝑛 ≥ 1. Then we have by the classical theory of Erdős–Rényi random graphs that each copy of 𝐾𝑛 contains a giant component with high probability, while the number of ‘horizontal’ edges connecting the two copies of 𝐾𝑛 converges in distribution to a Poisson(2) random variable and is therefore equal to zero with probability bounded away from zero as 𝑛 → ∞. Thus, the number of giant clusters in this example is unconcentrated and can be equal to either one or two, each with good probability. Our next main result shows that examples of roughly this form are the only transitive counterexamples to the supercritical uniqueness property. We will in fact characterise the failure of the supercritical uniqueness property for dense vertextransitive graphs in two equivalent ways. We say that an infinite set H ⊆ F is dense if lim inf 𝐺∈H |𝐸 (𝐺)|/|𝑉 (𝐺)| 2 > 0; this is equivalent to the vertex degree 𝑑 (𝐺) growing linearly in the number of vertices in the sense that lim inf 𝑛→∞ 𝑑 (𝐺)/|𝑉 (𝐺)| > 0. Definition 5.1.4. Let H ⊆ 𝐹 be a set. Given 𝑚 ∈ {2, 3, . . . }, we say that H is 𝑚-molecular if it is infinite and dense and there exists a constant 𝐶 < ∞ such that for each 𝐺 ∈ H there exists a set of edges 𝐹 ⊆ 𝐸 (𝐺) satisfying the following conditions: 1. 𝐺 \ 𝐹 has 𝑚 connected components; 2. 𝐹 is invariant under the action of Aut 𝐺; 3. |𝐹 | ≤ 𝐶 |𝑉 (𝐺)|. These conditions imply that the 𝑚 connected components of 𝐺 \ 𝐹 are dense, vertex-transitive, and isomorphic to each other. For example, the family of Cartesian products {𝐾𝑛 □𝐾2 : 𝑛 ≥ 1} discussed above is 2-molecular. We say H is molecular if it is 𝑚-molecular for some 𝑚 ≥ 2. Definition 5.1.5. Let 𝐺 = (𝑉, 𝐸) be a finite graph. For each set 𝐴 ⊆ 𝑉, we write 𝜕𝐸 𝐴 for the set of edges that have one endpoint in 𝐴 and the other in 𝑉 \ 𝐴. For each 𝜃 ∈ (0, 1/2], the quantity Separator(𝐺, 𝜃) is defined to be Separator(𝐺, 𝜃) := ( min |𝜕𝐸 𝐴| : 𝜃 ) ∑︁ 𝑣∈𝑉 163 deg(𝑣) ≤ ∑︁ 𝑣∈𝐴 deg(𝑣) ≤ (1 − 𝜃) ∑︁ 𝑣∈𝑉 deg(𝑣) . In other words, Separator(𝐺, 𝜃) is the minimal number of edges needed to cut 𝐺 into two pieces of roughly equal size. We say that a set H of isomorphism classes of finite, simple graphs has linear 𝜃-separators if H is infinite and lim sup𝐺∈H Separator(𝐺, 𝜃)/|𝑉 (𝐺)| < ∞. The following theorem provides a complete solution to the problem of supercritical uniqueness for Bernoulli bond percolation on finite vertex-transitive graphs and implies Theorem 5.1.2 as a special case. Theorem 5.1.6. For every infinite set H ⊆ F , the following are equivalent: (i) H does not have the supercritical uniqueness property; (ii) H contains a subset that is molecular; (iii) H contains a subset with linear 1/3-separators; (iv) H contains a dense subset with linear 𝜃-separators for some 𝜃 ∈ (0, 1/2]. The dense case of this result sharpens the transitive case of a theorem of Bollobás, Borgs, Chayes, and Riordan [Bol+10a], who proved supercritical uniqueness for any (not necessarily transitive) dense graph sequence converging to an irreducible graphon. In our language, their result states that a dense graph family has the supercritical uniqueness property whenever it does not have any subquadratic separators, i.e., whenever lim inf Separator(𝐺, 𝜃)/|𝑉 (𝐺)| 2 > 0 𝐺∈H for every 𝜃 ∈ (0, 1/2] (see [Bol+10a, Lemma 7]). In fact, since they also prove that the giant cluster density is zero at the percolation threshold, their results imply uniqueness of the giant cluster for all (not necessarily supercritical) assignments of parameters. The same authors also established a formula for the limiting critical probability of dense graph sequences that we will use to prove the implication (iv) ⇒ (i) of Theorem 8.1.1. Further comparison of our results with those of [Bol+10a] is given in remark 5.4.2. Remark 5.1.4. In [Eas22] the first author has built on the results of the present paper to characterise which infinite subsets of F admit a percolation threshold. The obstacle to having a percolation threshold turns out to be the presence of molecular subsets for infinitely many values of 𝑚 ∈ {2, 3, . . .}. In particular, every infinite subset of F that is sparse admits a percolation threshold. So a posteriori, for Theorem 5.1.2 it suffices to work with the original (more natural) definition 164 of supercritical assignments of parameters, which refers to the percolation threshold, rather than the more general definition. Note the surprising logical order here: we first proved uniqueness of the supercritical giant cluster (in the present paper), then this was used to prove that there exists a percolation threshold for the emergence of a giant cluster (in [Eas22]). About the proof and organization We now briefly overview the structure of the paper and outline the proofs of the main steps. Section 2: Lower bounds on point-to-point connection probabilities. In this section we prove that for 𝜀-supercritical percolation on any finite, simple, connected, vertex-transitive graph, we always have a uniform lower bound P 𝑝 (𝑥 ↔ 𝑦) ≥ 𝛿(𝜀) > 0 on the probability that any two given vertices are connected. This was previously known only in the bounded degree case, with constants depending on the degree. Our argument starts by partitioning the vertex set into classes within which we have such a lower bound then recursively merges these classes until a single class contains the entire graph. The merging step makes use of a new high-degree version of insertion-tolerance, which allows us to open a single edge in a sufficiently large random set of edges. Section 3: Uniqueness under the sharp density property. In this section we prove that the supercritical uniqueness property holds for any infinite set H ⊆ F satisfying the sharp density property, meaning that P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) has a sharp threshold for every 𝛼 ∈ (0, 1] in an appropriately uniform sense. This section is at the heart of the paper and contains the most significant new arguments. The proof has two parts. First, given any particular supercritical parameter 𝑝, we use the sharp density property to (non-constructively) deduce the existence of a smaller parameter 𝑞 ≤ 𝑝 such that under P𝑞 there is a giant cluster whose density is concentrated, i.e., lies in a small interval with high probability. The point-to-point connection probability lower bound easily implies that this giant cluster is the unique giant cluster under P𝑞 with high probability. The remainder of the proof consists in showing that non-uniqueness of the giant cluster under P 𝑝 would imply non-concentration of the density of the giant cluster under P𝑞 , establishing a contradiction. To this end we introduce a new object called a sandcastle, which is a large subgraph that is not resilient to 𝑞/𝑝-bond percolation. We observe that under the hypothesis that there are at least two giant clusters under P 𝑝 with good probability, at least one of these clusters must be a sandcastle with good probability. As we pass from P 𝑝 to P𝑞 in the standard monotone coupling of these measures, this sandcastle-cluster disintegrates into small clusters with good probability 165 by definition. On this event, the giant cluster under P𝑞 is constrained to the complement of this sandcastle-cluster from P 𝑝 , where edges are distributed (conditionally) independently. We argue that if this is the case then there must be a subset of 𝑉 with density significantly less than 1 that has good probability under P𝑞 to contain a cluster of approximately the same size as the typical global size of the largest cluster in 𝑉. Finally, we use the existence of this subset together with Harris’ inequality and the point-to-point lower bound to deduce that ∥𝐾1 ∥ is abnormally large with good probability under P𝑞 , contradicting the previously established concentration property. In the final subsection of this section, we verify that subalgebraic degree graphs have the sharp density property, completing the proof of the main theorems in this case and establishing the quantitative estimate Theorem 5.1.3. Section 4: Non-molecular graphs have the sharp density property. In this section we prove that the only way for an infinite subset of F to fail to have the sharp density property is for it to contain a molecular subset, completing the proof of the main theorem. Our argument uses a theorem of Bourgain [Fri99a] formalising the heuristic that increasing events without sharp thresholds are heavily influenced by the state of a bounded number of edges. In our case the event is the existence of a giant cluster of a given density. We apply a delicate sprinkling argument to iteratively reduce the size of this bounded-size set of edges until it contains a single edge. A novel trick in this induction is that during each iteration we use the second author’s universal tightness theorem [Hut21b] and the high-degree analogue of insertion-tolerance from Section 2 to stick large (but not necessarily giant) clusters to both endpoints of an edge in the current set, allowing the small number of sprinkled edges to have a disproportionately large effect. This is the most technical part of the paper. Once the set of edges reaches a singleton, we apply Russo’s formula to derive a contrasting lower bound on the sharpness of the threshold for our event. For this lower bound to not contradict our original upper bound, the graph in question must be dense, completing the proof of Theorem 5.1.2; the proof of the implication (i) ⇒ (ii) of Theorem 8.1.1 in the dense case relies on a second, rather subtle application of the sprinkling technology we develop to prove that the graph must in fact be molecular. Finally we show in Section 5.4 that the remaining non-trivial implication (iv) ⇒ (i) of Theorem 8.1.1 follows easily from the results of [Bol+10a]. We end the paper with some further discussion and closing remarks in Section 5.5. 166 5.2 Lower bounds on point-to-point connection probabilities The goal of this section is to prove that point-to-point connection probabilities are uniformly bounded away from zero in 𝜀-supercritical percolation on finite vertex-transitive graphs, where all relevant constants depend only on the parameter 𝜀 > 0. Given a finite graph 𝐺 and constants 0 < 𝛼, 𝛿 ≤ 1, we define  𝑝 𝑐 (𝛼, 𝛿) = 𝑝 𝐺 (5.2.1) 𝑐 (𝛼, 𝛿) = inf 𝑝 ∈ [0, 1] : P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝛿 , so that 𝑝 is 𝜀-supercritical if and only if (1 − 𝜀) 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) and |𝑉 (𝐺)| ≥ 2𝜀 −3 . By continuity and strict monotonicity, 𝑝 𝑐 (𝛼, 𝛿) is equivalently the unique parameter satisfying P 𝑝 𝑐 (𝛼,𝛿) (∥𝐾1 ∥ ≥ 𝛼) = 𝛿. Theorem 5.2.1. Let 𝐺 = (𝑉, 𝐸) be a finite, connected, simple, vertex-transitive graph and let 𝜀 > 0. If |𝑉 | ≥ 2𝜀 −3 and 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) then   P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏(𝜀) := exp −105 · 𝜀 −18 for every 𝑢, 𝑣 ∈ 𝑉. The exact value of this bound is not important for our purposes, and we have not attempted to optimize the relevant constants. For bounded degree graphs, a similar estimate follows from an argument essentially due to Schramm, which is recorded in [Ben01a] and in more detail in [HT21c, Lemma 2.1]. This argument yields in particular that if 𝐺 = (𝑉, 𝐸) is a finite vertex-transitive graph and 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) then      1 2 1 2 P 𝑝 (𝑢 ↔ 𝑣) ≥ exp −3 2 ∨ log 2 ∨ (5.2.2) 𝑝 𝑝 𝜀 𝜀 for every 𝑢, 𝑣 ∈ 𝑉. This bound is adequate for our purposes in the bounded degree case, in which the condition 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) bounds 𝑝 away from zero when |𝑉 | ≥ 2𝜀 −3 by lemma 5.2.6 below. As such, readers who are already familiar with (5.2.2) and are only interested in the bounded degree case of our results may safely skip the remainder of this section. The estimate (5.2.2) does not yield a uniform lower bound on the two-point function in the high-degree case however, making a more refined analysis necessary at this level of generality. Remark 5.2.1. The assumption that 𝐺 is simple is not really needed for this theorem to hold: the proof works whenever 𝐺 has degree at most |𝑉 |, and yields a similar statement (with different constants) under the assumption that the degrees are bounded by 𝐶 |𝑉 | for some constant 𝐶. No such uniform two-point lower bound holds without this assumption, as can be seen by taking the product 𝐾𝑛 □𝐾2 and replacing each edge of 𝐾𝑛 by a large number of parallel edges. 167 Remark 5.2.2. For infinite transitive graphs, it was proven by Lyons and Schramm [LS99] (in the unimodular case) and Tang [Tan19] (in the nonunimodular case) that there is a unique infinite cluster at 𝑝 if and only if inf 𝑥,𝑦 P 𝑝 (𝑥 ↔ 𝑦) > 0. As such, one might naively expect that we could deduce our main theorems on uniqueness directly from Theorem 5.2.1 via a similar argument. This does not appear to be the case: firstly, we note that there do exist large finite transitive graphs and values of 𝜀 such that there are multiple giant components with good probability at certain values of 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀), despite there always being a uniform lower bound on the two-point function at such values of 𝑝 by Theorem 5.2.1. Indeed, the product 𝐾𝑛 □𝐾2 and the elongated torus (Z/𝑛Z) × (Z/2𝑛 Z) both have this property. Secondly, the proofs of [LS99; Tan19] both rely essentially on indistinguishability theorems that are of an ergodic-theoretic nature and do not generalize to the finite-volume setting. We will deduce Theorem 5.2.1 as an analytic consequence of the following inductive lemma. Recall that we write ∥ 𝐴∥ = | 𝐴|/|𝑉 | for the density of a set of vertices in a finite graph. Lemma 5.2.2 (Two-point induction step). Let 𝐺 = (𝑉, 𝐸) be a finite, connected, vertex-transitive graph and let 𝑝 ≥ 1/(2|𝑉 |). If 𝜏 > 0 and 𝑘 ≥ 1 are such that ∥{𝑣 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏}∥ ≥ 2−𝑘 for every 𝑢 ∈ 𝑉 then there exists ℓ ∈ {0, . . . , 𝑘 − 1} such that n o 3 𝑘−ℓ 𝑣 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏 3 2 2−2ℓ−10 ≥ 2−ℓ for every 𝑢 ∈ 𝑉. Before proving this lemma we first state and prove some general facts that will be used in the proof. The first is a standard bound on the diameter of dense graphs whose proof we include for completeness. Lemma 5.2.3 (Dense graphs have bounded diameter). Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected graph. If every vertex of 𝐺 has degree at least 𝑎 |𝑉 | for some 𝑎 > 0 then diam 𝐺 ≤ (3 − 𝑎)/𝑎. Proof of Lemma 5.2.3. Let 𝑎 > 0, and let 𝐺 be a finite, connected graph with minimum vertex degree at least 𝑎 |𝑉 |. Let 𝑢 and 𝑣 be two vertices of 𝐺 and let 𝑢 = 𝑢 0 , 𝑢 1 , . . . , 𝑢 𝑘 = 𝑣 be a minimal length path from 𝑢 to 𝑣. Writing 𝑘 = 3𝑚 + 𝑟 where 𝑚 is a positive integer and 𝑟 ∈ {0, 1, 2}, it suffices to prove that 𝑚 + 1 ≤ 1/𝑎. Suppose for contradiction that this is not the case. Then 𝑚 ∑︁ 𝑖=0 deg 𝑢 3𝑖 ≥ (𝑚 + 1) · min deg 𝑢 3𝑖 > |𝑉 | , 0≤𝑖≤𝑚 168 (5.2.3) so, by the pigeonhole principle, we can find 𝑖, 𝑗 ∈ {0, . . . , 𝑚} with 𝑖 < 𝑗 such that 𝑢 3𝑖 and 𝑢 3 𝑗 have a common neighbour 𝑤. It follows that 𝑢 0 , 𝑢 1 , . . . , 𝑢 3𝑖 , 𝑤, 𝑢 3 𝑗 , . . . 𝑢 𝑘−1 , 𝑢 𝑘 is a shorter path from 𝑢 to 𝑣 than 𝑢 0 , . . . , 𝑢 𝑘 , a contradiction. □ The second ingredient we will require is a quantitative form of insertion-tolerance. In bounded degree contexts, insertion-tolerance usually refers to the fact that the conditional probability of an edge being included in the configuration given the status of every other edge is bounded away from zero. Such a statement need not be valid in regimes of interest in the high-degree case, where 𝑝 may be very small. Intuitively, the following proposition instead gives conditions in which we can insert exactly one edge into some sufficiently large set of edges with good probability. This proposition will be used again in the proof of Lemma 5.4.9. Proposition 5.2.4 (Quantitative insertion tolerance). Let 𝐺 = (𝑉, 𝐸) be a finite graph, let 𝑝 ∈ (0, 1), and let 𝐹 ⊆ 𝐸 be a collection of edges. Let 𝐴 ⊆ {0, 1} 𝐸 be an event, let 𝜂 > 0 and suppose that for each configuration 𝜔 ∈ 𝐴 there is a distinguished subset 𝐹 [𝜔] with 𝐹 [𝜔] ⊆ 𝐹 \ 𝜔 and |𝐹 [𝜔] | ≥ 𝜂 |𝐹 |. If we define 𝐴+ := {𝜔 ∪ {𝑒} : 𝜔 ∈ 𝐴 and 𝑒 ∈ 𝐹 [𝜔]} then P 𝑝 ( 𝐴+ ) ≥ 𝑝|𝐹 | 𝜂2 · · P 𝑝 ( 𝐴) 2 . 1 − 𝑝 𝑝|𝐹 | + 1 Note that the hypotheses of this proposition force there to be at most (1 − 𝜂)|𝐹 | open edges in 𝐹 [𝜔] whenever the event 𝐴 holds. The lower bound appearing here has not been optimized, and a more careful implementation of our argument would give a P 𝑝 ( 𝐴)/(− log P 𝑝 ( 𝐴)) term in place of the P 𝑝 ( 𝐴) 2 term above. The only important conclusion of this proposition for our purposes will be that if 𝑝|𝐹 |, P 𝑝 ( 𝐴), and 𝜂 are all bounded below by some constant 𝑐 > 0 then there exists 𝛿 = 𝛿(𝑐) > 0 such that P𝐺𝑝 ( 𝐴+ ) ≥ 𝛿. Proof of Proposition 6.6.4. We will abbreviate P = P𝐺𝑝 and E = E𝐺𝑝 . Given a set of edges 𝐻, we write 𝜔| 𝐻 for the configuration of open and closed edges in 𝐻, which by a standard abuse of notation we will think of both as a subset of 𝐻 and a function 𝐻 → {0, 1}. We can sample a configuration 𝜔 with law P using the following procedure: 1. Sample the restriction 𝜔| 𝐸\𝐹 of 𝜔 to 𝐸 \ 𝐹. 2. Sample a uniformly random permutation 𝜋 of the edges of 𝐹 independently of 𝜔| 𝐸\𝐹 . 3. Sample a binomial random variable 𝑁 ∼ Binomial(|𝐹 | , 𝑝) independently of 𝜔| 𝐸\𝐹 and 𝜋. 169 4. Set 𝜔| 𝐹 := 𝜋({1, . . . 𝑁 }). Let P̃ denote the joint measure of 𝜔, 𝜋, and 𝑁 sampled as in this procedure. Let 𝐴 and 𝐹 [𝜔] be as in the statement of the proposition. The assumption that |𝐹 [𝜔]| ≥ 𝜂|𝐹 | > 0 and 𝐹 [𝜔] ⊆ 𝐹 \ 𝜔 for every 𝜔 ∈ 𝐴 guarantees that 𝑁 ≤ (1 − 𝜂)|𝐹 | < |𝐹 | on the event that 𝜔 ∈ 𝐴. By construction we have that 𝜔 = 𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑁 }) so that for each 𝑛 ∈ {1, . . . , |𝐹 |} we can rewrite {𝜔 ∈ 𝐴+ } ∩ {𝑁 = 𝑛} = {𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑁 }) ∈ 𝐴+ } ∩ {𝑁 = 𝑛} = {𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑛}) ∈ 𝐴+ } ∩ {𝑁 = 𝑛}. (5.2.4) Note that the two events on the second line are independent. One way for the union 𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑛}) to belong to 𝐴+ is for 𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑛 − 1}) to belong to 𝐴 and for 𝜋(𝑛) to belong   to the set 𝐹 𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑛 − 1}) . Since |𝐹 [𝜈] | ≥ 𝜂 |𝐹 | for every configuration 𝜈 ∈ 𝐴, 𝜋(𝑛) belongs to this set with probability at least 𝜂 conditional on 𝜔| 𝐸\𝐹 and 𝜋({1, . . . , 𝑛 − 1}) and we deduce that

P̃ 𝜔 ∈ 𝐴+ and 𝑁 = 𝑛 ≥ 𝜂P̃ 𝜔| 𝐸\𝐹 ∪ 𝜋({1, . . . , 𝑛 − 1}) ∈ 𝐴 P̃ (𝑁 = 𝑛) (5.2.5) P̃ (𝑁 = 𝑛) . = 𝜂P̃ (𝜔 ∈ 𝐴 and 𝑁 = 𝑛 − 1) · P̃ (𝑁 = 𝑛 − 1) The ratio of probabilities appearing here is given by P̃ (𝑁 = 𝑛) = P̃ (𝑁 = 𝑛 − 1) |𝐹 |
𝑛 |𝐹 |−𝑛 𝑝(|𝐹 | − 𝑛 + 1) 𝑛 𝑝 (1 − 𝑝) =
|𝐹 | 𝑛−1 (1 − 𝑝) |𝐹 |−𝑛+1 (1 − 𝑝)𝑛 𝑛−1 𝑝 (5.2.6) and we deduce that P̃(𝜔 ∈ 𝐴+ ) ≥ 𝜂 |𝐹 | ∑︁ 𝑝(|𝐹 | − 𝑛 + 1) P̃ (𝜔 ∈ 𝐴 and 𝑁 = 𝑛 − 1) (1 − 𝑝)𝑛 𝑛=1   𝑝𝜂 |𝐹 | − 𝑁 = Ẽ 𝜔 ∈ 𝐴 P̃(𝜔 ∈ 𝐴), 1− 𝑝 𝑁 +1 (5.2.7) where we used that 𝑁 ≤ (1−𝜂)|𝐹 | < |𝐹 | whenever 𝜔 ∈ 𝐴 in the second line. Since (|𝐹 | −𝑥)/(𝑥 +1) is convex we may apply Jensen’s inequality to deduce that P̃(𝜔 ∈ 𝐴+ ) ≥ 𝑝𝜂 |𝐹 | − Ẽ[𝑁 | 𝜔 ∈ 𝐴] P̃(𝜔 ∈ 𝐴). 1 − 𝑝 Ẽ[𝑁 | 𝜔 ∈ 𝐴] + 1 170 (5.2.8) Using again that 𝑁 ≤ (1 − 𝜂)|𝐹 | whenever 𝜔 ∈ 𝐴 and using the bound Ẽ[𝑁 | 𝜔 ∈ 𝐴] ≤ Ẽ[𝑁]/P̃(𝜔 ∈ 𝐴) = 𝑝|𝐹 |/P̃(𝜔 ∈ 𝐴) we deduce that P̃(𝜔 ∈ 𝐴+ ) ≥ 𝑝|𝐹 | 𝜂2 · · P̃(𝜔 ∈ 𝐴) 2 , 1 − 𝑝 𝑝|𝐹 | + 1 concluding the proof. (5.2.9) □ We are now ready to prove lemma 5.2.2. Proof of lemma 5.2.2. Let 𝐺 ∗ be the graph with vertex set 𝑉 in which two vertices 𝑢 and 𝑣 are connected by an edge if and only if P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏. The graph 𝐺 ∗ is simple, vertex-transitive, and is dense in the sense that its vertex degrees are all at least 2−𝑘 |𝑉 |. Let 𝐶 be a connected component of 𝐺 ∗ , noting that 2−𝑘 ≤ ∥𝐶 ∥ ≤ 1. Applying Lemma 5.2.3 to the subgraph of 𝐺 ∗ induced by 𝐶 implies that diam(𝐶) ≤ (3 − 2−𝑘 /∥𝐶 ∥)/(2−𝑘 /∥𝐶 ∥) = (3∥𝐶 ∥ − 2−𝑘 )/2−𝑘 ≤ 3 · 2 𝑘 ∥𝐶 ∥. Thus, if 𝑢 and 𝑣 belong to the same connected component of 𝐺 ∗ then there exists a sequence of vertices 𝑢 = 𝑢 0 , 𝑢 1 , . . . , 𝑢 𝑛 = 𝑣 with 𝑛 ≤ 3 · 2 𝑘 ∥𝐶 ∥ such that 𝑢𝑖 is adjacent to 𝑢𝑖−1 in 𝐺 ∗ for every 1 ≤ 𝑖 ≤ 𝑛. It follows by the Harris-FKG inequality that P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑛 Ö P 𝑝 (𝑢𝑖−1 ↔ 𝑢𝑖 ) ≥ 𝜏 3·2 ∥𝐶 ∥ =: 𝜏1 𝑘 (5.2.10) 𝑖=1 for every 𝑢, 𝑣 in the same connected component of 𝐺 ∗ . If 𝐺 ∗ is connected then ∥𝐶 ∥ = 1 and the claim follows with ℓ = 0, so we may assume that 𝐺 ∗ is disconnected and that 2−ℓ−1 ≤ ∥𝐶 ∥ < 2−ℓ for some ℓ ∈ {1, . . . , 𝑘 − 1}. Since 𝐺 is connected there must exist at least one edge of 𝐺 connecting 𝐶 to 𝑉\𝐶. Since the connected-component equivalence relation on 𝐺 ∗ is invariant under the automorphisms of 𝐺, it follows by vertex-transitivity of 𝐺 that every vertex in 𝐶 belongs to an edge from 𝐶 to 𝑉\𝐶. Letting 𝜕𝐸 𝐶 be the set of edges of 𝐺 with one endpoint in 𝐶 and the other in 𝑉 \ 𝐶, it follows that |𝜕𝐸 𝐶 | ≥ |𝐶 |. Since there are |𝑉 |/|𝐶 | connected components of 𝐺 ∗ , it follows by the pigeonhole principle that there exists a connected component 𝐶 ′ ≠ 𝐶 of 𝐺 ∗ such that at least |𝐶 | 2 /|𝑉 | edges of |𝜕𝐸 𝐶 | have their other endpoint in 𝐶 ′. Let 𝐼 be the set of oriented edges 𝑒 of 𝐺 with tail 𝑒 − ∈ 𝐶 and head 𝑒 + ∈ 𝐶 ′, so that |𝐼 | ≥ |𝐶 | 2 /|𝑉 |. Fix vertices 𝑢 ∈ 𝐶 and 𝑣 ∈ 𝐶 ′ that are the endpoints of some edge in 𝐼. We claim that P 𝑝 (𝑢 ↔ 𝑣) ≥ 171 𝜏18 28 ∥𝐶 ∥ 2 . (5.2.11) Let 𝐿 be the random set of oriented edges 𝑒 ∈ 𝐼 such that 𝑢 ↔ 𝑒 − and 𝑣 ↔ 𝑒 + . For each 𝑒 ∈ 𝐼, the Harris-FKG inequality and eq. (5.2.10) imply that P 𝑝 (𝑢 ↔ 𝑒 − and 𝑣 ↔ 𝑒 + ) ≥ 𝜏12 and hence that E 𝑝 |𝐿| ≥ 𝜏12 |𝐼 |. Applying Markov’s inequality to |𝐼 \ 𝐿| we obtain that !   𝜏12 1 2 P 𝑝 |𝐿| ≥ |𝐼 | = 1 − P 𝑝 |𝐼 \ 𝐿| > (2 − 𝜏1 )|𝐼 | 2 2 (5.2.12) 2 − 2𝜏12 𝜏12 1 2 ≥ 1− = ≥ 𝜏 . 2 − 𝜏12 2 − 𝜏12 2 1 𝜏2 Let 𝐴 be the event that |𝐿| ≥ 21 |𝐼 | and that every edge of 𝐿 is closed, so that {𝑢 ↔ 𝑣} ⊇ {|𝐿| ≥ 𝜏12 |𝐼 |/2} \ 𝐴. If P 𝑝 ( 𝐴) ≤ 𝜏12 /4 then ! 𝜏12 1 (5.2.13) P 𝑝 (𝑢 ↔ 𝑣) ≥ P 𝑝 |𝐿| ≥ |𝐼 | − P 𝑝 ( 𝐴) ≥ 𝜏12 , 2 4 which is stronger than the claimed inequality, so we may assume that P 𝑝 ( 𝐴) ≥ 𝜏12 /4. In this case, applying Proposition 6.6.4 with 𝐹 = 𝐼, 𝐹 [𝜔] = 𝐿, and 𝜂 = 𝜏12 /2 yields that 𝜏18 𝜏18 𝑝|𝐼 | |𝐼 | |𝐶 | 2 · ≥ · ≥ · P 𝑝 (𝑢 ↔ 𝑣) ≥ P 𝑝 ( 𝐴 ) ≥ , 64 𝑝|𝐼 | + 1 64 |𝐼 | + 2|𝑉 | 64 |𝐶 | 2 + 2|𝑉 | 2 + 𝜏18 (5.2.14) 1 2 where we used the assumption 𝑝 ≥ 2|𝑉 | in the second inequality and the inequality |𝐼 | ≥ |𝐶 | /|𝑉 | in the third. Bounding |𝐶 | 2 + 2|𝑉 | 2 by 4|𝑉 | 2 completes the proof of (5.2.11). It follows from this inequality, (5.2.10), and a further application of Harris-FKG that P 𝑝 (𝑢 ↔ 𝑤) ≥ P 𝑝 (𝑢 ↔ 𝑣)P 𝑝 (𝑣 ↔ 𝑤) ≥ 𝜏19 28 ∥𝐶 ∥ 2 (5.2.15) for every 𝑤 ∈ 𝐶 ′. The same inequality also holds for every 𝑤 ∈ 𝐶 by (5.2.10). Thus, recalling that 𝑘 1 ≤ ℓ < 𝑘 is such that 2−ℓ−1 ≤ ∥𝐶 ∥ < 2−ℓ and using that 𝜏1 = 𝜏 3·2 ∥𝐶 ∥ , we deduce that n o 3 𝑘−ℓ 𝑤 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑤) ≥ 𝜏 3 2 2−2ℓ−10 ≥ ∥𝐶 ∪ 𝐶 ′ ∥ ≥ 2−ℓ , (5.2.16) completing the proof. (The reason why we have worked so hard to get an extra factor of 2 via ∥𝐶 ∪ 𝐶 ′ ∥ ≥ 2∥𝐶 ∥ in the final inequality above will become clear shortly.) □ lemma 5.2.2 implies the following general inequality by induction, from which we will deduce Theorem 5.2.1 as a special case. 172 1 Lemma 5.2.5. Let 𝐺 = (𝑉, 𝐸) be a finite, connected, vertex-transitive graph and let 𝑝 ≥ 2|𝑉 | . If 𝜏 > 0 and 𝑘 ≥ 0 are such that ∥{𝑣 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏}∥ ≥ 2−𝑘 for every 𝑢 ∈ 𝑉 then P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏 (54) 2−𝑘 (54) 𝑘 𝑘 for every 𝑢, 𝑣 ∈ 𝑉. 1 Proof of lemma 5.2.5. Fix 𝐺 = (𝑉, 𝐸) and 𝑝 ≥ 2|𝑉 | . Applying lemma 5.2.2 recursively implies that there exists a decreasing sequence of non-negative integers 𝑘 = 𝑘 0 > 𝑘 1 > · · · > 𝑘 𝑚 = 0 with 𝑚 ≤ 𝑘 such that if we define the sequence of positive real numbers 𝜏0 , . . . , 𝜏𝑚 recursively by 𝜏0 = 𝜏 and 3 𝑘𝑖 −𝑘𝑖+1 −2𝑘 𝜏𝑖+1 = 𝜏𝑖3 2 2 𝑖+1 −10 (5.2.17) for each 0 ≤ 𝑖 ≤ 𝑚 − 1 then ∥{𝑣 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜏𝑖 }∥ ≥ 2−𝑘 𝑖 (5.2.18) for each 1 ≤ 𝑖 ≤ 𝑚. It follows by induction on 𝑖 that 𝜏𝑖 = 𝜏 33𝑖 2 𝑘−𝑘𝑖 𝑖  Ö 2 −2𝑘 𝑗 −10  33(𝑖− 𝑗 ) 2𝑘 𝑗 −𝑘𝑖 (5.2.19) 𝑗=1 for every 0 ≤ 𝑖 ≤ 𝑚 and hence that 𝜏𝑚 = 𝜏 33𝑚 2 𝑘 𝑚  Ö 2 −2𝑘 𝑗 −10  33(𝑚− 𝑗 ) 2𝑘 𝑗 ≥𝜏 33𝑚 2 𝑘 𝑗=1 𝑚  Ö 2 −2𝑘−10  33(𝑚− 𝑗 ) 2𝑘 𝑗=1 = 𝜏3 3𝑚 2 𝑘 Í𝑚 3 −3 𝑗 𝑗=1 3𝑚 2 𝑘 2−(2𝑘+10)3 (5.2.20) 3𝑚 2 𝑘 2−3 (5.2.21) ≥ 𝜏3 3𝑚 𝑘2 𝑘 , Í where we used the inequality (2𝑘 + 10) 𝑚𝑗=1 3−3 𝑗 ≤ 12𝑘 · (1/26) ≤ 𝑘 for 𝑘 ≥ 1 to simplify the final expression. The claim follows since 𝑚 ≤ 𝑘 and 33 · 2 = 54. □ To deduce Theorem 5.2.1 from lemma 5.2.5 we will need the following elementary but useful lower bound on the critical probability. Lemma 5.2.6. If 𝐺 is a finite graph with maximum degree 𝑑 and 𝜀 > 0 is such that |𝑉 | ≥ 2𝜀 −3 then 𝑝 𝑐 (𝜀, 𝜀) ≥ 1/2𝑑. In particular, if 𝑝 is 𝜀-supercritical then 𝑝 ≥ 1/2𝑑. 173 Proof. Fix a vertex 𝑣 ∈ 𝑉. For each 𝑟, the expected number of open simple paths of length 𝑟 starting Í∞ 𝑟 𝑟 at 𝑣 is at most 𝑑 (𝑑 − 1) 𝑟−1 ≤ 𝑑 𝑟 and it follows that if 𝑝 < 1/2𝑑 then E|𝐾𝑣 | ≤ 𝑟=0 𝑝 𝑑 < 2. On the other hand, if 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) then ∑︁ E 𝑝 |𝐾𝑣 | ≥ E 𝑝 |𝐾1 | 2 ≥ 𝜀 3 |𝑉 | 2 , 𝑣∈𝑉 so if the inequalities 𝑝 < 1/2𝑑 and 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) both hold then |𝑉 | < 2𝜀 −3 . □ We are now ready to conclude the proof of Theorem 5.2.1. Proof of Theorem 5.2.1. Since 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀) we have that P 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀 and hence that P 𝑝 (∥𝐾𝑢 ∥ ≥ 𝜀) ≥ 𝜀P 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀 2 for every 𝑢 ∈ 𝑉 by vertex-transitivity. It follows in Í particular that 𝑣∈𝑉 P 𝑝 (𝑢 ↔ 𝑣) = E 𝑝 |𝐾𝑢 | ≥ 𝜀 3 |𝑉 | and hence by Markov’s inequality that n 𝑣 ∈ 𝑉 : P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜀3 o 𝜀3 ≥ . 2 2 (5.2.22) Moreover, since |𝑉 | ≥ 2𝜀 −3 and 𝐺 is simple it follows from lemma 5.2.6 that 𝑝 ≥ 1/2𝑑 ≥ 1/2|𝑉 |. Thus, applying lemma 5.2.5 with 𝜏 = 𝜀 3 /2 and 𝑘 = ⌈log2 (2/𝜀 3 )⌉ we deduce by elementary calculations that   2 3 ⌈log2 (2/𝜀 3 )⌉ ⌈log2 (2/𝜀 3 )⌉ P 𝑝 (𝑢 ↔ 𝑣) ≥ exp −(54) log 3 − ⌈log2 (2/𝜀 )⌉ (54) 𝜀  3 3 2 2 ≥ exp − 54 · (54) log2 (2/𝜀 ) log 3 − 54 · (54) log2 (2/𝜀 ) log2 3 𝜀 𝜀  log2 (2/𝜀 3 ) − 54 · (54)   2 log2 (2/𝜀 3 ) ≥ exp −162 · (54) log2 3 (5.2.23) 𝜀 for every 𝑢, 𝑣 ∈ 𝑉, where we used the inequality log(2/𝜀 3 ) ≤ log2 (2/𝜀 3 ) in the final inequality. We have by calculus that 𝑥54𝑥 ≤ 64𝑥 /(𝑒 log(64/54)) for every 𝑥 ≥ 0, and using that (64 · 162)/(𝑒 log(64/54)) = 22449.65 . . . ≤ 105 we deduce that   P 𝑝 (𝑢 ↔ 𝑣) ≥ exp −105 · 𝜀 −18 for every 𝑢, 𝑣 ∈ 𝑉 as claimed. (5.2.24) □ 174 5.3 Uniqueness under the sharp density property In this section we prove supercritical uniqueness under the assumption that our infinite set H ⊆ F satisfies the sharp density property, which we now introduce. This section is at the heart of the paper and contains the most significant new techniques. Definition 5.3.1. Let 𝐺 = (𝑉, 𝐸) be a finite graph and let Δ : (0, 1) → (0, 1/2] be decreasing. Recall from Section 5.2 that for each 0 < 𝛼, 𝛿 < 1 we define 𝑝 𝑐 (𝛼, 𝛿) = 𝑝 𝐺 𝑐 (𝛼, 𝛿) := inf{𝑝 ∈ [0, 1] : P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝛿}. We say 𝐺 has the Δ-sharp density property if 𝑝 𝑐 (𝛼, 1 − 𝛿) ≤ 𝑒𝛿 𝑝 𝑐 (𝛼, 𝛿) for every 0 < 𝛼 < 1 and Δ(𝛼) ≤ 𝛿 ≤ 1/2. Let H ⊆ F be an infinite set. Given a H -indexed family (Δ𝐺 )𝐺∈H of decreasing (i.e. nonincreasing) functions Δ𝐺 : (0, 1) → (0, 1/2] such that Δ𝐺 → 0 pointwise as 𝐺 → ∞ in H , we say that H has the (Δ𝐺 )𝐺∈H -sharp density property if 𝐺 has the Δ𝐺 -sharp density property for every 𝐺 ∈ H . We say that H has the sharp density property if there is some H -indexed family of decreasing functions (Δ𝐺 )𝐺∈H with Δ𝐺 : (0, 1) → (0, 1/2] and Δ𝐺 → 0 pointwise as 𝐺 → ∞ in H such that H has the (Δ𝐺 )𝐺∈H -sharp density property. Equivalently, H has the sharp density property if and only if 𝑝 𝐺 (𝛽, 1 − 𝛿) =1 lim sup 𝑐 𝐺 𝐺∈H 𝛽∈[𝛼,1] 𝑝 𝑐 (𝛽, 𝛿) for every 0 < 𝛼 < 1 and 0 < 𝛿 ≤ 1/2. Graphs with subalgebraic vertex degrees can straightforwardly be shown to satisfy the sharp density property using standard sharp threshold theorems [FK96; Bou+92; Tal94], all of which are proven via Fourier analysis on the hypercube. Indeed, applying these theorems in our setting leads to the following proposition, which will be used in the proof of Theorem 5.1.3 and whose proof is deferred to Section 5.3. Proposition 5.3.2. There exists a universal constant 𝐶 such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected vertex-transitive graph with vertex degree 𝑑. Then 𝐺 has the Δ-sharp density property with √︃  log 𝑑   21 ∧ 𝐶 log  if 𝛼 ≥ (2/|𝑉 |) 1/3 |𝑉 | Δ(𝛼) =   21 otherwise.  In particular, if (𝐺 𝑛 )𝑛≥1 = ((𝑉𝑛 , 𝐸 𝑛 ))𝑛≥1 is a sequence of finite, vertex-transitive graphs with |𝑉𝑛 | → ∞ and with subalgebraic degrees 𝑑𝑛 = |𝑉𝑛 | 𝑜(1) then (𝐺 𝑛 )𝑛≥1 has the sharp density property. 175 A more general proposition stating that an infinite subset of F has the sharp density property if and only if it does not have a molecular subsequence is proven in Section 5.4. For now we will focus on the consequences of the sharp density property, leaving the verification of this property to later sections. We now state the main quantitative result of this section. When applying this theorem we will think of 𝜀 > 0 as a fixed constant, the number of vertices |𝑉 | as being large, and Δ(𝜀) as being small. We have not attempted to optimise the universal constants appearing in this theorem. Theorem 5.3.3. Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected, vertex-transitive graph, let 𝜀 ∈ (0, 1], and let 𝜏(𝜀) > 0 be as in Theorem 5.2.1. If 𝐺 has the Δ-sharp density property for some Δ : (0, 1) → (0, 1/2] then    25 𝜀 200Δ(𝜀) + 2 ≤ P 𝑝 ∥𝐾2 ∥ ≥ 𝜆 3 𝜆 𝜀 𝜏(𝜀) 𝜀 𝜏(𝜀)|𝑉 | for every 𝜀-supercritical parameter 𝑝 and every 𝜆 ≥ 1. Corollary 5.3.4. Let H ⊆ F be an infinite set. If H has the sharp density property, then H has the supercritical uniqueness property. In this proof, we think of an event as holding with high probability if the probability of its complement is controlled by |𝑉 | −1 and Δ(𝑥) for some constant 𝑥. Similarly, we think of a realvalued random variable as being concentrated if the random variable lies in an interval of width controlled by |𝑉 | −1 and Δ(𝑥) for some constant 𝑥 with high probability. Fix a parameter 𝑝 that is 𝜀-supercritical with respect to 𝐺. Our plan is as follows. First, in Section 5.3, we show that it is possible to find a parameter 𝑞 with 𝑝 𝑐 (𝜀, 𝜀) ≤ 𝑞 ≤ 𝑝 such that ∥𝐾1 ∥ is concentrated in a small window under P𝑞 . Then, in Section 5.3, we deduce that ∥𝐾2 ∥ is small with high probability under P𝑞 . Finally, in Section 8.4 we introduce the notion of sandcastles and use this notion to prove that non-uniqueness of the giant cluster at the fixed parameter 𝑝 would contradict the established properties of percolation at the well-chosen lower parameter 𝑞. Concentration at a lower parameter Our first step is to use the sharp density property to find another parameter 𝑞 with 𝑝 𝑐 (𝜀, 𝜀) ≤ 𝑞 ≤ 𝑝 such that the largest cluster under P𝑞 is a giant whose density is concentrated in a small interval. Lemma 5.3.5. Let 𝐺 = (𝑉, 𝐸) be a finite graph with the Δ-sharp density property for some Δ. Then for every 𝜀 ∈ (0, 1] and every 𝜀-supercritical parameter 𝑝, there exists a parameter 176 𝑞 ∈ ( 𝑝 𝑐 (𝜀, 𝜀), 𝑝) and a density 𝛼 ≥ 𝜀 such that   1 4Δ(𝜀) + ≤ 2Δ(𝜀). P𝑞 |∥𝐾1 ∥ − 𝛼| ≥ |𝑉 | 𝜀 (5.3.1) Roughly speaking, the idea behind the following proof is that if we pick 𝑞 such that the median — or any other particular quantile — of the density of the largest cluster increases slowly across a small neighbourhood of 𝑞 then the sharp density property implies that the density of the giant at 𝑞 must be concentrated; such a 𝑞 can always be found since a bounded increasing function cannot increase rapidly everywhere. Proof of Lemma 5.3.5. We may assume 4Δ(𝜀) ≤ 𝜀, the lemma being trivial otherwise. Consider the increasing sequence of reals 𝑞 0 , 𝑞 1 , . . . given by 𝑞 𝑗 := 𝑒 𝑗Δ(𝜀) 𝑝 𝑐 (𝜀, 𝜀) for each 𝑗 ≥ 0, and let 𝑘 be the maximum integer such that 𝑞 2𝑘 ≤ 𝑝. We start by finding a simple lower bound for 𝑘. Since 𝑝 is 𝜀-supercritical, we know (1 − 𝜀) −1 · 𝑝 𝑐 (𝜀, 𝜀) ≤ 𝑝 and hence that 𝑘 ≥ 𝑟 for any integer 𝑟 satisfying 𝑒 2𝑟Δ(𝜀) ≤ (1 − 𝜀) −1 . It follows in particular that       1 log(1 + 𝜀) 𝜀 𝜀 1 log 1/(1 − 𝜀) · · , (5.3.2) ≥ ≥ ≥ 𝑘≥ 2 Δ(𝜀) 2 Δ(𝜀) 4Δ(𝜀) 8Δ(𝜀) where we used the inequality 1/(1 − 𝑥) ≥ 1 + 𝑥 for 0 ≤ 𝑥 < 1 in the first inequality, the inequality log(1 + 𝑥) ≥ 𝑥/2 for 0 ≤ 𝑥 ≤ 1 in the second inequality, and the assumption 4Δ(𝜀) ≤ 𝜀 in the final inequality. Now, for each 𝑖 ≥ 0 we define the density 𝜆𝑖 of 𝐾1 under P𝑞𝑖 by 𝜆𝑖 := max{𝛽 ∈ [0, 1] : P𝑞𝑖 (∥𝐾1 ∥ ≥ 𝛽) ≥ 𝜀}, so that 𝜆𝑖 ≥ 𝜆 0 = 𝜀 for every 𝑖 ≥ 0. Since 𝜆𝑖 is increasing in 𝑖 we have that 𝑘 ∑︁ |𝜆 2𝑖 − 𝜆 2(𝑖−1) | = 𝑖=1 𝑘 ∑︁ 𝜆 2𝑖 − 𝜆 2(𝑖−1) = 𝜆 2𝑘 − 𝜆 0 ≤ 1, 𝑖=1 and hence by the pigeonhole principle that there is some 𝑗 ∈ {1, . . . , 𝑘 } such that |𝜆 2 𝑗 − 𝜆 2( 𝑗−1) | = 𝜆 2 𝑗 − 𝜆 2( 𝑗−1) ≤ 1 8Δ(𝜀) ≤ , 𝑘 𝜀 where the final inequality follows from (5.3.2). We will argue that the values 𝑞 = 𝑞 2 𝑗−1 𝛼= and 177 𝜆 2 𝑗 + 𝜆 2( 𝑗−1) 2 satisfy the conclusions of the lemma. Indeed, by definition of 𝜆, we have that  
−1 < 𝜀. P𝑞2( 𝑗 −1) ∥𝐾1 ∥ ≥ 𝜆 2( 𝑗−1) ≥ 𝜀 but P𝑞2 𝑗 ∥𝐾1 ∥ ≥ 𝜆 2 𝑗 + |𝑉 | (5.3.3) We also have by assumption that 4Δ(𝜀) ≤ 𝜀 and 𝜀 ≤ 1/2, so that Δ(𝜀) ≤ 𝜀 ≤ 1 − Δ(𝜀) and hence by the definiton of the Δ-sharp density property that   𝑝 𝑐 (𝛽, 1 − Δ(𝜀)) 𝑝 𝑐 (𝛽, 𝜀) 𝑝 𝑐 (𝛽, 1 − Δ(𝜀))) max , ≤ ≤ 𝑒 Δ(𝜀) 𝑝 𝑐 (𝛽, 𝜀) 𝑝 𝑐 (𝛽, Δ(𝜀)) 𝑝 𝑐 (𝛽, Δ(𝜀)) for every 𝛽 ∈ [𝜀, 1]. Applying this inequality with the values 𝛽 = 𝜆 2( 𝑗−1) and 𝛽 = (𝜆 2 𝑗 + |𝑉 | −1 ) ∧ 1 and using that 𝑒 −Δ(𝜀) 𝑞 2 𝑗 = 𝑞 = 𝑒 Δ(𝜀) 𝑞 2( 𝑗−1) , we deduce from (5.3.3) that  
P𝑞 ∥𝐾1 ∥ ≥ 𝜆 2( 𝑗−1) ≥ 1 − Δ(𝜀) and P𝑞 ∥𝐾1 ∥ ≥ 𝜆2 𝑗 + |𝑉 | −1 ≤ Δ(𝜀). Since |𝛼 − 𝜆 2( 𝑗−1) | and |𝛼 − 𝜆 2 𝑗 | are both bounded by 4Δ(𝜀)/𝜀, it follows that   1 4Δ(𝜀) + ≤ 2Δ(𝜀) P𝑞 |∥𝐾1 ∥ − 𝛼| ≥ |𝑉 | 𝜀 as claimed. □ Concentration implies uniqueness By applying Lemma 5.3.5 with our fixed parameter 𝑝, we obtain a parameter 𝑞 ∈ ( 𝑝 𝑐 (𝜀, 𝜀), 1) and a density 𝛼 ≥ 𝜀 that satisfy (5.3.1). We next argue that concentration of ∥𝐾1 ∥ under P𝑞 implies uniqueness of the giant cluster under P𝑞 . Lemma 5.3.6. Let 𝐺 = (𝑉, 𝐸) be a finite graph, let 𝑞 ∈ (0, 1], and let 𝜏 := min𝑢,𝑣∈𝑉 P𝑞 (𝑢 ↔ 𝑣). The estimate   1 P𝑞 (∥𝐾2 ∥ ≥ 2𝛿) ≤ 1 + 2 P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) 4𝛿 𝜏 holds for every 𝛼, 𝛿 > 0. The idea is that by the two-point connection property (and positive association), on any increasing event, we can connect 𝐾1 and 𝐾2 to form a new largest cluster 𝐾1 with good probability. Thus, given that ∥𝐾1 ∥ is concentrated, ∥𝐾1 ⊔ 𝐾2 ∥ must be close to ∥𝐾1 ∥ with high probability. Equivalently, ∥𝐾2 ∥ = ∥𝐾1 ⊔ 𝐾2 ∥ − ∥𝐾1 ∥ must be close to zero with high probability. Proof of Lemma 5.3.6. By the union bound, P𝑞 (∥𝐾2 ∥ ≥ 2𝛿 and ∥𝐾1 ∥ ≥ 𝛼 − 𝛿) ≥ P𝑞 (∥𝐾2 ∥ ≥ 2𝛿) − P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) . 178 On the event that ∥𝐾2 ∥ ≥ 2𝛿 and ∥𝐾1 ∥ ≥ 𝛼 − 𝛿, every pair of vertices 𝑢, 𝑣 with 𝑢 ∈ 𝐾2 and 𝑣 ∈ 𝐾1 has ∥𝐾𝑢 ∪ 𝐾𝑣 ∥ ≥ 𝛼 + 𝛿, and there are at least 4𝛿2 |𝑉 | 2 such pairs. So, by linearity of expectation, max P𝑞 (∥𝐾𝑢 ∪ 𝐾𝑣 ∥ ≥ 𝛼 + 𝛿) ≥ 1 ∑︁ P𝑞 (∥𝐾𝑢 ∪ 𝐾𝑣 ∥ ≥ 𝛼 + 𝛿) |𝑉 | 2 𝑢,𝑣∈𝑉   ≥ 4𝛿2 P𝑞 (∥𝐾2 ∥ ≥ 2𝛿) − P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) . 𝑢,𝑣∈𝑉 (5.3.4) When ∥𝐾𝑢 ∪ 𝐾𝑣 ∥ ≥ 𝛼 + 𝛿 and 𝑢 ↔ 𝑣, we are guaranteed to have ∥𝐾1 ∥ ≥ 𝛼 + 𝛿 and hence that |∥𝐾1 ∥ − 𝛼| ≥ 𝛿. It follows by Harris’s inequality that P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) ≥ max P𝑞 (∥𝐾𝑢 ∪ 𝐾𝑣 ∥ ≥ 𝛼 + 𝛿) · P𝑞 (𝑢 ↔ 𝑣) 𝑢,𝑣∈𝑉   ≥ 4𝛿2 𝜏 P𝑞 (∥𝐾2 ∥ ≥ 2𝛿) − P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) , and the claim follows by rearranging. (5.3.5) □ Proof of Theorem 5.3.3 via sandcastles So far we have obtained good control over ∥𝐾1 ∥ and ∥𝐾2 ∥ under P𝑞 , where 𝑞 is a well-chosen parameter 𝑝 𝑐 (𝜀, 𝜀) ≤ 𝑞 ≤ 𝑝. We now need to convert this into an upper bound on the probability that the second largest cluster is large under P 𝑝 . We do this by introducing an object we call a sandcastle. This is defined in terms of the canonical monotone coupling (𝜔 𝑞 , 𝜔 𝑝 ) of the percolation measures P𝑞 and P 𝑝 with 𝑞 ≤ 𝑝 on any given graph, where each closed edge of 𝜔 𝑝 is also closed in 𝜔 𝑞 and each open edge of 𝜔 𝑝 is open in 𝜔 𝑞 with probability 𝑞/𝑝. We write P𝑞,𝑝 for the joint law of this coupling. (Recall that in this coupling, when we condition on 𝜔 𝑝 , the states of the edges in 𝜔 𝑞 are still independent of each other.) Informally, a sandcastle is a large connected subgraph of 𝐺 with the property that even knowing that the subgraph is entirely open in 𝜔 𝑝 , there remains a good condtional probability that it contains no large cluster for 𝜔 𝑞 . We fix this ‘good probability’ to be 1/2 in the following definition, but we could have used any other universal constant in (0, 1). Definition 5.3.7. Let 𝐺 = (𝑉, 𝐸) be a finite graph. Let 0 ≤ 𝑞 ≤ 𝑝 ≤ 1 and let 0 ≤ 𝛼, 𝛽 ≤ 1. A [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle is a connected subgraph 𝑆 ⊆ 𝐺 such that ∥𝑆∥ ≥ 𝛽 and P𝑞,𝑝
1 𝐾1 (𝜔 𝑞 ∩ 𝑆) < 𝛼 | 𝑆 ⊆ 𝜔 𝑝 ≥ . 2 We now show that non-uniqueness of the giant cluster under P 𝑝 and uniqueness of the giant cluster under P𝑞 together imply that some cluster must be a sandcastle with good probability under the measure P 𝑝 . 179 Lemma 5.3.8. Let 𝐺 = (𝑉, 𝐸) be a finite graph. For each 0 ≤ 𝑞 ≤ 𝑝 ≤ 1, and 0 < 𝛼, 𝛽 < 1 there exists a vertex 𝑢 such that P 𝑝 (𝐾𝑢 is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle)   ≥ 𝛽 P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) − 4P𝑞 (∥𝐾2 ∥ ≥ 𝛼) . Proof of Lemma 5.3.8. Consider a configuration 𝜈 ∈ {0, 1} 𝐸 in which ∥𝐾2 ∥ ≥ 𝛽 but no clusters are [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastles. Let 𝐴 := 𝐾1 (𝜈) and 𝐵 := 𝐾2 (𝜈). Since ∥ 𝐴∥ ≥ 𝛽 but 𝐴 is not a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle, we know by the definition of sandcastles that

1 P𝑞,𝑝 𝐾1 (𝜔 𝑞 ∩ 𝐴) ≥ 𝛼 | 𝜔 𝑝 = 𝜈 = P𝑞,𝑝 𝐾1 (𝜔 𝑞 ∩ 𝐴) ≥ 𝛼 | 𝐴 ⊆ 𝜔 𝑝 ≥ . 2 The same result holds for 𝐵. Since 𝐴 and 𝐵 are disjoint, the restrictions of 𝜔 𝑞 to 𝐴 and 𝐵 are conditionally independent given 𝜔 𝑝 , and hence
1 𝐾1 (𝜔 𝑞 ∩ 𝐴) ≥ 𝛼 and 𝐾1 (𝜔 𝑞 ∩ 𝐵) ≥ 𝛼 | 𝜔 𝑝 = 𝜈 ≥ . 4 The edges in the boundary of 𝐴 are all closed in 𝜈, disconnecting 𝐴 from 𝐵. Since 𝜔 𝑞 ≤ 𝜔 𝑝 , these edges are also closed in 𝜔 𝑞 when 𝜔 𝑝 = 𝜈. In particular, given that 𝜔 𝑝 = 𝜈, the subgraphs 𝐾1 (𝜔 𝑞 ∩ 𝐴) and 𝐾1 (𝜔 𝑞 ∩ 𝐵) are not connected to each other in 𝜔 𝑞 , and hence P𝑞,𝑝 P𝑞,𝑝 𝐾2 (𝜔 𝑞 ) ≥ 𝛼 | 𝜔 𝑝 = 𝜈

1 𝐾1 (𝜔 𝑞 ∩ 𝐴) ≥ 𝛼 and 𝐾1 (𝜔 𝑞 ∩ 𝐵) ≥ 𝛼 | 𝜔 𝑝 = 𝜈 ≥ . 4 Letting E be the event that ∥𝐾2 (𝜔 𝑝 )∥ ≥ 𝛽 but no cluster in 𝜔 𝑝 is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle, it follows since 𝜈 ∈ E was arbitrary that
1 P𝑞,𝑝 𝐾2 (𝜔 𝑞 ) ≥ 𝛼 | 𝜔 𝑝 ∈ E ≥ . 4 It follows from this and a union bound that
P𝑞 (∥𝐾2 ∥ ≥ 𝛼) ≥ P𝑞,𝑝 𝐾2 (𝜔 𝑞 ) ≥ 𝛼 | 𝜔 𝑝 ∈ E · P 𝑝 (E)  1 ≥ P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) 4  − P 𝑝 (some cluster is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle) , ≥ P𝑞,𝑝 which rearranges to give that P 𝑝 (some cluster is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle) ≥ P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) − 4P𝑞 (∥𝐾2 ∥ ≥ 𝛼) . (5.3.6) 180 Every cluster that is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle contains at least 𝛽 |𝑉 | vertices by definition, and we deduce by linearity of expectation that max P 𝑝 (𝐾𝑢 is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle) 𝑢∈𝑉 1 ∑︁ ≥ P 𝑝 (𝐾𝑢 is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle) |𝑉 | 𝑢∈𝑉 ≥ 𝛽P 𝑝 (some cluster is a [( 𝑝, 𝛽) → (𝑞, 𝛼)]-sandcastle) . The claimed inequality follows from this and (5.3.6). □ We want to use the fact that 𝐾𝑢 is a sandcastle with good probability to contradict the concentration of ∥𝐾1 ∥ under P𝑞 . The rough idea is as follows: as we pass from 𝜔 𝑝 to 𝜔 𝑞 , with good probability this sandcastle disintegrates into only small clusters, none of which are equal to the giant cluster 𝐾1 (𝜔 𝑞 ). Since the status of any edge that does not touch the cluster of the vertex 𝑢 in 𝜔 𝑝 remains conditionally distributed as Bernoulli percolation, this implies that there exists a large set of vertices whose complement contains, with good probability, an 𝜔 𝑞 cluster whose density is close to the typical density of the largest cluster in the whole graph. Using Harris’ inequality and uniqueness of the giant cluster in 𝜔 𝑞 , we deduce that 𝐾1 (𝜔 𝑞 ) is abnormally high with good probability, contradicting the concentration of the giant cluster’s density under P𝑞 . We now begin to make this argument precise. For each subgraph 𝐻 of 𝐺, let 𝐻 denote the set of all edges that have at least one endpoint in the vertex set of 𝐻. Lemma 5.3.9. Let 𝐺 = (𝑉, 𝐸) be a finite, vertex-transitive graph and let 𝑞 ∈ (0, 1]. The estimate   2𝛽P𝑞 (∥𝐾1 ∥ ≥ 𝛽) − 𝛽2
P𝑞 𝐾 1 𝜔 \ 𝐻 ≥ 𝛽 · 2 − 𝛽2    𝛽2 ≤ P𝑞 ∥𝐾1 ∥ ∉ 𝛽, 𝛽 + ∥𝐻 ∥ + P𝑞 (∥𝐾2 ∥ ≥ 𝛽) (5.3.7) 2 holds for every subgraph 𝐻 of 𝐺 and every 0 < 𝛽 < 1. Proof of lemma 5.3.9. Let 𝑋 be the set of vertices that are contained in clusters with density at least 𝛽, noting that P𝑞 (𝑋 ≠ 𝐾1 ) ≤ P𝑞 (∥𝐾1 ∥ ≤ 𝛽) + P𝑞 (∥𝐾2 ∥ ≥ 𝛽) (5.3.8) and hence that     𝛽2 𝛽2 P𝑞 ∥ 𝑋 ∥ ≥ 𝛽 + ∥𝐻 ∥ ≤ P𝑞 ∥𝐾1 ∥ ≥ 𝛽 + ∥𝐻 ∥ 2 2 + P𝑞 (∥𝐾1 ∥ ≤ 𝛽) + P𝑞 (∥𝐾2 ∥ ≥ 𝛽) . (5.3.9) 181 We have by Markov’s inequality applied to ∥𝐻 \ 𝑋 ∥ that       𝛽2 𝛽2 P𝑞 ∥ 𝑋 ∩ 𝐻 ∥ ≥ ∥𝐻 ∥ = 1 − P𝑞 ∥𝐻 \ 𝑋 ∥ > 1 − ∥𝐻 ∥ 2 2  −1  𝛽2 ∥𝐻 ∥ −1 E𝑞 ∥𝐻 \ 𝑋 ∥ ≥ 1− 1− 2   −1 𝛽2 =1− 1− P𝑞 (∥𝐾𝑢 ∥ < 𝛽), 2 (5.3.10) where 𝑢 is an arbitrary vertex and we used vertex-transitivity in the last line. Bounding P𝑞 (∥𝐾𝑢 ∥ ≥ 𝛽) ≥ 𝛽P𝑞 (∥𝐾1 ∥ ≥ 𝛽) we deduce that  P𝑞  2 − 2𝛽P𝑞 (∥𝐾1 ∥ ≥ 𝛽) 𝛽2 ∥𝑋 ∩ 𝐻∥ ≥ ∥𝐻 ∥ ≥ 1 − 2 2 − 𝛽2 2𝛽P𝑞 (∥𝐾1 ∥ ≥ 𝛽) − 𝛽2 = 2 − 𝛽2 and hence by Harris’ inequality that     𝛽2 𝛽2 ∥𝐻 ∥ P𝑞 ∥ 𝑋 ∥ ≥ 𝛽 + ∥𝐻 ∥ ≥ P𝑞 (∥ 𝑋 \ 𝐻 ∥ ≥ 𝛽) · P𝑞 ∥ 𝑋 ∩ 𝐻 ∥ ≥ 2 2   2𝛽P𝑞 (∥𝐾1 ∥ ≥ 𝛽) − 𝛽2 ≥ P𝑞 ∥𝐾1 (𝜔 \ 𝐻)∥ ≥ 𝛽 · . 2 − 𝛽2 The claim follows by combining (5.3.9) and (5.3.13). (5.3.11) (5.3.12) (5.3.13) □ Proof of Theorem 5.3.3. Write 𝜏 = 𝜏(𝜀) and Δ = Δ(𝜀), fix an 𝜀-supercritical parameter 𝑝, and let 𝑞 and 𝛼 satisfying 𝛼 ≥ 𝜀 be as in Lemma 5.3.5. Define 𝛿= 1 4Δ + 𝜀 |𝑉 | and 𝛽0 = 25𝛿 200Δ 25 = + , 𝜀2 𝜏 𝜀3 𝜏 𝜀 2 𝜏|𝑉 | and fix some 𝛽 ≥ 𝛽0 . This value of 𝛿 is chosen so that   P𝑞 ∥𝐾1 ∥ − 𝛼 ≥ 𝛿 ≤ 2Δ (5.3.14) by Lemma 5.3.5. We will refer to [( 𝑝, 𝛽) → (𝑞, 𝜀/2)]-sandcastles simply as sandcastles for the remainder of the proof. It suffices to prove that P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) < 𝛽0 200Δ ≤𝜀· , 2 𝛽 𝜀 𝜏𝛽 182 so we will suppose for contradiction that the reverse inequality P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) ≥ 200Δ 𝜀 2 𝜏𝛽 (5.3.15) holds. Since ∥𝐾2 ∥ ≤ 1, in this case we must have that 𝛽0 ≤ 𝛽 ≤ 1 and hence that 𝛿 ≤ 𝜀 2 /25 ≤ 𝜀/2 and Δ ≤ 1/200. Since 𝑞 ≥ 𝑝 𝑐 (𝜀, 𝜀), we can apply Theorem 5.2.1 to bound the minimal connection probability min𝑢,𝑣 P𝑞 (𝑢 ↔ 𝑣) ≥ 𝜏 = 𝜏(𝜀). Thus, applying Lemma 5.3.6 yields that     𝜀 4 𝜀 P𝑞 ∥𝐾2 ∥ ≥ ≤ 1 + 2 P𝑞 |∥𝐾1 ∥ − 𝛼| ≥ 2 2 𝜀 𝜏   (5.3.16) 10Δ 4 ≤ 1 + 2 · 2Δ ≤ 2 , 𝜀 𝜏 𝜀 𝜏 where we used the assumption 𝜀/2 ≥ 𝛿 and eq. (5.3.14) in the second inequality. Applying Lemma 5.3.8, we deduce that there exists a vertex 𝑢 such that P 𝑝 (𝐾𝑢 is a sandcastle) ≥ 𝛽P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) − 40𝛽Δ 𝛽 P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) , ≥ 2 𝜀2 𝜏 (5.3.17) where we used the assumption (5.3.15) in the final inequality. By vertex-transitivity, this holds for every vertex 𝑢 ∈ 𝑉. Fix a vertex 𝑢 ∈ 𝑉 and let S𝑢 be the event that 𝐾𝑢 (𝜔 𝑝 ) is a sandcastle. Since 𝜔 𝑞 ≤ 𝜔 𝑝 , no vertex in 𝐾𝑢 (𝜔 𝑝 ) is connected to a vertex of 𝑉 \ 𝐾𝑢 (𝜔 𝑝 ) in 𝜔 𝑞 , and using the fact that 𝛼 − 𝛿 ≥ 𝜀/2 (because 𝛼 ≥ 𝜀 and 𝛿 ≤ 𝜀/2), we have the inclusion of events n o n o 𝐾1 (𝜔 𝑞 ∩ 𝐾𝑢 (𝜔 𝑝 )) < 𝜀/2 ∩ 𝐾1 (𝜔 𝑞 ) ≥ 𝛼 − 𝛿 n   o ⊆ 𝐾1 𝜔 𝑞 \ 𝐾𝑢 (𝜔 𝑝 ) ≥ 𝛼 − 𝛿 . Taking probabilities and using the definition of sandcastles, we deduce that   P𝑞,𝑝 𝐾1 (𝜔 𝑞 \ 𝐾𝑢 (𝜔 𝑝 )) ≥ 𝛼 − 𝛿 S𝑢  
𝜀 ≥ P𝑞,𝑝 𝐾1 (𝜔 𝑞 ∩ 𝐾𝑢 (𝜔 𝑝 )) < S𝑢 − P𝑞,𝑝 𝐾1 (𝜔 𝑞 ) ≤ 𝛼 − 𝛿 | S𝑢 2
1 ≥ − P𝑞,𝑝 𝐾1 (𝜔 𝑞 ) ≤ 𝛼 − 𝛿 | S𝑢 . 2 (5.3.18) Using (5.3.1) and (5.3.17), we can bound the error term P𝑞,𝑝
P𝑞 (|∥𝐾1 ∥ − 𝛼| ≥ 𝛿) 𝐾1 (𝜔 𝑞 ) ≤ 𝛼 − 𝛿 | S𝑢 ≤ P 𝑝 (𝐾𝑢 is a sandcastle) 4Δ 1 ≤ ≤ 𝛽P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) 4 183 (5.3.19) (5.3.20) by the assumption that P 𝑝 (∥𝐾2 ∥ ≥ 𝛽) ≥ 200Δ/(𝜀 2 𝜏𝛽) ≥ 16Δ/𝛽, so that  1  P𝑞,𝑝 𝐾1 (𝜔 𝑞 \𝐾𝑢 (𝜔 𝑝 )) ≥ 𝛼 − 𝛿 | S𝑢 ≥ . 4 (5.3.21) Since the left hand side of (5.3.21) can be written as a weighted sum of conditional probabilities given that 𝐾𝑢 (𝜔 𝑝 ) is equal to a specific sandcastle, there must exist a sandcastle 𝑆 such that  
P𝑞,𝑝 𝐾1 𝜔 𝑞 \𝑆 ≥ 𝛼 − 𝛿 | 𝐾𝑢 (𝜔 𝑝 ) = 𝑆  1 
= P𝑞,𝑝 𝐾1 𝜔 𝑞 \𝐾𝑢 (𝜔 𝑝 ) ≥ 𝛼 − 𝛿 | 𝐾𝑢 (𝜔 𝑝 ) = 𝑆 ≥ . (5.3.22) 4 Since the event {𝐾𝑢 (𝜔 𝑝 ) = 𝑆} depends only on the status of edges in 𝑆, it is independent of the restriction of 𝜔 𝑞 to 𝐸\𝑆, and we deduce that     1
P𝑞 𝐾1 (𝜔\𝑆) ≥ 𝛼 − 𝛿 = P𝑞,𝑝 𝐾1 𝜔 𝑞 \𝑆 ≥ 𝛼 − 𝛿 | 𝐾𝑢 (𝜔 𝑝 ) = 𝑆 ≥ . (5.3.23) 4 On the other hand, using that Δ ≤ 1/200 and hence that 2P𝑞 (∥𝐾1 ∥ ≥ (𝛼 − 𝛿)) ≥ 2(1 − 2Δ) ≥ 𝛼 − 𝛿, lemma 5.3.9 implies that  P𝑞  2 − (𝛼 − 𝛿) 2 𝐾1 (𝜔\𝑆) ≥ 𝛼 − 𝛿 ≤ 2(𝛼 − 𝛿)P𝑞 (∥𝐾1 ∥ ≥ 𝛼 − 𝛿) − (𝛼 − 𝛿) 2      (𝛼 − 𝛿) 2  𝛽 + P𝑞 (∥𝐾2 ∥ ≥ 𝛼 − 𝛿) , (5.3.24) · P𝑞 ∥𝐾1 ∥ ∉ 𝛼 − 𝛿, 𝛼 − 𝛿 + 2 and since 𝛼 − 𝛿 ≥ 𝜀/2 ≤ 1/2 and 𝛽 ≥ 16𝜀 −2 𝛿, and P𝑞 (∥𝐾1 ∥ ≥ 𝛼 − 𝛿) ≥ 1 − 2Δ ≥ 99/100, it follows that   P𝑞 𝐾1 (𝜔\𝑆) ≥ 𝛼 − 𝛿 ≤ 8 4𝜀P𝑞 (∥𝐾1 ∥ ≥ 𝛼 − 𝛿) − 𝜀 2  
· P𝑞 ∥𝐾1 ∥ − 𝛼 ≥ 𝛿 + P𝑞 (∥𝐾2 ∥ ≥ 𝜀/2) . (5.3.25) Applying (5.3.14) and (5.3.16) to control the two probabilities appearing here we obtain that     8 10Δ 48Δ P𝑞 𝐾1 (𝜔\𝑆) ≥ 𝛼 − 𝛿 ≤ · 2Δ + 2 ≤ 3 , (5.3.26) 4𝜀(1 − 2Δ − 𝜀/4) 𝜀 𝜏 𝜀 𝜏 where we used that Δ ≤ 1/200 ≤ 1/8 in the final inequality. The two estimates (5.3.23) and (5.3.26) contradict each other since 200Δ/𝜀 3 𝜏 ≤ 𝛽0 ≤ 1. □ 184 Subalgebraic degree graphs have the sharp density property In this section we prove Proposition 5.3.2. As stated above, this proposition is a straightforward application of standard sharp-threshold theorems. The details of the implementation of the proof are somewhat technical but do not contain any significant new ideas. We will apply the following straightforward consequence of the results of Talagrand [Tal94]. We refer the reader to [Gri18, Chapter 4] and [ODo14] for general background on sharp threshold theorems. Theorem 5.3.10. Let 𝐸 be a finite set and let 𝐴 ⊆ {0, 1} 𝐸 be an increasing event. Let Γ be a group acting on 𝐸 and for each 𝑒 ∈ 𝐸 let Γ𝑒 = {𝛾𝑒 : 𝛾 ∈ Γ} be the orbit of 𝑒 under Γ. There exists a universal constant 𝑐 > 0 such that if 𝐴 is invariant under the action of Γ on {0, 1} 𝐸 then   −1   2 d P 𝑝 ( 𝐴) ≥ 𝑐 𝑝(1 − 𝑝) log P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) log 2 min |Γ𝑒| 𝑒∈𝐸 d𝑝 𝑝(1 − 𝑝) for every 𝑝 ∈ (0, 1). Proof of Theorem 5.3.10. The influence of an edge 𝑒 with respect to 𝐴 under P 𝑝 is defined to be 𝐼 𝑝 ( 𝐴, 𝑒) := P 𝑝 (𝜔 ∪ {𝑒} ∈ 𝐴, 𝜔 \ {𝑒} ∉ 𝐴). Russo’s formula states that if 𝐴 is an increasing event then ∑︁ d P 𝑝 ( 𝐴) = 𝐼 𝑝 ( 𝐴, 𝑒) d𝑝 𝑒∈𝐸 (5.3.27) for every 𝑝 ∈ [0, 1]. It is a theorem of Talagrand [Tal94] that there exists a universal constant 0 < 𝑐 ≤ 1 such that if 𝐴 is increasing then   ∑︁ 𝐼 𝑝 ( 𝐴, 𝑒) 2 𝑝(1 − 𝑝) log ≥ 𝑐 · P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) (5.3.28) 𝑝(1 − 𝑝) 𝑒∈𝐸 log 𝑝(1−𝑝)𝐼1 ( 𝐴,𝑒) 𝑝 and hence that ∑︁ 𝐼 𝑝 ( 𝐴, 𝑒) ≥ 𝑐 · P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) 𝑒∈𝐸  2 · 𝑝(1 − 𝑝) log 𝑝(1 − 𝑝)  −1 log 1 . (5.3.29) 𝑝(1 − 𝑝) max𝑒 𝐼 𝑝 ( 𝐴, 𝑒) (Note that Talagrand states his inequality in terms of open pivotals, so that his expression differs from ours by some factors of 1/𝑝.) Intuitively, this inequality implies that any event that does not depend too strongly on the status of any particular edge must have a sharp threshold, i.e., must have 185 probability changing rapidly from near 0 to near 1 over a short interval. Letting 𝑒 maximize the influence, we have by (5.3.27) and (5.3.29) that  d 1 P 𝑝 ( 𝐴) ≥ · max |Γ𝑒| 𝑝(1 − 𝑝)𝐼 𝑝 ( 𝐴, 𝑒), d𝑝 𝑝(1 − 𝑝) (5.3.30)  −1   2 1 . 𝑐 · P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) · log log 𝑝(1 − 𝑝) 𝑝(1 − 𝑝)𝐼 𝑝 ( 𝐴, 𝑒) Since the function 𝑓 (𝑥) = max{𝑎𝑥, 𝑏 log 1/𝑥} attains its minimum when 1𝑥 log 1𝑥 = 𝑎𝑏 , it follows that 𝑐 d P 𝑝 ( 𝐴) ≥ · P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) d𝑝 𝑝(1 − 𝑝)  · log 2 𝑝(1 − 𝑝)  −1 2 |Γ𝑒| log 𝑝(1−𝑝) 𝑊 𝑐 · P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) ! (5.3.31) where 𝑊 is the Lambert W-function (i.e., the inverse function of 𝑥𝑒 𝑥 ). The claim follows since □ P 𝑝 ( 𝐴) (1 − P 𝑝 ( 𝐴)) ≤ 1 and 𝑊 is increasing and satisfies 𝑊 (𝑥) ≥ 12 log 𝑥 for every 𝑥 ≥ 1. We now apply Theorem 5.3.10 to prove Proposition 5.3.2. Proof of Proposition 5.3.2. Let 𝐺 = (𝑉, 𝐸) be a finite vertex-transitive graph of degree 𝑑 ≥ 2. It follows from lemma 5.2.6 that 𝑝 𝑐 (𝛼, 𝛿) ≥ 1/2𝑑 for every 𝛼, 𝛿 ≥ 𝛼0 := (2/|𝑉 |) 1/3 . It suffices to show that there exists a universal constant 𝐶 ≥ 1 such that √︄ 𝑝 𝑐 (𝛼, 1 − 𝛿) 𝐶 log 𝑑 ≥ 𝑒𝛿 then 𝛿≤ . (5.3.32) If 𝛼, 𝛿 ≥ 𝛼0 and 𝑝 𝑐 (𝛼, 𝛿) log |𝑉 | Fix 𝛼0 ≤ 𝛼 ≤ 1 and 𝛼0 ≤ 𝛿 ≤ 1/2 and write 𝑝 0 = 𝑝 𝑐 (𝛼, 𝛿) and 𝑝 1 = 𝑝 𝑐 (𝛼, 1 − 𝛿). If 𝑝 0 ≤ 𝑝 ≤ 𝑝 1 then P 𝑝 (∥𝐾1 ∥ ≥ 𝛼)(1 − P 𝑝 (∥𝐾1 ∥ ≥ 𝛼)) ≥ 𝛿(1 − 𝛿) ≥ 21 𝛿 and it follows from Theorem 5.3.10 that there exists a universal constant 𝑐 > 0 such that ∫ 𝑝1 d 1 ≥ 1 − 2𝛿 = P 𝑝 (∥𝐾1 ∥ ≥ 𝛼)d𝑝 (5.3.33) 𝑝 0 d𝑝  −1 ∫ 𝑝1  𝑐𝛿 2 ≥ log |𝑉 | 𝑝(1 − 𝑝) log d𝑝, (5.3.34) 2 𝑝(1 − 𝑝) 𝑝0 where we used that every edge has at least |𝑉 |/2 edges in its Aut(𝐺) orbit on any vertex-transitive graph. To estimate this integral we first use the substitution 𝑝 = 𝜙(𝑥) := 𝑒 𝑥 /(𝑒 𝑥 + 1), which satisfies 186 d𝑝/d𝑥 = 𝑒 𝑥 /(𝑒 𝑥 + 1) 2 = 𝑝(1 − 𝑝), to write ∫ 𝑝1  𝑝0 2 𝑝(1 − 𝑝) log 𝑝(1 − 𝑝)  −1 ∫ 𝑥1  2(𝑒 𝑥 + 1) 2 d𝑝 = log 𝑒𝑥 𝑥0 ∫ 𝑥1 1 ≥ d𝑥, 𝑥 0 |𝑥| + log 8  −1 d𝑥 (5.3.35) where we write 𝑥𝑖 = 𝜙−1 ( 𝑝𝑖 ) = log 𝑝𝑖 /(1 − 𝑝𝑖 ) and use the elementary bound (𝑒 𝑥 + 1) 2 /𝑒 𝑥 = 𝑒 𝑥 + 2 + 𝑒 −𝑥 ≤ 4𝑒 |𝑥| in the final inequality. The logarithmic derivative of 𝜙(𝑥) is 1/(𝑒 𝑥 + 1) so that if 𝑝 1 ≥ 𝑒 𝛿 𝑝 0 then we have that ∫ 𝑥1 1 d𝑥 ≥ 𝛿 and hence that 𝑥 1 − 𝑥 0 ≥ (𝑒 𝑥0 + 1)𝛿. 𝑥 𝑥0 𝑒 + 1 It follows that if 𝑝 1 ≥ 𝑒 𝛿 𝑝 0 then ∫ 𝑥0 +(𝑒 𝑥0 +1)𝛿 𝑥0 2 1 d𝑥 ≤ , |𝑥| + log 8 𝑐𝛿 log |𝑉 | from which the claim may easily be proven via case analysis according to whether 𝑥 0 ≤ 0 or 𝑥 0 > 0, noting that 𝑥0 ≥ 𝜙−1 (1/2𝑑) ≥ − log 2𝑑 since 𝑝 0 ≥ 1/2𝑑. In the first case we use that |𝑥| + log 8 ≤ |𝑥 0 | + log 8 + 𝛿 ≤ |𝑥 0 | + 3 for every 𝑥 0 ≤ 𝑥 ≤ 𝑥 0 + 𝛿 to deduce that ∫ 𝑥0 +𝛿 𝛿 2 d𝑥 ≤ ≤ when 𝑥 0 ≤ 0, (5.3.36) 3 + log 2𝑑 |𝑥 0 | + 3 𝑐𝛿 log |𝑉 | 𝑥0 while in the case 𝑥0 > 0 we lower bound the integral by the minimum of the integrand times the length of the interval to obtain that (𝑒 𝑥0 + 1)𝛿 2 2 𝛿≤ ≤ 𝑥 0 2 + log 8 𝑥 0 + (𝑒 + 1)𝛿 + log 8 𝑐𝛿 log |𝑉 | Putting together (5.3.36) and (5.3.37) completes the proof. when 𝑥 0 > 0. (5.3.37) □ Proof of Theorem 5.1.3. The claim follows immediately from Theorems 5.2.1 and 5.3.3 and Proposition 5.3.2. □ 5.4 Non-molecular graphs have the sharp density property In this section we complete the proofs of our main theorems, Theorem 5.1.2 and Theorem 8.1.1. The most important remaining step is to deduce Theorem 5.1.2 and the implication (i) ⇒ (ii) of Theorem 8.1.1 from Corollary 5.3.4 by proving the following proposition. 187 Proposition 5.4.1. Let H ⊆ F be an infinite set. If H does not have the sharp density property, then H contains a molecular subset. Remark 5.4.1. It follows from Theorem 8.1.1 and Corollary 5.3.4 that the converse of this proposition also holds, that is, that molecular sequences do not have the sharp density property. Recall from Section 5.3 that a sequence of graphs is said to have the sharp density property if the emergence of a giant cluster of a given density always has a sharp threshold. As we saw in Section 5.3, it is an immediate consequence of standard sharp threshold results [FK96; Bou+92; Tal94] that this property holds whenever the graphs in question have bounded or subalgebraic vertex degrees. Indeed, these results imply that any increasing event depending in a sufficiently symmetric way on 𝑚 i.i.d. Bernoulli-𝑝 random variables has a sharp threshold provided that this threshold occurs around a value of 𝑝 that is subalgebraically small in 𝑚; lemma 5.2.6 implies that the latter condition is satisfied for the event {∥𝐾1 ∥ ≥ 𝛼} whenever 𝐺 𝑛 has subalgebraic degrees. Unfortunately it is not true in general that every symmetric increasing event has a sharp threshold without this condition on the location of the threshold. For example, the event that the Erdős–Rényi graph contains a triangle has a coarse threshold on the scale 𝑝 = Θ(𝑛−1 ) and the event that the Erdős–Rényi graph contains a tetrahedron has a coarse threshold on the scale 𝑝 = Θ(𝑛−2/3 ) [AS16, Chapter 10.1]. Thus, to prove Proposition 5.4.1 we will need to use specific properties of the event {∥𝐾1 ∥ ≥ 𝛼} on non-molecular graphs. Our proof will apply a theorem first established by Bourgain [Fri99a] and sharpened by Hatami [Hat12a], which, roughly speaking, states that any event that does not have a sharp threshold must be heavily influenced by the status of a small number of edges. Throughout this section, the prime in e.g. P′𝑝 will always refer to a 𝑝-derivative. We say that an event A is non-trivial if A ≠ ∅ and A 𝑐 ≠ ∅. (The following theorem actually only requires that A ≠ ∅.) Theorem 5.4.2 (Hatami 2012, Corollary 2.10). Let 𝐺 = (𝑉, 𝐸) be a finite graph, and let A ⊆ {0, 1} 𝐸 be a non-trivial increasing event. For every 𝑝 ∈ (0, 1/2] and 𝜖 > 0, there is a set of edges 𝐹 ⊆ 𝐸 such that P 𝑝 (A | 𝐹 ⊆ 𝜔) ≥ 1 − 𝜀 and    2 |𝐹 | ≤ exp 1013 𝑝 · P′𝑝 (A) P 𝑝 (A) −2 𝜖 −2 . In particular, for each 𝜀 > 0 there exists a constant 𝐶 (𝜀) < ∞ such that if P 𝑝 (A) ≥ 𝜀 and 𝑝 · P′𝑝 (A) ≤ 𝜀 −1 then there is a set of edges 𝐹 ⊆ 𝐸 such that P 𝑝 (A | 𝐹 ⊆ 𝜔) ≥ 1 − 𝜀 and |𝐹 | ≤ 𝐶 (𝜀). 188 The relevance of this theorem to sharp-threshold phenomena is made clear by the following elementary lemma, which shows that when P 𝑝 (A) does not have a sharp threshold there must be a good supply of parameters where P 𝑝 ( 𝐴) is not close to 0 or 1 and 𝑝 · P′𝑝 (A) is not large. Note that if A ⊆ {0, 1} 𝐸 is a non-trivial increasing event then its probability P 𝑝 (A) is a strictly increasing function of 𝑝 ∈ [0, 1] by [Gri99, Theorem 2.38] and hence defines an invertible increasing homeomorphism [0, 1] → [0, 1]. Lemma 5.4.3. Let 𝐺 = (𝑉, 𝐸) be a finite graph, and let A ⊆ {0, 1} 𝐸 be a non-trivial increasing event. Define 𝑓 : [0, 1] → [0, 1] by 𝑓 ( 𝑝) := P 𝑝 (A). If 𝑓 −1 (1 − 𝛿) ≥ (1 + 𝜀) 𝑓 −1 (𝛿) for some 𝜀 ∈ (0, 1] and 0 < 𝛿 ≤ 1/2 then    1  4 −1 ′ ≥ L 𝑓 −1 [𝛿, 1 − 𝛿] , L 𝑝 ∈ 𝑓 [𝛿, 1 − 𝛿] : 𝑝 𝑓 ( 𝑝) ≤ 𝜀 2 where L denotes the Lebesgue measure on [0, 1]. Proof of Lemma 5.4.3. Let 𝐼 := 𝑓 −1 [𝛿, 1−𝛿]. The function 𝑓 is differentiable, as it is a polynomial, and satisfies ∫ ∫ ′ −1 𝑝 𝑓 ( 𝑝) d𝑝 ≤ 𝑓 (1 − 𝛿) 𝑓 ′ ( 𝑝) d𝑝 = (1 − 2𝛿) 𝑓 −1 (1 − 𝛿) ≤ 𝑓 −1 (1 − 𝛿). 𝐼 𝐼 On hand, rearranging our hypothesis 𝑓 −1 (1 − 𝛿) ≥ (1 + 𝜀) 𝑓 −1 (𝛿) gives 𝑓 −1 (1 − 𝛿) ≤  the other   1 + 𝜀1 𝑓 −1 (1 − 𝛿) − 𝑓 −1 (𝛿) ≤ 𝜀2 L (𝐼) and hence that 1 L (𝐼) ∫ 𝑝 𝑓 ′ ( 𝑝) d𝑝 ≤ 𝐼 2 . 𝜀 The result follows by applying Markov’s inequality to the normalised Lebesgue measure on 𝐼. □ When the edges in the set 𝐹 given by Theorem 5.4.2 are open, the event A occurs with conditional probability at least 1 − 𝜀. Since P 𝑝 (A | 𝐹 ⊆ 𝜔) ≥ P 𝑝 (A) by Harris’s inequality, this result is only interesting when P 𝑝 (A) is significantly smaller than 1 − 𝜀. In this case, the state of 𝐹 plays a decisive role in determining whether A occurs in the sense that the event {𝜔 ∉ A but 𝜔 ∪ 𝐹 ∈ A} occurs with good probability. This motivates our definition of a subgraph 𝐻 that activates an event A, a generalisation of being a closed pivotal edge. Definition 5.4.4. Let 𝐺 = (𝑉, 𝐸) be a finite graph, let 𝐻 ⊆ 𝐺 be a subgraph, and let A ⊆ {0, 1} 𝐸 be an increasing event. We say 𝐻 activates the event A in a configuration 𝜔 if 𝜔 ∉ A but 𝜔 ∪ 𝐻 ∈ A. For every density 𝛼, we simply say 𝐻 activates 𝛼 to mean 𝐻 activates the event {∥𝐾1 ∥ ≥ 𝛼}, and we label this event Act𝛼 (𝐻). 189 Corollary 5.4.5. For every 𝛿 > 0 there exists 𝜀 > 0 such that if 𝐺 = (𝑉, 𝐸) is a finite, simple, vertextransitive graph and 𝛼, 𝑝 ∈ (0, 1] are such that P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) ∈ [𝛿, 1−𝛿] and 𝑝 ·P′𝑝 (∥𝐾1 ∥ ≥ 𝛼) ≤ 𝛿−1 then there exists a subgraph 𝐻 of 𝐺 such that |𝐸 (𝐻)| ≤ 𝜀 −1 and P 𝑝 (Act𝛼 (𝐻)) ≥ 𝜀. Proof of Corollary 5.4.5. The case 𝑝 ≤ 1/2 follows trivially from Theorem 5.4.2. Now assume 𝑝 ≥ 1/2. Since {∥𝐾1 ∥ ≥ 𝛼} is invariant under automorphisms of 𝐺 and 𝐺 is vertex-transitive we can apply Theorem 5.3.10 to obtain that there exists a universal constant 𝑐 > 0 such that P′𝑝 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝑐 · P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) P 𝑝 (∥𝐾1 ∥ < 𝛼) log |𝑉 | . Plugging in our assumed bounds on 𝑝, 𝑝 · P′𝑝 (∥𝐾1 ∥ ≥ 𝛼), and P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) gives |𝑉 | ≤ 𝑒 2/𝑐𝛿 , 3 3 in which case |𝐸 | ≤ 𝑒 4/𝑐𝛿 and the result holds trivially with 𝜀 = 𝛿 ∧ 𝑒 −4/𝑐𝛿 . □ 3 When the event {∥𝐾1 ∥ ≥ 𝛼} has a coarse (i.e., not sharp) threshold on 𝐺 = (𝑉, 𝐸), this lemma gives us a subgraph 𝐻 ⊆ 𝐺 that has a good probability of activating 𝛼. When 𝐻 has only a single edge 𝑒, we can use this in the other direction to establish a lower bound on the sharpness of the phase transition. Indeed, 𝑒 activates 𝛼 if and only if 𝑒 is closed and pivotal for the event {∥𝐾1 ∥ ≥ 𝛼}. So, by Russo’s formula and lemma 5.2.6 we have that ∑︁ P 𝑝 ( 𝑓 is pivotal for {∥𝐾1 ∥ ≥ 𝛼}) 𝑝 · P′𝑝 (∥𝐾1 ∥ ≥ 𝛼) 𝑝 𝑓 ∈𝐸 ≳ |Orb(e)| P 𝑝 (Act𝛼 (𝑒)) , deg 𝐺 where Orb(e) denotes the orbit of the edge 𝑒 under the action of the automorphism group Aut 𝐺. Contrasting this with the assumed upper bound on the sharpness of the phase transition with which we started, we can extract information about the underlying graph 𝐺. For example, we immediately deduce that 𝐺 is dense, and we are only one step away from concluding that 𝐺 is part of a molecular sequence. Our main challenge is to reduce to this case, that is to say, to show that we can take 𝐻 to be a subgraph with a single edge. During the proof we will want to apply a sprinkling argument, where by slightly increasing the parameter 𝑝 we can make strict subsets of an activator 𝐻 become activators. One difficulty is that as we increase the percolation parameter, we may form a giant cluster with density at least 𝛽, in which case nothing can be an activator. As such, we must carefully choose the amount that we sprinkle by at each step. To this end, we will work only with a special sequence of such values constructed by the following lemma. 190 For the remainder of this section, given a finite, connected, vertex-transitive graph 𝐺 = (𝑉, 𝐸), we write 𝑑 = deg(𝐺) and 𝛼0 = (2/|𝑉 |) 1/3 . Given 𝛽 > 0 and 𝛿 ∈ (0, 1/2] we write 𝐼 = 𝐼 (𝐺, 𝛽, 𝛿) = [ 𝑝 𝑐 (𝛽, 𝛿), 𝑝 𝑐 (𝛽, 1 − 𝛿)] and 𝑄 = 𝑄(𝐺, 𝛽, 𝛿) = {𝑝 ∈ 𝐼 : 𝑝 · P′𝑝 (∥𝐾1 ∥ ≥ 𝛽) ≤ 4𝛿 }. Lemma 5.4.6 (A good sequence for sprinkling). Let 𝐺 = (𝑉, 𝐸) be a finite, connected, vertextransitive graph, let 𝛿 > 0, and let 0 < 𝛽 ≤ 1. There exists a sequence ( 𝑝 𝑛 )𝑛≥1 in 𝑄 such that 𝑝 𝑛+1 − 𝑝 𝑛 ≥ 3−(𝑛+1) L (𝑄) and P 𝑝 𝑛+1 (∥𝐾1 ∥ ≥ 𝛽) − P 𝑝 𝑛 (∥𝐾1 ∥ ≥ 𝛽) ≤ 2−𝑛 for every 𝑛 ≥ 0. Proof of Lemma 5.4.6. We will prove more generally that if 𝑋 ⊆ [0, 1] is a non-empty closed set and 𝑓 : 𝑋 → [0, 1] is an increasing (but not necessarily continuous) function then there exists a sequence (𝑥 𝑛 )𝑛≥1 in 𝑋 such that 𝑥 𝑛+1 − 𝑥 𝑛 ≥ 3−(𝑛+1) L (𝑋) and 𝑓 (𝑥 𝑛+1 ) − 𝑓 (𝑥 𝑛 ) ≤ 2−𝑛 for every 𝑛 ≥ 0. The claim is trivial if L (𝑋) = 0 since we can take 𝑥 𝑛 constant in this case, so we may assume that L (𝑋) > 0. First consider the case 𝑋 = [0, 1]. Let 𝑥 0 := 0 and define (𝑥 𝑛 )𝑛≥1 recursively as follows. Assume we have defined 𝑥 𝑛 for some 𝑛 ≥ 0. Set 𝑥 𝑛,𝑖 := 𝑥 𝑛 + 𝑖3−(𝑛+1) for each 𝑖 ∈ {1, 2, 3} and define    𝑥 𝑛,1 if 𝑓 (𝑥 𝑛,2 ) − 𝑓 (𝑥 𝑛,1 ) ≤ 𝑓 (𝑥 𝑛,3 ) − 𝑓 (𝑥 𝑛,2 ),  𝑥 𝑛+1 :=   𝑥 𝑛,2 otherwise.  For each 𝑛 ≥ 0 write 𝑦 𝑛 := 𝑥 𝑛,3 so that 𝑥 𝑛 ≤ 𝑦 𝑚 for every 𝑛 ≥ 𝑚 ≥ 0. It follows by induction that 𝑥 𝑛 ∈ [0, 1], 𝑥 𝑛+1 − 𝑥 𝑛 ≥ 3−(𝑛+1) , and 𝑓 (𝑦 𝑛 ) − 𝑓 (𝑥 𝑛 ) ≤ 2−𝑛 for every 𝑛 ≥ 0, and the claim follows since 𝑓 (𝑥 𝑛+1 ) ≤ 𝑓 (𝑦 𝑛 ) for every 𝑛 ≥ 0. Now let 𝑋 ⊆ [0, 1] be an arbitrary closed set with L (𝑋) > 0. Since 𝑋 is closed, we can define an increasing function 𝜙 : [0, 1] → 𝑋 such that L ([0, 𝜙(𝑥)] ∩ 𝑋) = 𝑥L (𝑋) for all 𝑥 ∈ [0, 1]. Construct a sequence (𝑥 𝑛 )𝑛≥1 by the above procedure but for the function 𝑓 ◦ 𝜙 instead of 𝑓 . Then the sequence (𝜙(𝑥 𝑛 ))𝑛≥1 has the properties that 𝜙(𝑥 𝑛 ) ∈ 𝑋, 𝜙(𝑥 𝑛+1 ) − 𝜙(𝑥 𝑛 ) ≥ (𝑥 𝑛+1 −𝑥 𝑛 )L (𝑋) ≥ 3−(𝑛+1) L (𝑋), and 𝑓 (𝜙(𝑥 𝑛+1 )) − 𝑓 (𝜙(𝑥 𝑛 )) ≤ 2−𝑛 for every 𝑛 ≥ 0 as required. □ We now state our key technical sprinkling proposition. Proposition 5.4.7 (Reducing to a single edge by sprinkling). Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected, vertex-transitive graph, let 𝛼0 ≤ 𝛿 ≤ 1/2 and 𝛼0 ≤ 𝛼 ≤ 𝛽 ≤ 1, and suppose that 191 𝑝 𝑐 (𝛽, 1 − 𝛿) > 𝑒 𝛿 𝑝 𝑐 (𝛽, 𝛿). Let ( 𝑝 𝑛 )𝑛≥1 be as in Lemma 5.4.6. For each 𝜀 > 0 there exists 𝑁 = 𝑁 (𝛼, 𝛿, 𝜀) such that for each 𝑛 ≥ 𝑁 there exists 𝜂𝑛 = 𝜂𝑛 (𝛼, 𝛿, 𝜀) > 0 such that if 𝐻 is a subgraph of 𝐺 with |𝐸 (𝐻)| ≤ 𝜀 −1 and P 𝑝 𝑛 (Act 𝛽 (𝐻)) ≥ 𝜀 then there exists 𝑒 ∈ 𝐸 (𝐻) such that P 𝑝 𝑚 (Act 𝛽 (𝑒)) ≥ 𝜂𝑛 for every 𝑚 ≥ 𝑛 + 𝑁. Note that the choice of edge 𝑒 ∈ 𝐸 (𝐻) may depend on the choice of 𝑛 ≥ 𝑁 and that the constants 𝑁 and 𝜂𝑛 are independent of 𝐺 and 𝛽. We now show how Proposition 5.4.1 follows from Corollary 5.4.5 and Proposition 5.4.7, deferring the proof of Proposition 5.4.7 to Section 5.4. Proof of Proposition 5.4.1 given Proposition 5.4.7. Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected, vertex-transitive graph of degree 𝑑 and let 𝛼0 = (2/|𝑉 |) 1/3 . Let 𝛼0 ≤ 𝛼 ≤ 𝛽 ≤ 1, let 𝛼0 ≤ 𝛿 ≤ 1/2, and suppose that 𝑝 𝑐 (𝛽, 1 − 𝛿) > 𝑒 𝛿 𝑝 𝑐 (𝛽, 𝛿). It suffices to prove that there exist positive constants 𝑐 = 𝑐(𝛼, 𝛿) and 𝐶 = 𝐶 (𝛼, 𝛿) such that if |𝑉 | ≥ 𝐶 then deg 𝐺 ≥ 𝑐|𝑉 | and there exists an automorphism-invariant set of edges 𝐹 with |𝐹 | ≤ 𝐶 |𝑉 | such 𝐺 \ 𝐹 has at most 𝐶 connected components. Let ( 𝑝 𝑛 )𝑛≥1 be as in Lemma 5.4.6. It follows from Corollary 5.4.5 that there exists 𝜂1 = 𝜂1 (𝛿) such that for each 𝑝 ∈ 𝑄 there exists a subgraph 𝐻 of 𝐺 such that |𝐸 (𝐻)| ≤ 𝜂1−1 and P 𝑝 (Act 𝛽 (𝐻)) ≥ 𝜂1 . Applying this fact with 𝑝 = 𝑝 𝑛 for an appropriately large constant 𝑛, it follows from Proposition 5.4.7 that there exist positive constants 𝜂2 = 𝜂2 (𝛼, 𝛿) and 𝑁1 = 𝑁1 (𝛼, 𝛿) and an edge 𝑒 0 ∈ 𝐸 such that P 𝑝 𝑛 (Act 𝛽 (𝑒 0 )) ≥ 𝜂2 for every 𝑛 ≥ 𝑁1 . Since the edge 𝑒 0 activates 𝛽 if and only if 𝑒 0 is closed and pivotal for the event {∥𝐾1 ∥ ≥ 𝛽}, we have by Russo’s formula that ∑︁ P′𝑝 𝑛 (∥𝐾1 ∥ ≥ 𝛽) = P 𝑝 𝑛 (𝑒 is pivotal for {∥𝐾1 ∥ ≥ 𝛽}) 𝑒∈𝐸
≥ |Orb(𝑒 0 )| P 𝑝 𝑛 Act 𝛽 (𝑒 0 ) ≥ 𝜂2 |Orb(𝑒 0 )| for every 𝑛 ≥ 𝑁1 . Since 𝑝 𝑁1 ≥ 𝑝 𝑐 (𝛽, 𝛿) and 𝛼, 𝛿 ≥ 𝛼0 it follows from lemma 5.2.6 that 𝑝 𝑁1 ≥ 1/2𝑑 and hence that |Orb(𝑒 0 )| 4 ≤ 𝑝 𝑁1 · P′𝑝 𝑁 (∥𝐾1 ∥ ≥ 𝛽) ≤ 𝜂2 1 2𝑑 𝛿 192 for every 𝑛 ≥ 𝑁1 , where the upper bound follows since 𝑝 𝑛 ∈ 𝑄 for every 𝑛 ≥ 1. Since 𝐺 is vertex-transitive |Orb(𝑒 0 )| ≥ 12 |𝑉 | and it follows that the constant 𝑐 2 = 𝑐 2 (𝛼, 𝛿) = 𝛿𝜂2 /16 satisfies 𝑑 ≥ 𝑐 2 |𝑉 |. This establishes the desired density of 𝐺. All that remains is to find an Aut 𝐺-invariant linear-sized set of edges 𝐹 ⊆ 𝐸 that disconnects 𝐺 into a bounded number of components. Let 𝐶1 = 6/𝑐 2 , let 𝜂3 = 𝜂2 ∧ 𝐶1−1 , and let 𝑁2 = 𝑁 (𝛼, 𝛿, 𝜂3 ) and 𝜂4 = 𝜂 𝑁1 ∨𝑁2 (𝛼, 𝛿, 𝜂3 ) be as in Proposition 5.4.7. We claim that if we set 𝑘 = 𝑁1 ∨ 𝑁2 + 𝑁2 then there exists a constant 𝐶2 = 𝐶2 (𝛼, 𝛿) such that the set
𝐹 := {𝑒 ∈ 𝐸 : P 𝑝 𝑘 Act 𝛽 (𝑒) ≥ 𝜂4 } has the desired properties when |𝑉 | ≥ 𝐶2 . This set 𝐹 is clearly Aut 𝐺-invariant. Moreover, by Russo’s formula, since 𝑝 𝑘 · P′𝑝 𝑘 (∥𝐾1 ∥ ≥ 𝛽) ≤ 4𝛿 and 𝑝 𝑘 ≥ 1/2𝑑 ≥ 1/2|𝑉 |, we have that |𝐹 | ≤ = 1 ∑︁ P 𝑝 (𝑒 is pivotal for {∥𝐾1 ∥ ≥ 𝛽}) 𝜂4 𝑒∈𝐸 𝑘 8 1 ′ P 𝑝 𝑘 (∥𝐾1 ∥ ≥ 𝛽) ≤ |𝑉 |, 𝜂4 𝛿𝜂4 so that |𝐹 | is at most linear in |𝑉 |. Since |𝐹 | ≤ 𝛿𝜂8 4 |𝑉 | and 𝑑 = deg 𝐺 ≥ 𝑐 2 |𝑉 |, we have that deg(𝐺 \ 𝐹) ≥ 𝑐 2 |𝑉 | − 16 . 𝛿𝜂4 Thus, there exists a constant 𝐶2 = 𝐶2 (𝛼, 𝛿) such that if |𝑉 | ≥ 𝐶2 then deg(𝐺 \ 𝐹) ≥ 𝑐2 |𝑉 |. 2 It now suffices to prove that 𝐺 \ 𝐹 is not connected when |𝑉 | ≥ 𝐶2 . Indeed, once this is shown it follows automatically that 𝐺 \ 𝐹 has a bounded number of components since if 𝐺 \ 𝐹 has 𝑚 components then 𝑐42 |𝑉 | 2 ≤ |𝐸 | ≤ 𝑚(|𝑉 |/𝑚) 2 + |𝐹 |, and hence there exists a constant 𝐶4 = 𝐶4 (𝛼, 𝛿) such that 𝑚 ≤ 1 + 4/𝑐 2 when |𝑉 | ≥ 𝐶4 . Suppose for contradiction that |𝑉 | ≥ 𝐶2 and that 𝐺 \ 𝐹 is connected. Since 𝐺 \ 𝐹 is connected and deg(𝐺 \ 𝐹) ≥ 𝑐22 |𝑉 |, it follows from Lemma 5.2.3 that diam(𝐺 \ 𝐹) ≤ 6/𝑐 2 = 𝐶1 . Let 𝑃 be a path of length at most 𝐶1 connecting the endpoints of 𝑒 0 in 𝐺 \ 𝐹. If 𝑒 0 activates 𝛽 then so does the set 𝑃, and it follows that |𝑃| ≤ 𝐶1 ≤ 𝜂3−1 and P 𝑝 𝑁1 ∨𝑁2 (Act 𝛽 (𝑃)) ≥ P 𝑝 𝑁1 ∨𝑁2 (Act 𝛽 (𝑒 0 )) ≥ 𝜂2 ≥ 𝜂3 . 193 As such, it follows from the definitions of the quantities 𝑁2 = 𝑁 (𝛼, 𝛿, 𝜂3 ) and 𝜂4 = 𝜂 𝑁1 ∨𝑁2 (𝛼, 𝛿, 𝜂3 ) as in Proposition 5.4.7 that there exists an edge 𝑒 1 ∈ 𝑃 such that P 𝑝 𝑘 (Act 𝛽 (𝑒 1 )) ≥ 𝜂4 . This implies that 𝑒 1 ∈ 𝐹, a contradiction. □ Proof of Proposition 5.4.7 In this section we complete the proof of Proposition 5.4.1 by proving Proposition 5.4.7. The proof will proceed inductively, showing that — by changing to a different value of 𝑝 ∈ 𝑄 if necessary — we can reduce the number of edges in the subgraph 𝐻 given by Corollary 5.4.5 while keeping P 𝑝 (Act𝛼 (𝐻)) bounded away from zero. More precisely, we will deduce Proposition 5.4.7 as an inductive consequence of the following lemma. Lemma 5.4.8 (Removing one edge by sprinkling). Let 𝐺 = (𝑉, 𝐸) be a finite, simple, connected, vertex-transitive graph, let 𝛼0 ≤ 𝛿 ≤ 1/2 and 𝛼0 ≤ 𝛼 ≤ 𝛽 ≤ 1, and suppose that 𝑝 𝑐 (𝛽, 1 − 𝛿) > 𝑒 𝛿 𝑝 𝑐 (𝛽, 𝛿). Let ( 𝑝 𝑛 )𝑛≥1 be as in Lemma 5.4.6. For each 𝜀 > 0 there exists 𝑁 = 𝑁 (𝛼, 𝛿, 𝜀) such that if 𝑛 ≥ 𝑁 then there exists 𝜂𝑛 = 𝜂𝑛 (𝛼, 𝛿, 𝜀) > 0 such that if 𝐻 is a subgraph of 𝐺 with |𝐸 (𝐻)| ≤ 𝜀 −1 and P 𝑝 𝑛 (Act 𝛽 (𝐻)) ≥ 𝜀 then there exists a subgraph 𝐻 ′ of 𝐻 with |𝐸 (𝐻 ′)| ≤ max{1, |𝐸 (𝐻)| − 1} such that P 𝑝 𝑚 (Act 𝛽 (𝐻 ′)) ≥ 𝜂𝑛 for every 𝑚 > 𝑛. Note that the choice of subgraph 𝐻 ′ of 𝐻 may depend on the choice of 𝑛 ≥ 𝑁 and that the constants 𝑁 and 𝜂𝑛 are independent of 𝐺 and 𝛽. Also, note that 𝑁 and 𝜂𝑛 here are not the same 𝑁 and 𝜂𝑛 as in the statemenet of Proposition 5.4.7. Proof of Proposition 5.4.7 given Lemma 5.4.8. Applying Lemma 5.4.8 iteratively ⌊1/𝜀⌋ times yields the claim. □ For the remainder of this section we fix a finite, simple, connected, vertex-transitive graph 𝐺 = (𝑉, 𝐸) of degree 𝑑, fix 𝛼0 ≤ 𝛿 ≤ 1/2 and 𝛼0 ≤ 𝛼 ≤ 𝛽 ≤ 1 such that 𝑝 𝑐 (𝛽, 1 − 𝛿) > 𝑒 𝛿 𝑝 𝑐 (𝛽, 𝛿), and let ( 𝑝 𝑛 )𝑛≥1 be as in Lemma 5.4.6. We also continue to write 𝐼 = 𝐼 (𝐺, 𝛽, 𝛿) = [ 𝑝 𝑐 (𝛽, 𝛿), 𝑝 𝑐 (𝛽, 1−𝛿)] and 𝑄 = 𝑄(𝐺, 𝛽, 𝛿) = {𝑝 ∈ 𝐼 : 𝑝 · P′𝑝 (∥𝐾1 ∥ ≥ 𝛽) ≤ 4𝛿 }. When 𝐺 has bounded vertex degrees, and hence critical parameters are bounded away from zero, Lemma 5.4.8 could be proven easily by (classical) insertion-tolerance. The problem is rather more delicate in general. The idea is to use vertex-transitivity of 𝐺 to find many copies of 𝐻 that each activate 𝛽 simultaneously, then sprinkle, i.e. open a small number of additional edges by slightly increasing the percolation parameter, and argue that, after sprinkling, many of the copies of 𝐻 have strict subgraphs that are activators. To ensure that sprinkling reduces the number of edges 194 necessary to activate 𝛽 in a positive proportion of the copies of 𝐻, we need these copies to be well-connected to each other in the open subgraph. We guarantee this by sticking a large cluster to each endpoint of an edge in 𝐻, using the next lemma. Lemma 5.4.9. For every 𝜀 > 0 there exists 𝜂 = 𝜂(𝛼, 𝛿, 𝜀) > 0 such that if 𝑝 ∈ 𝑄 and 𝐻 is a
subgraph of 𝐺 with |𝐸 (𝐻)| ≤ 𝜀1 and P 𝑝 Act 𝛽 (𝐻) ≥ 𝜀 then there is an edge 𝑒 ∈ 𝐸 (𝐻) with endpoints 𝑢 and 𝑣 such that
P 𝑝 Act 𝛽 (𝐻) ∩ {∥𝐾𝑢 ∥ ≥ 𝜂} ∩ {|𝐾𝑣 | ≥ 𝜂𝑑} ≥ 𝜂. The proof of this lemma uses the quantitative insertion-tolerance estimate of Proposition 6.6.4 together with the following theorem of the second author [Hut21b, Theorem 2.2], which guarantees that the size of the largest intersection of a cluster with a fixed set of vertices is always of the same order as its mean with high probability. We state a special case of the theorem that is adequate for our purposes. Theorem 5.4.10 (Universal Tightness). There exist universal constants 𝐶, 𝑐 > 0 such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a countable, locally finite graph, let Λ ⊆ 𝑉 be a finite nonempty set of vertices, and let 𝑝 ∈ [0, 1] be a parameter. Set |𝑀 | := max{|𝐾𝑣 ∩ Λ| : 𝑣 ∈ 𝑉 }. Then

and P 𝑝 |𝑀 | ≤ 𝜀E 𝑝 |𝑀 | ≤ 𝐶𝜀 P 𝑝 |𝑀 | ≥ 𝛼E 𝑝 |𝑀 | ≤ 𝐶𝑒 −𝑐𝛼 for every 𝛼 ≥ 1 and 0 < 𝜀 ≤ 1. Proof of Lemma 5.4.9. We may assume that 𝜀 ≤ 𝛿. We may also assume that 𝐻 has no isolated points, so that we have the bound |𝑉 (𝐻)| ≤ 2 |𝐸 (𝐻)| ≤ 𝜀2 . When 𝐻 activates 𝛽 we must have that Ð 𝑢∈𝑉 (𝐻) 𝐾𝑢 ≥ 𝛽 and hence by the pigeonhole principle that there exists 𝑢 ∈ 𝑉 (𝐻) such that ∥𝐾𝑢 ∥ ≥ 𝛽 𝛼𝜀 ≥ . |𝑉 (𝐻)| 2 It follows that there exists a fixed vertex 𝑢 ∈ 𝑉 (𝐻) such that  n
1 𝜀 𝜀2 𝛼𝜀 o P 𝑝 Act 𝛽 (𝐻) ∩ ∥𝐾𝑢 ∥ ≥ ≥ P 𝑝 Act 𝛽 (𝐻) ≥ ≥ . |𝑉 (𝐻)| 2 |𝑉 (𝐻)| 2 (5.4.1) Since 𝐻 has no isolated points, 𝑢 is the endpoint of some edge 𝑒 ∈ 𝐸 (𝐻). Let 𝑣 be the other endpoint of 𝑒, let 𝑁 be the set of neighbours of 𝑣 in 𝐺, and let 𝑋 be an 𝜔-connected subset of 𝑁 195 (i.e. a subset of 𝑁 that is contained in a single 𝜔-cluster) of maximum size. Since 𝑝 ≥ 𝑝 𝑐 (𝛼, 𝛿) and 𝐺 is vertex-transitive, E 𝑝 |𝑋 | ≥ E 𝑝 [|𝐾1 ∩ 𝑁 |] = |𝑁 | E 𝑝 ∥𝐾1 ∥ ≥ 𝛼𝛿𝑑. (5.4.2) Applying Theorem 5.4.10 it follows that there exists a positive constant 𝑐 1 = 𝑐 1 (𝛼, 𝛿, 𝜀) such that 2 P 𝑝 (|𝑋 | ≤ 𝑐 1 𝑑) ≤ 𝜀4 and hence by a union bound that  n  𝜀2 𝛼𝜀 o P 𝑝 Act 𝛽 (𝐻) ∩ ∥𝐾𝑢 ∥ ≥ ∩ {|𝑋 | ≥ 𝑐 1 𝑑} ≥ . 2 4 (5.4.3) To obtain a similar bound with |𝐾𝑣 | in place of |𝑋 |, we use insertion tolerance to open an edge connecting 𝑣 to 𝑋, which forces |𝐾𝑣 | ≥ |𝑋 |. Unfortunately we cannot argue this way directly since opening this edge may produce a cluster with density at least 𝛽, in which case Act 𝛽 (𝐻) would no longer hold. To avoid this issue, we first claim that there exists a constant 𝐶1 = 𝐶1 (𝛼, 𝛿, 𝜀) such that if |𝑉 | ≥ 𝐶1 then  𝜀2  n 𝛼𝜀 o ∩ {|𝑋 | ≥ 𝑐 1 𝑑} ∩ Act 𝛽 (𝑣𝑋) 𝑐 ≥ , P 𝑝 Act 𝛽 (𝐻) ∩ ∥𝐾𝑢 ∥ ≥ 2 8 (5.4.4) where 𝑣 𝑋 denotes the set of edges with one endpoint equal to 𝑣 and the other in 𝑋. (Note that since 𝑋 is 𝜔-connected, each edge in 𝑣𝑋 individually activates 𝛽 if and only if the entire set 𝑣𝑋 activates 𝛽.) Indeed, suppose that (5.4.4) does not hold. We have by (5.4.3) and a union bound that
𝜀2 P 𝑝 {|𝑋 | ≥ 𝑐 1 𝑑} ∩ Act 𝛽 (𝑣𝑋) ≥ 8 and hence that ∑︁ P 𝑝 Act 𝛽 (𝑒) ≥ E 𝑝 |𝑋 | 1Act𝛽 (𝑣 𝑋)
  𝑒∈𝐸: 𝑒∋𝑣 𝑐1 𝜀2 ≥ 𝑑. 8 Applying Russo’s formula and using that 𝐺 is vertex-transitive, it follows that P′𝑝 (∥𝐾1 ∥ ≥ 𝛽) ≥ 𝑐1 𝜀2 𝑑 |𝑉 | . 16 Since 𝑝 ≥ 𝑝 𝑐 (𝛼0 , 𝛼0 ) we also have that 𝑝 ≥ 1/2𝑑 by lemma 5.2.6 and hence that 𝑝 · P′𝑝 (∥𝐾1 ∥ ≥ 𝛽) ≥ 𝑐1 𝜀2 |𝑉 | . 32 Since 𝜀 ≤ 𝛿, this contradicts the hypothesis that 𝑝 ∈ 𝑄 whenever |𝑉 | ≥ 𝐶1 := 128/(𝑐 1 𝜀 3 ), completing the proof of the claim. 196 There are finitely many graphs with |𝑉 | ≤ 𝐶1 , hence the lemma holds trivially in this case and we may assume without loss of generality that (5.4.4) holds. Since we have that 𝑝 ≥ 1/2𝑑 as above, we can use insertion tolerance Proposition 6.6.4 to open an edge in 𝑣𝑋 in the event appearing on the left hand side of (5.4.4), giving that   n 𝛼𝜀 o ∩ {|𝐾𝑣 | ≥ 𝑐 1 𝑑} ≥ 𝜂, P 𝑝 Act 𝛽 (𝐻) ∩ ∥𝐾𝑢 ∥ ≥ 2 for some 𝜂 = 𝜂(𝛼, 𝛿, 𝜀) > 0 as claimed. □ Once we have many copies of 𝐻 that activate 𝛽 and have large clusters stuck to the copies of 𝑢 and 𝑣, we use the following easy fact about equivalence relations to deduce that the copies of 𝑒 are well-connected to each other. More precisely, we will use this lemma to show that we can find many large disjoint sets of copies of the edge 𝑒 in which any two copies 𝛾1 (𝑒) and 𝛾2 (𝑒) of 𝑒 are 𝜔-connected by paths 𝛾1 (𝑢) ↔ 𝛾2 (𝑢) and 𝛾1 (𝑣) ↔ 𝛾2 (𝑣). Lemma 5.4.11. Let 𝑋 be a non-empty finite set, let ∼ be an equivalence relation on 𝑋, and let 𝑌 ⊂ 𝑋 be a non-empty subset of 𝑋. For each 𝑥 ∈ 𝑋 write [𝑥] for the equivalence class of 𝑥 under | ∼. If min𝑦∈𝑌 | [𝑦]| ≥ 2|𝑋 |𝑌 | then there exists a collection (𝑍𝑖 )𝑖∈𝐼 of disjoint subsets of 𝑌 such that each 𝑍𝑖 is contained in an equivalence class of ∼ and that satisfies the inequalities |𝑍𝑖 | ≥ |𝑌 | min | [𝑦]| 2 |𝑋 | 𝑦∈𝑌 and |𝐼 | ≥ |𝑋 | . 4 min𝑦∈𝑌 | [𝑦] | Proof of Lemma 5.4.11. Write 𝑚 = min𝑦∈𝑌 |[𝑦]| and define   |𝑌 | | [𝑦] | 𝑌− := 𝑦 ∈ 𝑌 : | [𝑦] ∩ 𝑌 | ≤ and 2 |𝑋 | 𝑌+ := 𝑌 \ 𝑌− . Observe that if C denotes the set of equivalence classes of ∼ then ∑︁ |[𝑦]| ∑︁ 2|𝑋 | |𝑌− | ≤ = |𝐶 | 1 (𝐶 ∩ 𝑌 ≠ ∅) ≤ |𝑋 |, |𝑌 | |[𝑦] ∩ 𝑌 | 𝑦∈𝑌 𝐶∈C so that |𝑌− | ≤ 12 |𝑌 | and |𝑌+ | ≥ 12 |𝑌 |. Let (𝑍𝑖 )𝑖∈𝐼 be a maximal collection ofmdisjoint subsets of 𝑌+ l such that each 𝑍𝑖 is contained in an equivalence class of ∼ and |𝑍𝑖 | = element 𝑦 ∈ 𝑌+ has     |𝑌 | |𝑌 | | [𝑦] ∩ 𝑌 | ≥ |[𝑦]| ≥ 𝑚 . 2 |𝑋 | 2 |𝑋 | 197 |𝑌 | 2|𝑋 | 𝑚 for each 𝑖 ∈ 𝐼. Every Ð So, by maximality, the union 𝑖∈𝐼 𝑍𝑖 contains at least half the elements in [𝑦] ∩ 𝑌 for every 𝑦 ∈ 𝑌+ . Ð So the union 𝑖∈𝐼 𝑍𝑖 contains at least half the elements in 𝑌+ , and we deduce that |𝐼 | ≥ l |𝑌 | /4 |𝑋 | m ≥ , |𝑌 | 4𝑚 𝑚 2|𝑋 | | where we used the hypothesis 𝑚 ≥ 2|𝑋 |𝑌 | in the final inequality. □ We are now ready to complete the proof of Lemma 5.4.8. The final step to reduce the number of edges in 𝐻 is to open a small number of these well-connected copies of 𝑒 by slightly increasing the percolation parameter. When 𝐻 activates 𝛽 and 𝑢 ↔ 𝑣, 𝐻 \ {𝑒} also activates 𝛽. Since the copies of 𝑒 are well-connected, opening this small number of edges is actually sufficient to ensure a positive proportion of the copies of 𝐻 \ {𝑒} activate 𝛽. By linearity of expectation, we conclude that at this higher parameter the set 𝐻 \ {𝑒} activates 𝛽 with good probability, as required. Proof of Lemma 5.4.8. Since 𝛼, 𝛿 ≥ 𝛼0 we have by Lemma 5.4.3 and Lemma 5.2.6 that L (𝑄) ≥ 1 1 𝛿 1 L (𝐼) = ( 𝑝 𝑐 (𝛽, 1 − 𝛿) − 𝑝 𝑐 (𝛽, 𝛿)) ≥ (𝑒 𝛿 − 1) 𝑝 𝑐 (𝛽, 𝛿) ≥ , 2 2 2 4𝑑 where we used that 𝑒 𝛿 − 1 ≥ 𝛿 in the final inequality. Fix 𝜀 > 0 and 𝑛 ≥ 1, and suppose that 𝐻 is
a finite subgraph of 𝐺 with |𝐸 (𝐻)| ≤ 𝜀 −1 such that P 𝑝 𝑛 Act 𝛽 (𝐻) ≥ 𝜀. By Lemma 5.4.9, we can find 𝜀1 = 𝜀1 (𝛼, 𝛿, 𝜀) and an edge 𝑒 ∈ 𝐸 (𝐻) with endpoints 𝑢 and 𝑣 such that the event A := Act 𝛽 (𝐻) ∩ {∥𝐾𝑢 ∥ ≥ 𝜀1 } ∩ {|𝐾𝑣 | ≥ 𝜀1 𝑑} satisfies P 𝑝 𝑛 (A) ≥ 𝜀1 . For each 𝑥 ∈ 𝑉, pick an automorphism 𝜙𝑥 ∈ Aut 𝐺 such that 𝜙𝑥 (𝑣) = 𝑥, so that 𝜙𝑥 (𝑢) is a neighbour of 𝑥 for each 𝑥 ∈ 𝑉. These maps exist because 𝐺 is vertex-transitive. Define 𝑋 := {𝑥 ∈ 𝑉 : 𝜙−1 𝑥 (𝜔) ∈ A}. We have by linearity of expectation that E 𝑝 𝑛 ∥ 𝑋 ∥ = P 𝑝 𝑛 (A) ≥ 𝜀 1 and hence by Markov’s inequality that P 𝑝 𝑛 (∥ 𝑋 ∥ ≥ 12 𝜀1 ) ≥ 21 𝜀1 . We will now prove that we can take 𝑁 = ⌈log2 (8/𝜀 1 )⌉, so assume from now on that 𝑛 ≥ log2 (8/𝜀1 ). Write B := {∥ 𝑋 ∥ ≥ 12 𝜀1 } and consider an arbitrary configuration 𝜔 ∈ B. Every 𝜙𝑥 (𝑢) with 𝑥 ∈ 𝑋 has ∥𝐾 𝜙 𝑥 (𝑢) ∥ ≥ 𝜀1 . So, by the pigeonhole principle, we can find a subset 𝑌 ⊆ 𝑋 with ∥𝑌 ∥ ≥ 12 𝜀 12 such that {𝜙𝑥 (𝑢) : 𝑥 ∈ 𝑌 } is contained in a single cluster of 𝜔. By definition of 𝑋 and A, every vertex 𝑦 ∈ 𝑌 has 𝐾 𝑦 ≥ 𝜀1 𝑑. We next claim that there exists a constant 𝐶1 = 𝐶1 (𝛼, 𝛿, 𝜀) such that if |𝑉 | ≥ 𝐶1 then 𝜀1 𝑑|𝑌 | ≥ 4|𝑉 |. Indeed, since ∥𝑌 ∥ ≥ 21 𝜀12 , if this inequality does not hold then we must have that 𝑑 ≤ 8𝜀1−3 . In this 198 case, since 𝑝 𝑛 ∈ 𝑄 and 𝑝 𝑛 ≥ 1/2𝑑, and since every edge has Aut 𝐺-orbit of size at least |𝑉 | /2 by transitivity, it follows from Theorem 5.3.10 that there exists a constant 𝑐 1 = 𝑐 1 (𝛼, 𝛿, 𝜀) such that 4 ≥ 𝑝 𝑛 · P′𝑝 𝑛 (∥𝐾1 ∥ ≥ 𝛽) ≥ 𝑐 1 log |𝑉 |, 𝛿 which rearranges to give the claim. Since graphs with |𝑉 | < 𝐶1 can be handled trivially, we may assume throughout the rest of the proof that |𝑉 | ≥ 𝐶1 and hence that 𝜀1 𝑑|𝑌 | ≥ 4|𝑉 |. By construction, every vertex 𝑦 ∈ 𝑌 has 𝐾 𝑦 ≥ 𝜀1 𝑑. So by splitting clusters, we can find an 𝜔 equivalence relation that is a refinement of ← → in which the equivalence class of each vertex 𝑦 ∈ 𝑌 has between 𝜀1 𝑑/2 and 𝜀1 𝑑 total vertices. We now apply Lemma 5.4.11 to this equivalence relation with the sets 𝑉 and 𝑌 in place of 𝑋 and 𝑌 . (The hypothesis of Lemma 5.4.11 is met because 𝜀1 𝑑 |𝑌 | ≥ 4 |𝑉 |.) This yields a collection of disjoint 𝜔-connected subsets (𝑍𝑟 )𝑟∈𝑅 of 𝑌 such that for every 𝑟, 𝜀13 𝑑 |𝑉 | |𝑉 | 𝜀1 𝑑 |𝑌 | |𝑅| ≥ |𝑍𝑟 | ≥ and ≥ . ≥ 4 |𝑉 | 8 4𝜀1 𝑑 4𝑑 Whenever 𝐻 activates 𝛽 and 𝑢 ↔ 𝑣, 𝐻 \ {𝑒} also activates 𝛽. On the event B we must have that 𝑥 ↔ 𝑦 and 𝜙𝑥 (𝑢) ↔ 𝜙 𝑦 (𝑢) whenever 𝑥, 𝑦 both belong to 𝑍𝑟 for some 𝑟 ∈ 𝑅. Thus, on the event B, if there exist 𝑟 ∈ 𝑅 and 𝑥 ∈ 𝑍𝑟 such that 𝑥 ↔ 𝜙𝑥 (𝑢) then 𝜙𝑥 ′ (𝐻 \ {𝑢𝑣}) activates 𝛽 for every 𝑥 ′ ∈ 𝑍𝑟 . In view of this fact, our next step will be to increase the percolation parameter to open an edge 𝜙𝑥 (𝑒) with 𝑥 ∈ 𝑍𝑟 for a positive proportion of the indices 𝑟 ∈ 𝑅, thus making a positive proportion Ð of the copies 𝜙𝑥 (𝐻 \ {𝑒}) with 𝑥 ∈ 𝑟∈𝑅 𝑍𝑟 activate 𝛽. Let 𝑚 > 𝑛 ≥ ⌈log2 (8/𝜀1 )⌉, and let P be the joint law of the standard monotone coupling (𝜔, 𝜔′) of percolation with the two parameters 𝑝 𝑛 ≤ 𝑝 𝑚 . It suffices to prove that there exists a constant 𝜂𝑛 = 𝜂𝑛 (𝛼, 𝛿, 𝜀) > 0 such that P 𝑝 𝑚 (Act 𝛽 (𝐻 \ {𝑒})) ≥ 𝜂𝑛 . (5.4.5) Recall that the increasing sequence ( 𝑝 𝑛 )𝑛≥1 was defined so that 𝑝 𝑚 − 𝑝 𝑛 ≥ 3−(𝑛+1) L (𝑄) ≥ 3−(𝑛+1) 𝛿 4𝑑 and P 𝑝 𝑚 (∥𝐾1 ∥ ≥ 𝛽) − P 𝑝 𝑛 (∥𝐾1 ∥ ≥ 𝛽) ≤ 2−𝑛+1 199 for every 𝑚 > 𝑛. The assumption that 𝑚 > 𝑛 ≥ log2 (8/𝜀1 ) implies that P 𝑝 𝑚 (∥𝐾1 ∥ ≥ 𝛽) − P 𝑝 𝑛 (∥𝐾1 ∥ ≥ 𝛽) ≤ 𝜀41 , and since P 𝑝 𝑛 (B) ≥ 𝜀21 a union bound gives that P (𝜔 ∈ B and ∥𝐾1 (𝜔′)∥ < 𝛽) ≥ P 𝑝 𝑛 (B) − P (∥𝐾1 (𝜔)∥ < 𝛽 ≤ ∥𝐾1 (𝜔′)∥) = P 𝑝 𝑛 (B) − (5.4.6) P 𝑝 𝑚 (∥𝐾1 (𝜔)∥ ≥ 𝛽) − P 𝑝 𝑛 (∥𝐾1 (𝜔)∥ ≥ 𝛽) 𝜀1 𝜀1 𝜀1 − = , ≥ 2 4 4
(5.4.7) and hence in particular that P (∥𝐾1 (𝜔′)∥ < 𝛽 | 𝜔 ∈ B) ≥ 𝜀1 . 4 (5.4.8) Consider an arbitrary configuration 𝜉 ∈ B and let (𝑍𝑟 )𝑟∈𝑅 be a collection of sets as defined via Lemma 5.4.11 above, which we take to be a function of 𝜉. (Note in particular that the index set 𝑅 depends on 𝜉.) For each 𝑟 ∈ 𝑅 and 𝑥 ∈ 𝑍𝑟 we have that P (𝜙𝑥 (𝑒) ∈ 𝜔′ | 𝜔 = 𝜉) ≥ 𝑝𝑚 − 𝑝𝑛 𝛿 ≥ 3−(𝑛+1) , 1 − 𝑝𝑛 4𝑑 𝜀3 and since |{𝜙𝑥 (𝑒) : 𝑒 ∈ 𝑍𝑟 }| ≥ 21 |𝑍𝑟 | ≥ 161 𝑑 (where the factor of 1/2 accounts for the distinction between oriented and unoriented edges, with 𝜙𝑥 acting bijectively on the former), we deduce that there exists 𝜀2 = 𝜀2 (𝛼, 𝛿, 𝜀) > 0 such that  ′ P (∃𝑥 ∈ 𝑍𝑟 : 𝜙𝑥 (𝑒) ∈ 𝜔 | 𝜔 = 𝜉) ≥ 1 − 1 − 3 "  𝜀3 𝑑/16 1 −(𝑛+1) 𝛿 ≥ 1 − exp −3 4𝑑 𝜀3 𝛿 −(𝑛+1) 1 64 # ≥ 𝜀2 3−𝑛 , (5.4.9) where we used the inequality 1 − 𝑥 ≤ 𝑒 −𝑥 in the second line. Condition on the event 𝜔 = 𝜉 and 𝜔′ 𝜔 consider the random set 𝐽 := {𝑟 ∈ 𝑅 : 𝑍𝑟 ← → 𝑌 }. Note that for all 𝑟 1 , 𝑟 2 ∈ 𝑅 with 𝑍𝑟1 ̸← → 𝑌 and 𝜔 𝑍𝑟2 ̸← → 𝑌 , we have {𝜙𝑥 (𝑒) : 𝑥 ∈ 𝑍𝑟1 } ∩ {𝜙𝑥 (𝑒) : 𝑥 ∈ 𝑍𝑟2 } = ∅. So by independence, it follows that there exists a constant 𝐶2 = 𝐶2 (𝛼, 𝛿, 𝜀) such that if |𝑅| ≥ 𝐶2 3𝑛 then   𝜀2 𝜀1 (5.4.10) P |𝐽 | ≥ 3−𝑛 |𝑅| 𝜔 = 𝜉 ≥ 1 − . 2 8 We will now proceed by case analysis according to whether |𝑅(𝜉)| ≥ 𝐶2 3𝑛 for every 𝜉 ∈ B or |𝑅(𝜉)| < 𝐶2 3𝑛 for some 𝜉 ∈ B. First assume that |𝑅(𝜉)| ≥ 𝐶2 3𝑛 for every 𝜉 ∈ B. On the event 200 𝜔 ∈ B, we write (𝑍𝑟 )𝑟∈𝑅 and 𝐽 for the corresponding sets defined with respect to 𝜔. We have by (5.4.8) and (5.4.10) that  𝜀  𝜀1 𝜀1 𝜀2 −𝑛 1 ′ − = , (5.4.11) P |𝐽 | ≥ 3 |𝑅| and ∥𝐾1 (𝜔 )∥ < 𝛽 𝜔 ∈ B ≥ 2 4 8 8 and hence that  𝜀2 𝜀2 −𝑛 ′ P 𝜔 ∈ B, |𝐽 | ≥ 3 |𝑅| , and ∥𝐾1 (𝜔 )∥ < 𝛽 ≥ 1 . (5.4.12) 2 16 Consider a pair of configurations (𝜔, 𝜔′) satisfying the event whose probability is estimated in (5.4.12) and pick 𝑟 ∈ 𝐽 and 𝑦 ∈ 𝑍𝑟 . Since 𝜙 𝑦 (𝐻) activates 𝛽 in 𝜔 and ∥𝐾1 (𝜔′)∥ < 𝛽, we also have that 𝜙 𝑦 (𝐻) activates 𝛽 in 𝜔′. By definition of 𝐽, we can find 𝑥 ∈ 𝑍𝑟 such that 𝜙𝑥 (𝑒) is open in 𝜔′. Since 𝑥, 𝑦 ∈ 𝑍𝑟 we have by definition that 𝑥 and 𝑦 are connected in 𝜔, and since 𝑥, 𝑦 ∈ 𝑌 we have by definition of 𝑌 that 𝜙𝑥 (𝑢) and 𝜙 𝑦 (𝑢) are also connected in 𝜔. Since 𝜙𝑥 (𝑒) is open in 𝜔′ and 𝜙𝑥 (𝑣) = 𝑥, we deduce that 𝑥, 𝑦, 𝜙𝑥 (𝑢), and 𝜙 𝑦 (𝑢) all belong to the same cluster of 𝜔′. Since 𝜙 𝑦 (𝐻) activates 𝛽 in 𝜔′ and 𝑦 is connected to 𝜙 𝑦 (𝑢) in 𝜔′, we deduce that 𝜙 𝑦 (𝐻 \ {𝑒}) activates 𝛽 in 𝜔′ also. Since this holds for every 𝑟 ∈ 𝐽 and 𝑦 ∈ 𝑍𝑟 , we have  |𝑊 | := 𝑥 ∈ 𝑋 : 𝜔′ ∈ Act 𝛽 (𝜙𝑥 (𝐻 \ {𝑒}) ≥ |𝐽 | min |𝑍𝑟 | .  𝑟∈𝐽 𝜀3 𝜀3 𝜀2 1 Since |𝐽 | ≥ 𝜀22 3−𝑛 |𝑅| ≥ 3−𝑛 𝜀82 |𝑉𝑑 | and every 𝑍𝑟 has |𝑍𝑟 | ≥ 81 𝑑, we have that ∥𝑊 ∥ ≥ 64 3−𝑛 and hence that 𝜀 15 𝜀2 −𝑛 E ∥𝑊 ∥ ≥ 10 3 . 2
𝜀5 𝜀2 Thus, it follows by vertex-transitivity that P 𝑝 𝑚 Act 𝛽 (𝐻 \ {𝑒}) ≥ 2110 3−𝑛 . This bound has the required form, completing the proof of (5.4.5) in this case. It remains to consider the case that |𝑅(𝜉)| < 𝐶2 3𝑛 for some 𝜉 ∈ B. Since we always have |𝑅| ≥ |𝑉 | /4𝑑, we must have in this case that |𝑉 | ≤ 4𝐶2 3𝑛 𝑑. Observe that if 𝜔 ∈ B and ∥𝐾1 (𝜔′) ∥ < 𝛽 then 𝜔′ ∈ B also. Thus, by (5.4.8) and the fact that P 𝑝 𝑛 (B) ≥ 𝜀21 , we have that 𝜀12 P 𝑝 𝑚 (B) ≥ . 8 On the event B, pick one of the sets 𝑍𝑟 that are guaranteed to exist by our earlier argument, call it 𝑍, and let 𝑈 := {𝑥 ∈ 𝑍 : 𝜔 ∈ Act 𝛽 (𝜙𝑥 (𝑒))}. If   1  1 P 𝑝 𝑚 B ∩ ∥𝑈 ∥ ≥ ∥𝑍 ∥ ≥ P 𝑝 𝑚 (B) 2 2 201 then we have by vertex-transitivity and the bound |𝑍 | ≥ 𝜀13 𝑑/8 that 𝜀13 𝑑 𝜀15 𝜀12 P 𝑝 𝑚 Act 𝛽 (𝑒) ≥ E 𝑝 𝑚 [∥𝑈 ∥ 1 (B)] ≥ · ≥ · 3−𝑛 16|𝑉 | 16 210𝐶2
as required. Conversely, if   1 1 P 𝑝 𝑚 B ∩ ∥𝑈 ∥ ≥ ∥𝑍 ∥ ≤ P 𝑝 𝑚 (B) , 2 2  𝜀3 then we use insertion-tolerance (Proposition 6.6.4 - using the fact that ∥𝑍 ∥ ≥ 64𝐶12 3𝑛 ) to open a single edge in {𝜙𝑥 (𝑒) : 𝑥 ∈ 𝑍 \ 𝑈} on the event B ∩ {∥𝑈 ∥ ≤ 21 ∥𝑍 ∥}. This does not create a cluster with density at least 𝛽, since none of the edges 𝜙𝑥 (𝑒) with 𝑥 ∈ 𝑍 \𝑈 activates 𝛽. Arguing as before, after opening this edge, every set 𝜙𝑥 (𝐻 \ {𝑒}) with 𝑥 ∈ 𝑍 activates 𝛽. Thus, it follows by the same vertex-transitivity argument as above that there exists a positive constant 𝜀3 (𝑛) = 𝜀3 (𝛼, 𝛿, 𝜀, 𝑛) such that
P 𝑝 𝑚 Act 𝛽 (𝐻 \ {𝑒}) ≥ 𝜀3 . This completes the proof. □ Completing the proof of the main theorems In this section we complete the proof of Theorem 8.1.1 and hence of Theorem 5.1.2. Proof of Theorem 8.1.1. The implication (i) ⇒ (ii) follows immediately from Corollary 5.3.4 and Proposition 5.4.1, while the implication (iii) ⇒ (iv) is trivial. As such, it remains only to prove (ii) ⇒ (iii) and (iv) ⇒ (i). We begin with the implication (ii) ⇒ (iii), i.e., the claim that molecular sets admit linear 1/3separators. To see this, note that if H is 𝑚-molecular for some 𝑚 ≥ 2 then there exists a constant 𝐶 such that for each 𝐺 ∈ H there is an automorphism-invariant set of edges 𝐹 such that |𝐹 | ≤ 𝐶 |𝑉 (𝐺)| and 𝐹 disconnects 𝐺 into 𝑚 ≥ 2 connected components each of size |𝑉 (𝐺)|/𝑚. If we take 𝐴 to be the union of ⌈𝑚/2⌉ of these components then |𝑉 (𝐺)|/3 ≤ | 𝐴| = ⌈𝑚/2⌉|𝑉 (𝐺)|/𝑚 ≤ 2|𝑉 (𝐺)|/3 and |𝜕𝐸 𝐴| ≤ |𝐹 | ≤ 𝐶 |𝑉 (𝐺)| so that H has linear 1/3-separators as claimed. We next prove the implication (iv) ⇒ (i). We will use the following theorem [Bol+10a] identifying the location of the critical threshold for (not necessarily transitive) dense graph sequences. Theorem 5.4.12 (Bollobás, Borgs, Chayes, Riordan 2010). Let (𝐺 𝑛 ) be a dense sequence of finite, simple graphs with |𝑉 (𝐺 𝑛 )| → ∞, and for each 𝑛 ≥ 1 let 𝜆 𝑛 be the largest eigenvalue of the 202 adjacency matrix of 𝐺 𝑛 . For each 𝑐 > 0 lim P𝐺 𝑛 (∥𝐾1 ∥ ≥ 𝑐) = 0, 𝑛→∞ 𝜆 𝑛−1 and for each 𝜀 > 0 there exists 𝛿 > 0 such that lim P𝐺 𝑛 𝑛→∞ (1+𝜀)𝜆 𝑛−1 (∥𝐾1 ∥ ≥ 𝛿) = 1. Note that if 𝐴 is the adjacency matrix of a finite graph then 𝐴 is self-adjoint and the largest eigenvalue (a.k.a. the Perron–Frobenius eigenvalue) of 𝐴 coincides with the 𝐿 2 operator norm of 𝐴. This norm satisfies ∥ 𝐴∥ ≥ ⟨𝐴1, 1⟩/⟨1, 1⟩ = 2|𝐸 |/|𝑉 | where 1 is the constant-one function, and hence if (𝐺 𝑛 )𝑛≥1 is dense then the sequence of largest eigenvalues 𝜆 𝑛 satisfies lim inf |𝑉 (𝐺 𝑛 )| −1𝜆 𝑛 > 0. Suppose that H is dense and admits linear 𝜃-separators for some 𝜃 ∈ (0, 1/2], so that there exists a constant 𝐶1 such that for each 𝐺 ∈ H there exists a set 𝐴(𝐺) ⊆ 𝑉 (𝐺) with 𝜃|𝑉 (𝐺)| ≤ | 𝐴(𝐺)| ≤ (1 − 𝜃)|𝑉 (𝐺)| and |𝜕𝐸 𝐴(𝐺)| ≤ 𝐶1 |𝑉 (𝐺)|. For each 𝐺 ∈ H let 𝐻 (𝐺) and 𝐻 (𝐺) 𝑐 denote the subgraphs of 𝐺 induced by 𝐴(𝐺) and 𝐴(𝐺) 𝑐 respectively. Since every vertex of 𝐺 has degree 2|𝐸 (𝐺)|/|𝑉 (𝐺)| we have that min{|𝐸 (𝐻 (𝐺))|, |𝐸 (𝐻 (𝐺) 𝑐 )|} ≥ 𝜃|𝐸 (𝐺)| − |𝜕𝐸 𝐴(𝐺)|. For large 𝑛 we have that 𝜃|𝐸 (𝐺)| ≫ |𝜕𝐸 𝐴(𝐺)| and hence that (𝐻 (𝐺))𝐺∈H and (𝐻 (𝐺) 𝑐 )𝐺∈H are both dense. Thus, it follows from Theorem 5.4.12 that there exists a constant 𝐶2 such that if we set 𝑝(𝐺) = 𝐶2 /|𝑉 (𝐺)| for each 𝐺 ∈ H then both 𝐻 (𝐺) and 𝐻 (𝐺) 𝑐 contain a giant component with high probability under percolation with parameter 𝑝(𝐺). The same also holds at 𝑞(𝐺) = 2𝐶2 /|𝑉 (𝐺)|, which is 𝜀-supercritical for H for an appropriate choice of 𝜀 > 0. But at this same parameter 𝑞(𝐺) the expected number of edges connecting 𝐴(𝐺) and 𝐴(𝐺) 𝑐 is bounded by 2𝐶1𝐶2 , so that by Poisson approximation the probability that there are no such edges is bounded away from zero uniformly over 𝐺 ∈ H . Thus the probability that 𝑞(𝐺)-percolation on 𝐺 contains at least two giant components is bounded away from zero uniformly over 𝐺 ∈ H , and hence H does not have the supercritical uniqueness property. It follows that the supercritical uniqueness property fails for any set H ′ with H ⊆ H ′ ⊆ F , by extending 𝑞 from H to H ′ by setting 𝑞(𝐺) := 1 for all 𝐺 ∈ H ′\H . □ 203 K2n nα edges per vertex nα Kn edges per vertex K2n K2n nα nα nα nα edges Kn edges Kn edges Kn edges per per per per vertex vertex vertex vertex K2n Figure 5.1: Schematic illustration of the graphs discussed in remark 5.4.2 Remark 5.4.2. The proof of (iv) ⇒ (i) above shows more generally that any sequence of finite, simple graphs with linear minimal degree and linear 𝜃-separators for some 𝜃 ∈ (0, 1/2] fails to have the supercritical uniqueness property. On the other hand, the results of [Bol+10a] imply that dense graphs without subquadratic separators have the supercritical uniqueness property. It is natural to wonder in light of Theorem 8.1.1 whether the failure of supercritical uniqueness in graphs of linear minimal degree is always characterized by the existence of linear separators, without the assumption of vertex-transitivity. This is not the case. Indeed, let 0 < 𝛼 < 1/2 and suppose that we take two copies of 𝐾2𝑛 and one copy of 𝐾𝑛 arranged in a line with the two copies of 𝐾2𝑛 at the end and the copy of 𝐾𝑛 in the middle. We may glue these copies together in such a way that each vertex is connected to each of the complete graphs adjacent to its own complete graph by between 𝑛𝛼 and 3𝑛𝛼 edges. It is easily verified that the smallest separators in the resulting graph sequence are of order 𝑛1+𝛼 and that there will exist two distinct giant clusters with high probability when 𝑝 = 3/4𝑛. We focus on the second claim, which is more involved. For such 𝑝 the two copies of 𝐾2𝑛 are supercritical and each contains a giant cluster, while the copy of 𝐾𝑛 is subcritical and has largest cluster of order log 𝑛 with high probability. Thus, when we add in the edges between the various complete graphs, the probability that there exists a cluster in the copy of 𝐾𝑛 that has an edge connecting it to both of its neighbouring copies of 𝐾2𝑛 is small: a 𝐾𝑛 -cluster of size 𝑚 = 𝑂 (log 𝑛) has both such edges adjacent to it with probability of order 𝑚 2 𝑛2𝛼−2 , and since there are at most 2𝑛 such clusters the total conditional probability is 𝑂 ((log 𝑛) 2 𝑛2𝛼−1 ) = 𝑜(1) with high probability, yielding the claim. This gives an example of a linear minimal-degree graph sequence that does not have linear separators but does not have the supercritical uniqueness property either. By considering longer chains of copies of 𝐾𝑛 connecting the two copies of 𝐾2𝑛 as in fig. 5.1, one can obtain similar examples where the minimal size of a separator scales like an arbitrary power of 𝑛 between 𝑛 and 𝑛2 . 204 5.5 Closing remarks Counterexamples. We now discuss examples demonstrating that Theorem 8.1.1 does not extend to arbitrary insertion-tolerant percolation models on the torus or to critical percolation. Example 5.5.1 (Multiple giants at criticality). The cycle Z/𝑛Z (with its standard generating set) has multiple giant components with good probability when 𝑝 = 1 − 𝜆/𝑛, and the set of closed edges converges to a Poisson process on the circle as 𝑛 → ∞. As observed by Alon, Benjamini, and Stacey [ABS04b], by considering the highly asymmetric torus 𝑇𝑛 := (Z/2𝑛 Z) × (Z/𝑛Z) (with its standard generating set) one can obtain similar behaviour at values of 𝑝 that are bounded away from 1. Since these authors did not include a proof, let us now very briefly indicate how the analysis of this example works. We assume for notational simplicity that 𝑛 is a power of 2 and hence is a factor of 2𝑛 . Let 𝑋 be the set of cylinders in (Z/2𝑛 Z) × (Z/𝑛Z) of the form [𝑘𝑛, (𝑘 + 1)𝑛] × (Z/𝑛Z) whose vertices are incident to some simple cycle of dual edges that wrap around the torus. When 𝑝 > 1/2, it follows by sharpness of the phase transition on Z2 that there exists a constant 𝑐 𝑝 > 0 with 𝑐 𝑝 → ∞ as 𝑝 → 1 such that each particular cylinder has probability at most 𝑒 −𝑐 𝑝 𝑛 to belong to 𝑋, this probability being bounded by the probability that a box of size 𝑛 in Z2 intersects a dual cluster of diameter at least 𝑛. On the other hand, since the correlation length on Z2 diverges as 𝑝 ↑ 1/2, there exists a fixed 𝑝 > 1/2 such that E 𝑝 |𝑋 | → ∞ as 𝑛 → ∞. Using this one can prove that if we define 𝑝 𝑛 to be the unique value such that E 𝑝 𝑛 |𝑋 | = 1 then lim inf 𝑝 𝑛 > 1/2 and lim sup 𝑝 𝑛 < 1. It is fairly straightforward to prove that there are multiple giant clusters with positive probability at 𝑝 𝑛 . Indeed, using the same exponential decay estimates one can prove that non-neighbouring cylinders are highly de-correlated, so that E 𝑝 𝑛 |𝑋 | 2 = 𝑂 (1) and there is a good probability for 𝑋 to contain at least two elements that are well-spaced around the torus. On this event there must be at least two giant components with high probability: since 𝑝 𝑛 is bounded away from 𝑝 𝑐 (Z2 ) = 1/2, each vertex of the torus has good probability to belong to a cluster that wraps around the torus, and any two such vertices can be disconnected only if there are two closed dual cut-cycles separating them. The events that two distant vertices belong to such wrapping clusters are highly de-correlated, so that the number of vertices belonging to wrapping clusters is linear with high probability and the claim follows. Example 5.5.2 (Insertion-tolerance on the torus is not enough). Consider the symmetric torus (Z/10𝑛Z) 2 with its standard generating set. Consider the model defined as follows: First, select a pair of vertical strips of the form [𝑘, 𝑘 + 𝑛] × (Z/10𝑛Z) and [𝑘 + 5𝑛, 𝑘 + 6𝑛] × (Z/10𝑛Z) uniformly from among the 𝑛 available possibilities. Declare each edge belonging to one of these strips open with probability 1/4 and each edge not belonging to one of these strips open with probability 3/4. 205 Using standard properties of subcritical and supercritical percolation on Z2 , one easily obtains that this model contains exactly two giant clusters with high probability: the two large high-density strips will each contain a giant cluster with high probability, while there will be no clusters crossing either of the two thin low-density strips with high probability. By also applying a random rotation in {0, 𝜋/2, 𝜋, 3𝜋/4} one obtains an automorphism invariant, uniformly insertion-tolerant, percolationin-random-environment model on (Z/10𝑛Z) 2 with the same properties. As such, one should not expect any uniqueness-of-the-giant-component results to hold on finite graphs at anywhere near the same generality as found in the Burton–Keane theorem [BK89], even when restricting to symmetric tori of fixed dimension. By taking the relative width of the low-density strips to go to zero in a well-chosen manner as 𝑛 → ∞, one can construct a similar example in which the number of giant clusters is either one or two each with good probability and any two vertices are connected with good probability. The supercritical existence property. Theorem 8.1.1 can be though of a geometric characterisation of the infinite sets H ⊆ F for which supercritical percolation has at most one giant cluster with high probability. We now briefly address the complementary problem of whether there is at least one giant cluster with high probability in supercritical percolation, noting that the definitions only ensure that such a cluster exists with good probability (i.e., with probability bounded away from zero). Let H ⊆ F be an infinite set. We say that H has the supercritical existence property if for every supercritical assignment 𝑝 : H → [0, 1] there exists a constant 𝛼 > 0 such that lim inf P𝐺𝑝(𝐺) (the largest cluster contains at least 𝛼 |𝑉 (𝐺)| vertices) = 1. 𝐺∈H Notice that the sharp density property immediately implies the supercritical existence property. However the converse is false because (as we will show below) molecular graphs also have the supercritical existence property. This might lead one to suspect that the supercritical existence property always holds. In fact, this is not the case, and the counterexamples can once again be exactly characterised in terms of molecular graphs. Let us note that the weaker supercritical existence property lim lim inf P𝐺𝑝(𝐺) (the largest cluster contains at least 𝛼 |𝑉 (𝐺)| vertices) = 1 𝛼↓0 𝐺∈H always holds (even without transitivity) as an immediate consequence of the universal tightness theorem [Hut21b]. The following is a fairly straightforward consequence of our results and those of [Bol+10a]. 206 Corollary 5.5.3. An infinite set H ⊆ F has the supercritical existence property if and only if it is not the case that there exist arbitrarily large integers 𝑚 for which H contains an 𝑚-molecular subset. Sketch of proof. Applying the results of [Bol+10a] as in the proof of Theorem 8.1.1 easily yields that if H is 𝑚-molecular then lim inf P𝐺(1−𝜀) −1 𝑝 𝐺 (𝛼,𝜀) (∥𝐾1 ∥ ≤ 1/𝑚) > 0 𝐺∈H 𝑐 for every 0 < 𝛼, 𝜀 < 1, yielding the forward implication of the claim. We now suppose H does not have the supercritical existence property and argue that it contains 𝑚-molecular subsequences for arbitrarily large values of 𝑚. It is an immediate consequence of Proposition 5.4.1 that H has at least one molecular subsequence. Suppose for contradiction that the supremal value of 𝑚 such that H has an 𝑚-molecular subsequence is finite, and denote this supremum by 𝑀. Since H does not have the supercritical existence property, there exists an infinite subset S ⊆ H such that   𝜀2 𝐺 lim sup P (1−𝜀) −1 𝑝 𝐺 (𝜀,𝜀) ∥𝐾1 ∥ ≥ < 1, 𝑐 4𝑀 𝐺∈S and applying Proposition 5.4.1 as before we may assume that this subset is 𝑚-molecular for some 2 ≤ 𝑚 ≤ 𝑀. By taking a further infinite subset and changing 𝑚 if necessary, we may assume that this subset has density deg(𝐺)/|𝑉 (𝐺)| converging to some constant 𝑐 > 0 and does not have a further subset that is 𝑘-molecular for any 𝑘 > 𝑚. By definition there exists a constant 𝐶 such that for each 𝐺 ∈ S there exists an automorphisminvariant set 𝐹𝐺 ⊆ 𝐸 (𝐺) with |𝐹𝐺 | ≤ 𝐶 |𝑉𝐺 | such that 𝐺 \ 𝐹𝐺 has 𝑚 connected components. For each 𝐺 ∈ S, let 𝐻𝐺 be a graph isomorphic to each of the 𝑚 connected components of 𝐺\𝐹𝐺 . Since S does not admit a subset that is 𝑘-molecular for any 𝑘 > 𝑚, A := {𝐻𝐺 : 𝐺 ∈ S} cannot itself contain a molecular subset. Thus, it follows from Proposition 5.4.1 and Theorem 8.1.1 that A has the sharp density property and the supercritical uniqueness property. On the other hand, the vertex degrees of 𝐻𝐺 with 𝐺 ∈ S are asymptotically equal to those of 𝐺 in the sense that the ratio tends to 1, and Theorem 5.4.12 allows us to compute the location of the percolation thresholds for these transitive dense graph sequences in terms of their vertex degrees. (Recall that the largest eigenvalue for the adjacency matrix of a regular graph is equal to its vertex degree.) So any assignment 𝑝˜ : A → [0, 1] built from the assignment 𝑝 : S → [0, 1] with 𝑝(𝐺) := (1 − 𝜀) −1/2 · 𝑝 𝐺 𝑐 (𝜀, 𝜀) by arbitrarily picking 𝑝(𝐻) ˜ ∈ {𝑝(𝐺) : 𝐻𝐺 = 𝐻} for each 𝐻 ∈ A is itself supercritical for A. We deduce from Theorem 8.1.1 that 𝐺 ∥𝐾2 ∥ = 0. lim E𝐻𝑝(𝐺) 𝐺∈S 207 In order for a vertex to belong to the largest cluster in some 𝐺 ∈ S, either it must belong to the largest cluster in its copy of 𝐻𝐺 , or this is not the case and there is an open edge of 𝐹𝐺 incident to its cluster. By vertex transitivity every vertex of 𝐺 is incident to at most 𝐶 edges of 𝐹𝐺 . It follows that, writing 𝐾𝑖 for the 𝑖 th largest cluster, " # ∑︁ 𝐺 E𝐺𝑝(𝐺) ∥𝐾1 ∥ ≤ E𝐻𝑝(𝐺) ∥𝐾1 ∥ + 𝐶 𝑝(𝐺)∥𝐾𝑖 ∥ · |𝐾𝑖 | 𝑖≥2 (𝐺) 𝐻𝐺 ≤ E 𝑝(𝐺) ∥𝐾1 ∥ + 𝐶 𝑝(𝐺)|𝑉 (𝐺)|E𝐻𝑝(𝐺) ∥𝐾2 ∥. Since 𝑝(𝐺) is of order |𝑉 (𝐺)| −1 the second term tends to zero as 𝐺 → ∞ with 𝐺 ∈ S, and we 2 𝐺 deduce that E𝐻𝑝(𝐺) ∥𝐾1 ∥ ≥ 21 E𝐺𝑝(𝐺) ∥𝐾1 ∥ ≥ 𝜀2 for all but finitely many 𝐺 ∈ S. Define 𝑝′ : S → [0, 1] by 𝑝′ (𝐺) := (1 − 𝜀) −1/2 · 𝑝(𝐺) = (1 − 𝜀) −1 · 𝑝 𝐺 𝑐 (𝜀, 𝜀). Since A does not have any molecular subsets, it follows by Markov’s inequality and Proposition 5.4.1 that     𝜀2 𝜀2 𝐻𝐺 𝐺 lim inf P 𝑝 ′ (𝐺) ∥𝐾1 ∥ ≥ ≥ lim inf P 𝑝 ′ (𝐺) ∥𝐾1 ∥ ≥ = 1, 𝐺∈S 𝐺∈S 4𝑀 4 a contradiction. □ 208 Chapter 6 209 SUPERCRITICAL PERCOLATION ON FINITE TRANSITIVE GRAPHS II: CONCENTRATION, LOCALITY, AND EQUICONTINUITY OF THE GIANT’S DENSITY Joint with Tom Hutchcroft Abstract In the previous paper of this series, we showed that supercritical percolation on a large finite transitive graph typically has exactly one giant cluster. In the present paper, we simultaneously establish that the density of this unique giant cluster is concentrated around its mean and that this mean is equicontinuous with respect to the (suitably scaled) percolation parameter and is determined by the “local geometry” (suitably interpreted) of 𝐺. For example, consider the torus T𝑛𝑑 := (Z/𝑛Z𝑛 ) 𝑑 . Our general arguments recover the well-known fact that the supercritical giant cluster density for (T𝑛𝑑 )𝑛≥1 (𝑑 ≥ 2 fixed) converges to the infinite cluster density on Z𝑑 , whereas the supercritical giant cluster density for (T𝑛𝑑 ) 𝑑≥1 (𝑛 ≥ 2 fixed) converges to the survival probability of a Poisson branching process. Our proof relies on a new perspective on how to use sharp threshold theory in percolation: to exploit the fact that certain events of interest (such as the event 𝑜 ↔ ∞) do not to undergo sharp thresholds. These arguments apply equally well to infinite transitive graphs, yielding analogous, new results in this context too. For example, we show that for all 𝜀 > 0 and 𝑑 ∈ N, the function given by the restriction of 𝜃 𝐺 ( 𝑝) := P𝐺𝑝 (𝑜 ↔ ∞) to 𝑝 ∈ [ 𝑝 𝑐 (𝐺) + 𝜀, 1] is uniformly equicontinuous as 𝐺 varies over all infinite transitive graphs with vertex degree 𝑑. 6.1 Introduction This paper is the second in a series investigating the supercritical phase of Bernoulli bond percolation on finite, connected, vertex-transitive graphs. The overarching goal of both papers is to obtain results that hold for all such graphs, in contrast to earlier works that have focused on particular geometric settings (such as complete graphs, tori, or expanders) and used methods specific to those examples. In the first paper of this series, we answered the most basic question about the geometry of clusters in this phase by showing that there is typically a unique giant cluster with high probability. In this paper, we study the density of this giant cluster. We will show that as the volume of the graph tends to infinity, the density of this giant cluster concentrates around a deterministic value, unless the graph belongs to an explicit family of counterexamples known as molecular graphs which we introduced in the first paper. We then investigate the continuity properties of this limiting density, determining the senses in which it is determined by the local geometry of 𝐺 and continuous in 𝑝. As in [EH21a], the theory we develop applies without any constraints on the degree, but is also new in the bounded degree case. Setting the scene Before stating our results, we first briefly overview the results of the first paper in the series [EH21a] and the first author’s companion paper [Eas22]. In addition to providing important context for our new results, this will also give us an opportunity to introduce relevant definitions and notation. Let G be the set of all isomorphism classes of connected, simple (i.e., not containing loops or multiple edges), locally finite, vertex-transitive graphs, and let F = {𝐺 ∈ G : 𝐺 finite}. (We work with isomorphism classes partly to make sure these really are sets; F is countably infinite while G has the cardinality of the continuum. We will usually suppress the distinction between graphs and their isomorphism classes as much as possible when this does not cause any confusion.) Given an infinite set H ⊆ F , a function 𝜙 : H → R, and 𝛼 ∈ R we write lim𝐺∈H 𝜙(𝐺) = 𝛼 or “𝜙(𝐺) → 𝛼 as 𝐺 → ∞ in H ” to mean that for each 𝜀 > 0 there exists 𝑁 such that |𝜙(𝐺) − 𝛼| ≤ 𝜀 for every 𝐺 ∈ H with at least 𝑁 vertices, or equivalently that 𝜙(𝐺 𝑛 ) → 𝛼 for some (and hence every) enumeration H = {𝐺 1 , 𝐺 2 , . . .} of H . Similar conventions apply to define lim sup𝐺∈H , lim inf 𝐺∈H , and limits that may be equal to +∞ or −∞. Given two positive functions 𝑓 and 𝑔 on H , we write “ 𝑓 (𝐺) ∼ 𝑔(𝐺) as 𝐺 → ∞ in H ” to mean that lim𝐺∈H 𝑓 (𝐺)/𝑔(𝐺) = 1. Given a countable graph 𝐺 = (𝑉, 𝐸) and 𝑝 ∈ [0, 1], we write P 𝑝 = P𝐺𝑝 for the law of Bernoulli-𝑝 bond percolation on 𝐺, i.e., the random subgraph of 𝐺 in which each edge is included independently at random with inclusion probability 𝑝. Given a percolation configuration 𝜔 ∈ {0, 1} 𝐸 on 𝐺, the connected components of 𝜔 are called clusters. We write 𝐾𝑢 to denote the cluster containing the vertex 𝑢 and write 𝑢 ↔ 𝑣 for the event that 𝐾𝑢 = 𝐾𝑣 . Given a subset 𝑊 of 𝑉, the volume of 𝑊 is the number of vertices in 𝑊, denoted |𝑊 |, while if 𝐺 is finite, the density of 𝑊 is defined to be the ratio ∥𝑊 ∥ := |𝑊 | /|𝑉 |. We write 𝐾1 , 𝐾2 , . . . for the clusters of 𝜔 in decreasing order of volume 210 (breaking ties arbitrarily). Given an infinite set H ⊆ F , we say that an assignment of parameters 𝑝 𝑐 : H → [0, 1] is a percolation threshold if 1. lim𝐻∈H P (1−𝜀) 𝑝 𝑐 (∥𝐾1 ∥ ≥ 𝑐) = 0 for every 𝜀, 𝑐 > 0, and 2. For every 𝜀 > 0 there exists 𝛼 > 0 such that lim𝐻∈H P (1+𝜀) 𝑝 𝑐 (∥𝐾1 ∥ ≥ 𝛼) = 1, where we set P 𝑝 = P1 for 𝑝 ≥ 1. Note that critical thresholds are not unique when they exist, but any two percolation thresholds 𝑝 𝑐 , 𝑝˜𝑐 : H → [0, 1] must satisfy 𝑝 𝑐 (𝐺) ∼ 𝑝˜𝑐 (𝐺) as 𝐺 → ∞ in H . When a percolation threshold 𝑝 𝑐 : H → [0, 1] exists, we say that 𝑝 : H → [0, 1] is supercritical if the set H ′ = {𝐺 ∈ H : 𝑝(𝐺) < 1} is infinite and satisfies lim inf ′ 𝐺∈H 𝑝(𝐺) >1 𝑝 𝑐 (𝐺) or H ′ is finite. We generalise this definition to include the case that a threshold 𝑝 𝑐 does not exist by saying that 𝑝 is supercritical if H ′ is infinite and there exists 𝜀 > 0 such that lim inf P (1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀, ′ 𝐺∈H or H ′ is finite. Note that these two definitions of supercriticality coincide when H admits a threshold function, and in particular that the definition of supercriticality does not depend on the choice of threshold function. This definition also guarantees that every infinite set H ⊆ F admits a supercritical assignment 𝑝 : H → [0, 1], namely the trivial supercritical assignment 𝑝(𝐺) ≡ 1. We say that H has the supercritical uniqueness property if lim P 𝑝 (∥𝐾2 ∥ ≥ 𝜀) = 0 𝐺∈H for every supercritical 𝑝 : H → [0, 1] and every constant 𝜀 > 0. The first paper in this series [EH21a] together with the related work of the first author [Eas22] give simple characterizations of those transitive graph families that have the supercritical uniqueness property and that admit percolation thresholds respectively. Theorem 6.1.1 ([EH21a, Theorem 1.2]). An infinite set H ⊆ F has the supercritical uniqueness property if and only if it does not contain an infinite molecular subset. 211 Theorem 6.1.2 ([Eas22, Theorem 2]). An infinite set H ⊆ F admits a percolation threshold if and only if it does not contain an infinite 𝑚-molecular subset for infinitely many values of 𝑚. Here, given an integer 𝑚 ≥ 2, we say that an infinite set H ⊆ F is 𝑚-molecular if it has the following properties, where 𝐸 (𝐺) and 𝑉 (𝐺) denote the set of edges and the set of vertices of a given graph 𝐺: 1. H is dense, meaning that lim inf 𝐺∈H |𝐸 (𝐺)|2 > 0. |𝑉 (𝐺)| 2. There exists a constant 𝐶 < ∞ such that for each 𝐺 = (𝑉, 𝐸) ∈ H there exists a set of edges 𝐹 ⊆ 𝐸 satisfying the following conditions: a) 𝐺 \ 𝐹 has exactly 𝑚 connected components; b) 𝐹 is invariant under Aut(𝐺); c) |𝐹 | ≤ 𝐶 |𝑉 |. The stated conditions on 𝐺 and 𝐹 imply that the 𝑚 connected components of 𝐺 \ 𝐹 are dense, vertex-transitive, and isomorphic to each other. We say that an infinite set H ⊆ F is molecular if it is 𝑚-molecular for some 𝑚 ≥ 2. For example, the set of Cartesian products of complete graphs {𝐾𝑛 □𝐾𝑚 : 𝑛 ≥ 1} is 𝑚-molecular for each 𝑚 ≥ 2. Applying the analysis of percolation on dense graphs carried out in [Bol+10b], it is fairly easy to see that the supercritical uniqueness property does not hold for molecular sets of graphs. For example, if we take 𝑝 : {𝐾𝑛 □𝐾𝑚 : 𝑛 ≥ 1} → [0, 1] defined by 𝑝(𝐾𝑛 □𝐾𝑚 ) = 2/𝑛 then 𝑝 is supercritical since each copy of 𝐾𝑛 will contain a giant cluster by the classical theory of Erdős-Rényi random graphs, but the total number of edges between distinct copies of 𝐾𝑛 converges to a Poisson random variable and hence is zero with positive probability. The main result of [EH21a] is that constructions of this form are the only way to get a non-unique giant in supercritical percolation on a transitive graph. These theorems imply in particular that if H ⊆ F is an infinite set that is sparse, meaning that lim𝐺∈H |𝐸 (𝐺)|2 = 0, then H has the supercritical uniqueness property and admits a percolation |𝑉 (𝐺)| threshold function. An important example of such an infinite set H is F𝑑 (for each 𝑑 ≥ 2), the set of all (isomorphism classes of) finite, connected, simple, vertex-transitive graphs with degrees bounded by 𝑑. Concentration: The density is well-defined An important quantity associated to percolation on an infinite transitive graph at a given parameter 𝑝 is the infinite cluster density, the proportion of vertices contained in infinite clusters. In this 212 context, if we define the expectation 𝜃 ( 𝑝, 𝐺) := P𝐺𝑝 (𝑜 ↔ ∞), where 𝑜 denotes an arbitrary vertex of 𝐺, then it is easily shown that this quantity accurately captures the density of infinite clusters in the sense that if 𝐴 is any finite set of vertices then |{𝑥 ∈ 𝐴 : 𝑥 ↔ ∞}| = (𝜃 ± 𝑜(1))| 𝐴| with high probability when 𝐴 is large. (Indeed, the variance of this random variable is easily seen to be 𝑜(| 𝐴| 2 ). Getting sharp quantitative bounds on the fluctuations of this random variable in general geometry is a very interesting problem closely related to those of [HH19, Section 5.3].) Moreover, for percolation on Z𝑑 , it is an immediate consequence of the ergodic theorem that 1 (2𝑛 + 1) 𝑑 ∑︁ 1 (𝑥 ↔ ∞) → 𝜃 𝑥∈[−𝑛,𝑛] 𝑑 almost surely as 𝑛 → ∞. As such, for infinite graphs, the relationship between the almost-sure and in-expectation density of infinite clusters is trivial, and one instead focuses on questions concerning e.g. the dependence of the density on 𝑝, with continuity at 𝑝 𝑐 being a famous open problem for three-dimensional lattices. For finite transitive graphs, the most natural analogue of the infinite cluster density is the giant cluster density. In this setting, the ergodic theorem no longer applies and the analogous question on the relation between expected and almost-sure densities become much more subtle. Of course, the picture we would naively expect is that as 𝑝 increases across some threshold value 𝑝 𝑐 (𝐺), a unique macroscopic cluster should emerge, and the density of this macroscopic cluster should be concentrated around its mean 𝜃 ( 𝑝, 𝐺) := P𝐺𝑝 (𝑜 ∈ 𝐾1 ). Unfortunately this is not always the case: In the product of an 𝑛-vertex complete graph with an edge with 𝑝 = 𝜆/𝑛 where 𝜆 > 1, it can be shown the largest cluster either has density close to either mf (𝜆) or 21 mf (𝜆) with high probability, where mf (𝜆) is the limiting density of the giant cluster in the 𝑝 = 𝜆/𝑛 Erdős-Rényi 2 graph, with probability approximately 1 − 𝑒 −𝜆 mf (𝜆) to have density close to mf (𝜆); the resulting 2 density 𝜃 ∼ (1 − 21 𝑒 −𝜆 mf (𝜆) ) mf(𝜆) does not adequately capture the bimodal nature of this limiting distribution. Even for bounded degree graphs, it is possible for similar behaviour to hold at the critical point as we see on the long torus (Z/𝑛Z) × (Z/2𝑛 Z) [EH21a, Example 5.1]. Still it seems reasonable to conjecture that the density should be concentrated if we restrict to the supercritical case and impose some mild geometric conditions on the graph at hand. In order to address this question, we first introduce a relevant definition: We say that an infinite set H ⊆ F has the supercritical concentration property if
lim P 𝑝 ∥𝐾1 ∥ − E 𝑝 ∥𝐾1 ∥ ≥ 𝜀 = 0 𝐺∈H 213 for every supercritical 𝑝 : H → [0, 1] and every constant 𝜀 > 0. (Note that if concentration occurs at all then it must be around the mean by the bounded convergence theorem.) As with the uniqueness problem, it is reasonably easy to see that this property does not hold for molecular sets of graphs, for similar reasons to the concrete example of the product of a complete graph with an edge as discussed above. Our first main theorem states that, as with the supercritical uniqueness property, molecular sets are the only obstruction to the supercritical concentration property. Theorem 6.1.3. Let H be an infinite set of (isomorphism classes of) finite, simple, connected, vertex-transitive graphs. The following are equivalent: 1. H has the supercritical concentration property. 2. H has the supercritical uniqueness property. 3. H does not contain an infinite molecular subset. Note that the equivalence 2. ⇔ 3. and the implication 1. ⇒ 2. were established in [EH21a, Theorem 1.2 and Lemma 3.6]; the new content of the theorem is the implication 2&3. ⇒ 1. Corollary 6.1.4. Let H be an infinite set of (isomorphism classes of) finite, connected, vertextransitive graphs. If H is sparse, then it has the supercritical concentration property. This corollary is new even for H = F𝑑 , the set of all finite, simple, connected, vertex-transitive graphs with degrees bounded by 𝑑. (Again, we stress that for bounded degree graphs the giant can exist and fail to be unique or have concentrated density at the critical point, as part of a discontinuous phase transition [EH21a, Example 5.1].) Locality of the density Now that we know that 𝜃 ( 𝑝, 𝐺) = P𝐺𝑝 (𝑜 ∈ 𝐾1 ) accurately captures the density of the supercritical giant for large non-molecular finite transitive graphs with high-probability, we would like to understand the nature of the dependency of this quantity on 𝑝 and 𝐺. In particular, we would like to understand whether we can compute the density of the giant in terms of some appropriate local limit object as the size of our graph diverges. Before providing our complete answer to this question, let us introduce the analogous statements in the simpler setting of infinite transitive graphs. 214 Infinite graphs The most important open problem in the theory of percolation on infinite transitive graphs is to establish that the map 𝜃 (·, 𝐺) (i.e. 𝑝 ↦→ 𝜃 ( 𝑝, 𝐺)) is continuous for every fixed 𝐺 ∈ G ∗ , where G ∗ is the set of all infinite transitive graphs that are not quasi-isometric to Z. This map is always continuous on [0, 1]\{𝑝 𝑐 (𝐺)} [Sch99], but continuity at 𝑝 𝑐 (𝐺) remains open for many graphs, including 𝐺 = Z3 . In this paper, we are interested in the orthogonal question: Endow G ∗ with the local (Benjamini-Schramm) topology. Now is the map 𝜃 ( 𝑝, ·) continuous for every fixed 𝑝? This variant of continuity arises naturally in the study of percolation on large finite transitive graphs (see below). More generally, is the map 𝜃 : [0, 1] × G ∗ → [0, 1] jointly continuous, as a function of two variables? In fact, since 𝜃 (·, 𝐺) is monotone, this joint continuity is nothing more than continuity in each argument individually. Underlying both of these versions of continuity is the problem of showing that 𝜃 ( 𝑝, 𝐺) is uniformly well-approximated by 𝜃 𝑛 ( 𝑝, 𝐺) := P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛, 𝑜 ↮ ∞). Indeed, let 𝑑 ∈ N, let G𝑑∗ be the (compact) subset of G ∗ of graphs with vertex degree 𝑑, and consider some 𝑝 ∈ [0, 1] and 𝐺 ∈ G𝑑∗ . Now by Dini’s theorem, the functions 𝜃 (·, 𝐺), 𝜃 ( 𝑝, ·), and 𝜃 (·, ·) 𝑛→∞ are continuous if and only if 𝜃 𝑛 −−−−→ 𝜃 uniformly on [0, 1] × {𝐺}, {𝑝} × G𝑑∗ , and [0, 1] × G𝑑∗ , respectively. An obstacle to establishing the required uniform convergence on {𝑝} × G𝑑∗ for fixed 𝑝 is that for some choices of 𝐺, the parameter 𝑝 might be supercritical (i.e. 𝑝 > 𝑝 𝑐 (𝐺)), whereas for others, 𝑝 might be subcritical or critical. It is often easier to build arguments that are tailored to studying percolation in just one of these three phases at a time. In this paper, we directly establish that 𝜃 𝑛 → 𝜃 uniformly on the supercritical region {( 𝑝, 𝐺) : 𝑝 ≥ 𝑝 𝑐 (𝐺) + 𝜀} for every fixed 𝜀 > 0. This immediately yields the following statement. (There is no hypothesis that the graphs are not one-dimensional because this condition does not appear in our argument and because the result holds trivially when the graphs are one-dimensional.) Theorem 6.1.5. Let (𝐺 𝑛 )𝑛≥1 be a sequence of infinite transitive graphs converging locally to some infinite transitive graph 𝐺. Then lim 𝜃 ( 𝑝, 𝐺 𝑛 ) = 𝜃 ( 𝑝, 𝐺) 𝑛→∞ for all 𝑝 > sup 𝑝 𝑐 (𝐺 𝑛 ). 𝑛≥1 This theorem in particular recovers the fact that that 𝑝 𝑐 (𝐺) ≤ lim inf 𝑛→∞ 𝑝 𝑐 (𝐺 𝑛 ), and hence that the map 𝑝 𝑐 : G ∗ → [0, 1] is lower semi-continuous. This had previously been established as a consequence of the sharpness of the phase transition [Pet14, §14.2]. 215 In [EH23a], we established that 𝜃 𝑛 → 𝜃 uniformly on the subcritical region {( 𝑝, 𝐺) : 𝑝 ≤ 𝑝 𝑐 (𝐺) − 𝜀} for every fixed 𝜀 > 0. By the same reasoning as above, this implies that the map 𝑝 𝑐 : G ∗ → [0, 1] is upper semi-continuous. Since, as mentioned, 𝑝 𝑐 was known to be lower semi-continuous, our work established that 𝑝 𝑐 is continuous. The continuity of 𝑝 𝑐 had been known as Schramm’s locality conjecture. In stark contrast to our proof of Schramm’s locality conjecture, our proof of Theorem 6.1.5 is short and handles all (unimodular) transitive graphs via single argument. By combining these two results, we obtain the following concerning the continuity of 𝜃 ( 𝑝, ·). This corollary was previously known in special cases such as when all of the graphs in question have polynomial growth [CMT22]. We were not able to remove the hypothesis that 𝑝 ≠ 𝑝 𝑐 (𝐺); if we could, then by approximating Z3 by toroidal slabs, we could also prove that 𝜃 (·, Z3 ) is continuous by applying the results of [DST14]. Corollary 6.1.6. Let (𝐺 𝑛 )𝑛≥1 be a sequence of non-one-dimensional infinite transitive graphs converging locally to some infinite transitive graph 𝐺. Then lim 𝜃 ( 𝑝, 𝐺 𝑛 ) = 𝜃 ( 𝑝, 𝐺) 𝑛→∞ for all 𝑝 ∈ [0, 1]\{𝑝 𝑐 (𝐺)}. Bounded-degree finite graphs Continuity-in-𝐺 questions arise naturally in the study of percolation on large, bounded-degree, finite transitive graphs. Indeed, since every infinite set H of finite transitive graphs with bounded degrees is relatively compact in the local topology, we can for many purposes assume without loss of generality that H converges locally to some infinite transitive graph 𝐺. It is then natural to ask whether lim𝐻∈H 𝜃 ( 𝑝, 𝐻) = 𝜃 ( 𝑝, 𝐺) for a fixed 𝑝. All of our earlier discussion relating continuity properties of 𝜃 to uniform convergence 𝜃 𝑛 → 𝜃 can be adapted to finite graphs mutatis mutandis, where we define 𝜃 𝑛 ( 𝑝, 𝐺) := P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛, 𝑜 ∉ 𝐾1 ) when 𝐺 is a finite transitive graph. In particular, our proof of Theorem 6.1.5 yields the following for supercritical percolation on finite transitive graphs. Theorem 6.1.7. Let (𝐺 𝑛 )𝑛≥1 be a sequence of finite transitive graphs converging locally to some infinite transitive graph 𝐺. Let 𝑝 𝑐 be a percolation threshold function for {𝐺 𝑛 : 𝑛 ≥ 1}. Then lim 𝜃 ( 𝑝, 𝐺 𝑛 ) = 𝜃 ( 𝑝, 𝐺) 𝑛→∞ for all 𝑝 > lim sup 𝑝 𝑐 (𝐺 𝑛 ). 𝑛→∞ In [Eas24], the first author combined [EH23a; Eas22; EH21a] to prove an analogue of Schramm’s locality conjecture for finite graphs. The trouble here is that the non-one-dimensionality condition becomes more subtle. We need to exclude finite graphs are long and thin, like a circle. Formally, 𝜋 1 let distGH (𝑆 1 , diam 𝐺 𝐺) denote the Gromov-Hausdorff distance between the unit circle 𝑆 and the 216 𝜋 graph metric on 𝐺 after it has been rescaled by diam 𝐺 , where diam 𝐺 is the diameter of 𝐺. Now we would like to exclude sequences of graphs for this distance tends to zero. (In fact, for the following theorem, we may allow this distance to tend to zero, so long as it tends to zero not too quickly (log diam 𝐺) 1/9 (i.e. faster that 𝑒 diam 𝐺 ).) By combining this finite graph locality result with Theorem 6.1.5, we obtain the following result describing the asymptotic behaviour of 𝜃 on large bounded-degree finite transitive graphs. Corollary 6.1.8. Let (𝐺 𝑛 )𝑛≥1 be a sequence of finite transitive graphs converging locally to some infinite transitive graph 𝐺. Suppose that   𝜋 1 inf distGH 𝑆 , 𝐺 𝑛 > 0. 𝑛≥1 diam 𝐺 𝑛 Then lim 𝜃 ( 𝑝, 𝐺 𝑛 ) = 𝜃 ( 𝑝, 𝐺) 𝑛→∞ for all 𝑝 ∈ [0, 1]\{𝑝 𝑐 (𝐺)}. High degree finite graphs Let us now consider an infinite set H of finite transitive graphs with lim𝐺∈H deg 𝐺 = ∞, such as the sequence of complete graphs or the sequence of hypercubes. We would like to again relates the asymptotic behaviour of 𝜃 to some kind of local limit object for H . The problem is that since H has diverging vertex degrees, H cannot converge in the local topology. So the continuity-in-𝐺 questions above do not readily extend. On the other hand, the (equivalent) questions about the uniform convergence of 𝜃 𝑛 ( 𝑝, 𝐺) → 𝜃 ( 𝑝, 𝐺) do! For example, the analogue of continuity-in-𝐺 is that for a given sequence of parameters 𝑝 : H → [0, 1], lim sup |𝜃 ( 𝑝, 𝐺) − 𝜃 𝑛 ( 𝑝, 𝐺)| = 0. 𝑛→∞ 𝐺∈H (6.1.1) By considering a step-by-step exploration of the cluster at 𝑜, it is easy to see that for each fixed 𝑛, as 𝐺 → ∞ in H , the probability 𝜃 𝑛 ( 𝑝, 𝐺) tends to the probability that a Poi( 𝑝 · deg 𝐺) branching process contains 𝑛 vertices. In particular, writing mf (𝜆) for the survival probability of a Poi(𝜆) branching process, (6.1.1) is equivalent to having the following mean-field approximation: lim |𝜃 ( 𝑝, 𝐺) − mf( 𝑝 deg 𝐺)| = 1. 𝐺∈H Thus the analogue of the conclusion of Theorem 6.1.7 is that this mean-field approximation holds whenever 𝑝 is a supercritical sequence of parameter. In this paper, we will characterise for which families of finite transitive graphs this approximation holds. Since a Poi(𝜆)-branching process is critical at 𝜆 = 1, the analogue of the locality of the critical point is that 𝐺 ↦→ deg1 𝐺 is the percolation threshold for H , under some non-one-dimensionality hypothesis on 𝐺. We have nothing to say here about this question. 217 Unfortunately, the mean-field approximation does not always hold for supercritical percolation on high-degree finite transitive graphs. For example, consider the product 𝐾𝑛 × (Z/𝑛Z) 2 of a complete graph with a two-dimensional torus. It is easy to see that the failure of the mean-field approximation in this example extends to any family of graphs that can be automorphism-invariantly decomposed into a collection of dense graphs by deleting 𝑂 (|𝑉 |) edges. We call such graphs macromolecular. Here is the precise definition. Note that every molecular graph is macromolecular. Definition 6.1.9. We say that a finite transitive graph 𝐺 = (𝑉, 𝐸) is 𝜀-macromolecular, where 𝜀 > 0, if there exists an Aut 𝐺-invariant set of edges 𝐹 ⊆ 𝐸 with 𝜀 |𝐹 | ≤ |𝑉 | such that (𝑉, 𝐸\𝐹) is not connected, and the connected component (𝑉 ′, 𝐸 ′) of 𝑜 in (𝑉, 𝐸\𝐹) satisfies |𝐸 ′ | ≥ 𝜀 |𝑉 | 2 . We say that an infinite set H of finite transitive graphs is macromolecular limH deg 𝐺 = ∞ and there exists a constant 𝜀 > 0 such that all but finitely many of the graphs in H are 𝜀-macromolecular. We will show that macromolecular graphs are in fact the only obstacles to the mean-field approximation for high degree finite transitive graphs. Theorem 6.1.10. Let H be a set of (isomorphism classes of) finite, connected, vertex-transitive graphs, and suppose that deg(𝐺) → ∞ as 𝐺 → ∞ in H . Then the mean-field approximation lim |𝜃 ( 𝑝, 𝐺) − mf ( 𝑝 deg 𝐺)| = 1, 𝐺∈H holds for every supercritical assignment 𝑝 : H → [0, 1] if H does not have any infinite subsets that are macromolecular. The converse also holds, provided that H admits least one supercritical assignment 𝑝 : H → [0, 1] that is non-trivial in the sense that lim inf H 𝜃 ( 𝑝, 𝐺) < 1. What governs the asymptotic density 𝜃 for supercritical percolation on a high-degree macromolecular graph? We can think of percolation on the product of a complete graph with a torus as roughly “simulating percolation” on the torus, by first revealing the states of the edges in the complete graphs — producing a unique giant cluster in each complete graph — then revealing the remaining edges. This perspective can be used to show that 𝜃 (𝜆/𝑛, 𝐾𝑛 × (Z/𝑛Z) 2 ) ∼ 𝜃 (𝜆∗, (Z/𝑛Z) 2 ) as 𝑛 → ∞, where 𝜆∗ = (1 − 𝑒 −𝜆 ) mf (𝜆) 2 . In Section 6.9, we will show that in fact for all macromolecular graphs, the supercritical density 𝜃 is determined “locally” by the graph obtained by contracting each dense graph to a vertex. Note that this contraction can produce a torus (as above), and similarly any bounded-degree graph, but also large complete graphs and hypercubes. In particular, 218 the macromolecular case is at least as hard as the general bounded-degree case and the complete graph and hypercube cases, combined. However, crucially, this contraction cannot produce a macromolecular sequence of graphs. Equicontinuity Our results also yield the following general result about the equicontinuity of the supercritical giant cluster density on any finite transitive graph. Recall that F is the set of all finite transitive graphs. Theorem 6.1.11. Let 𝑝 : F → [0, 1] be supercritical. For each 𝐺 ∈ F , consider the function 𝑓𝐺 : [ 𝑝, 1] → [0, 1] given by 𝑓𝐺 (𝑞) := 𝜃 (𝑞/𝑝, 𝐺). Then the family of functions { 𝑓𝐺 : 𝐺 ∈ F } is uniformly equicontinuous. By specialising to families of bounded-degree infinite transitive graphs, we also obtain the following. (This time, we can use 𝑞 in place of 𝑞/𝑝, since 𝑝 is bounded away from zero.) Theorem 6.1.12. Let H be an infinite set of infinite transitive graphs with uniformly bounded vertex degrees. Let 𝜀 > 0, and for each 𝐺 ∈ H , consider the function 𝑓 𝑝 : [ 𝑝, 1] → [0, 1] given by 𝑓𝐺 (𝑞) := 𝜃 (𝑞, 𝐺). Then the family of functions { 𝑓𝐺 : 𝐺 ∈ H } is uniformly equicontinuous. Proof sketch Consider supercritical percolation P 𝑝 on a large finite transitive graph 𝐺 that is not molecular, and say that we are tasked with proving that the giant cluster density is concentrated. The natural subgoal is to prove the existence of a uniform tail on the distributution of non-giant clusters, i.e.
𝑛→∞ P 𝑝 |𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant −−−−→ 0 | {z } uniformly in 𝐺. (6.1.2) ♥ Indeed, all of the following implications are rather easy to prove: Uniform tail ⇐⇒ Locality =⇒ Equicontinuity =⇒ Concentration. See Section 6.11 for details. Unfortunately, both of the missing reverse implications in this chain are false. So our strategy to prove concentration by first proving eq. (6.1.2) is doomed to fail in general. Let us anyway see how far this plan will take us. We split into cases according to whether the vertex degrees of 𝐺 are bounded above or tending to infinity. 219 Bounded degrees Our starting point is the elementary observation that the required uniform tail on finite clusters holds for typical choices of 𝑝 when some neighbour 𝑢 of 𝑜 happens to belong to the giant, i.e.  𝑛→∞ P 𝑝 |𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant and 𝑢 ∈ giant −−−−→ 0 uniformly in 𝐺, | {z } ♦ because a simple mass-transport argument shows that P 𝑝 (♦) ≤ 𝜃 ′ ( 𝑝, 𝐺) , 𝑛 where 𝜃 ′ ( 𝑝, 𝐺) denotes derivative with respect to 𝑝, and this derivative must trivially be bounded above for most choices of 𝑝 because 𝑝 ↦→ 𝜃 ( 𝑝, 𝐺) is an increasing function taking values in [0, 1]. In principle, this approach could fail because the parameter 𝑝 that we were given was carefully chosen to belong a small set of parameters where this derivative is large. This turns out not to be a serious obstacle: if we could prove that eq. (6.1.2) holds whenever this derivative is bounded, we could quite easily deduce that eq. (6.1.2) holds in general. The real challenge is therefore to prove that for all 𝜀 > 0 there exists 𝛿 > 0 such that P 𝑝 (♥) ≥ 𝜀 =⇒ P 𝑝 (♦) ≥ 𝛿. (6.1.3) Here is an initial, naive approach: by an easy exploration argument (in which we first reveal the edges incident to 𝐾𝑜 , then reveal the remaining edges), with probability at least P 𝑝 (♥) · 𝜃 ( 𝑝, 𝐺) there is an edge 𝑢𝑣 in the boundary of 𝐾𝑜 whose outer endpoint 𝑣 belongs to the giant and whose inner endpoint 𝑢 satisfies |𝐾𝑢 | ≥ 𝑛 but does not belong to the giant. The problem is that we do not have any control over the location of this edge 𝑢𝑣. If only we knew that 𝑢𝑣 could be found inside of some bounded-size, deterministic set of edges 𝐹... An important theme in boolean analysis is that an increasing boolean function typically undergoes a sharp threshold1 if and only if it does not depend too much on a small number of bits. In percolation theory, the ‘only if ’ direction is typically applied to show that some connectivity event of interest does undergo a sharp threshold. Our new idea is to apply the ‘if ’ direction to the event {𝑜 ∈ giant} 1 See Section 6.3 for full background, including the definition of sharp thresholds. 220 which — in the cases of interest — trivially does not undergo a sharp threshold. This guarantees the existence of a bounded-size, deterministic set of edges 𝐹 such that if 𝐹 is entirely open, then 𝑜 is highly likely to belong to the giant cluster. Using this and insertion tolerance, we are able to fix the naive argument above to show a suitable edge 𝑢𝑣 can indeed be found within 𝐹 with good probability, as required. Unbounded degrees There are two problems with the above approach when it comes to graphs with large degrees. The first is the use of insertion tolerance: typically, we will be working with parameters that on the order of 1/deg 𝐺, and in particular, close to 0. Therefore, the cost to open all but one of the edges in 𝐹 is no longer constant. This turns out to be solvable by applying an adaptation of the sprinkling and surgery argument we introduced in the previous paper of the series to prove Theorem 6.2.1. This argument, which relies heavily on transitivity, lets us reduce the size of 𝐹 to a singleton, as required, by paying only a constant cost. A slightly technical point is that for this argument to work, which involves repeatedly sprinkling an unboundedly large number of times, it is helpful to work with an event that is increasing. For this reason, in section Section 6.5, we will show that bounds on the distribution of non-giant clusters (“germs”) — which may not be monotone in 𝑝 — can be converted into bounds on the probability that the giant cluster does not intersect given large deterministic sets (“holes”) — which clearly is monotone in 𝑝, and vice versa. The second problem is that the argument naively yields an upper bound on ♣1 := P 𝑝 (|𝐾𝑜 | ≥ 𝑛 deg 𝐺, 𝑜 ∉ giant) for a large constant 𝑛, rather than on ♣2 := P 𝑝 (|𝐾𝑜 | ≥ 𝑛, 𝑜 ∉ giant) . This is a more substantial problem because for macromolecular graphs, we can indeed have that |𝐾𝑜 | is on the order of deg 𝐺 with good probability, without belonging to the giant cluster. We will define a suitable automorphism-invariant relation on the vertices on 𝐺, called “being friends”, where, roughly speaking, 𝑢 and 𝑣 are friends if it is unlikely that the giant cluster contains many of the vertices in the neighbourhood of 𝑢 but not of 𝑣, or vice versa. Then, we show that if we cannot convert our control on ♣1 into a control on ♣2 , then the equivalence relation given by the transitive closure of this “friend" relation must actually induce a macromolecular decomposition. Thus, we can split into two cases: either we have the required control on ♣2 , or our graph is macromolecular. In the former case we are done as in the bounded-degree case. In the 221 macromolecular case, we relate percolation on the graph 𝐺 to percolation on the quotient graph obtained by contracting each equivalence class of 𝐺 to a point. This essentially reduces the problem back to the bounded-degree, since clusters of size much larger than constant in this quotient graph correspond to clusters of size much larger than deg 𝐺 in the original graph 𝐺, which we are able to control by ♣1 . More precisely, we use this coupling to show that the density 𝜃 ( 𝑝, 𝐺) on 𝐺 can be related to the microscopic cluster distribution in the quotient graph, and thus that this density is uniformly equicontinuous. From this equicontinuity, we then obtain concentration via Theorem 6.2.1. Note that a key challenge and novelty arising in our work, in contrast to previous works investigating percolation on general graphs under some isoperimetric conditions, is that we do not obtain a general bound on the size of microscopic clusters. Indeed, at our level of generality, such a bound does not exist. (Relatedly, we cannot give a non-trivial upper bound on the variance of ∥giant∥ when this density is concentrated around its mean, since by approximating molecular graphs, the rate of convergence could be arbitrarily small.) This leads us to use a expansion-vs-structure dichotomy — either we have good isoperimetric properties and therefore have sufficiently strong control of small clusters, or we have some non-trivial rigid structure (being macromolecular) that we can exploit. This is similar in spirit to the strategy often used in the study of probability on general infinite transitive graphs, for example in our proof of Schramm’s locality conjecture, which uses an expansion-vs-structure dichotomy with graphs with rapid volume growth having good expansion, and graphs with slow volume growth having approximately the structure of the Cayley graph of a nilpotent group (thanks to Gromov’s theorem about groups of polynomial growth and related results). Remark 6.1.1. In our first draft of this paper, we were only able to prove our results for families of graphs uniformly bounded vertex degrees. Our proof was quite different and less elementary, combining the second author’s two-ghost inequality with a standard application sharp threshold theory - namely, using Talagrand’s inequality to prove that certain connectivity events do have sharp thresholds. Realising that these methods could be relevant to Schramm’s locality conjecture, we paused on this project to complete our proof of Schramm’s conjecture in [EH23a]. These original methods make up [EH23a, Section 2], and we sketched in [EH23a, Section 7.2] how these methods can be used to establish uniqueness, concentration, locality, and equicontinuity for bounded-degree finite transitive graphs. However, these methods break down completely when trying to analyse the supercritical giant cluster density for finite transitive graphs with large vertex degrees. More precisely, these methods 222 fail to prove locality and equicontinuity as soon as graphs have even slowly diverging vertex degrees, and they fail to prove uniqueness and concentration once the vertex degrees grow algebraically with respect to the total number of vertices. Indeed, it is clear that these methods must fail in general because they do not detect whether a graph is macromolecular. 6.2 Notation Graphs Let 𝐺 be a graph. For us, this means that 𝐺 is an isomorphism class of connected, simple2 , locally finite, countable graphs. When the choice of graph (or multigraph) 𝐺 is clear from context, we write 𝑉 and 𝐸 for the vertex and edge sets of 𝐺; when it is not clear we instead write 𝑉 (𝐺) and 𝐸 (𝐺) to be explicit. We will adopt the convention that the volume of 𝐺 is the number of vertices in 𝐺, denoted |𝐺 | := |𝑉 (𝐺)|. If 𝐺 is (vertex-)transitive, then we write 𝑜 to denote some fixed vertex of 𝐺, which we refer to as the origin. We write dist(·, ·) for the graph metric on 𝐺 and define diam 𝐺 := max𝑢,𝑣 dist(𝑢, 𝑣). Given 𝑢 ∈ 𝑉 and 𝑛 ≥ 0, we define 𝐵𝑛 (𝑢) := {𝑣 ∈ 𝑉 : dist(𝑢, 𝑣) ≤ 𝑛}, and when 𝐺 is transitive, we set 𝐵𝑛 := 𝐵𝑛 (𝑜). Classes of graphs We write G and F to denote the set of all infinite and finite transitive graphs, respectively. Later, we will also write U and N for set of all unimodular and nonunimodular transitive graphs, respectively. The definition of unimodularity is recalled in Section 6.4. We say that an infinite transitive graph 𝐺 ∈ G is one-dimensional if G is roughly-isometric to Z, and we define G ∗ := {𝐺 ∈ G : 𝐺 is not one-dimensional}. For each 𝑑 ≥ 1, we write G𝑑 to denote the set of all graphs in G having vertex degree exactly 𝑑, and we analogously define F𝑑 , U𝑑 , N𝑑 and G𝑑∗ . We endow the whole of G ∪ F with the local (aka Benjamini-Schramm) topology. This obviously makes G𝑑 compact and F𝑑 relatively compact (for each 𝑑 ≥ 1), but in fact, the spaces G𝑑∗ , U𝑑 , and N𝑑 are also compact. Dense graphs Let 𝐺 = (𝑉, 𝐸) be a finite graph. Let 𝜀 > 0. We say that 𝐺 is 𝜀-dense if |𝐸 | ≥ 𝜀 |𝑉 | 2 . If 𝐺 is transitive, this is equivalent to deg 𝐺 ≥ 𝜀2 |𝑉 |, where deg 𝐺 denotes the common vertex degree of 𝐺. We say that an infinite set of finite graphs H is dense if there exists 𝜀 > 0 such that all but finitely many graphs in H are 𝜀-dense, and we say that H is sparse if instead lim𝐺∈H |𝐸 (𝐺)|2 = 0. Note that every infinite set of finite graphs must contain an infinite subset that |𝑉 (𝐺)| is sparse or dense. 2 This assumption can be replaced throughout the paper with the assumption that there are a bounded number of edges between any two vertices. In particular, all our results about bounded degree graphs do not really require the assumption of simplicity. 223 Macromolecular graphs Let 𝐺 = (𝑉, 𝐸) be a finite transitive graph. We say that a pair ( 𝐴, 𝐵) is a macromolecular decomposition for 𝐺 if there exists a non-trivial equivalence relation3 on 𝑉 that is invariant under the action of Aut 𝐺 such that 𝐴 is the subgraph of 𝐺 induced by the equivalence class containing 𝑜, and 𝐵 = 𝐺/∼ is the multigraph obtained from 𝐺 by contracting each equivalence class to a (distinct) vertex. Given 𝜀 > 0, an 𝜀-macromolecular decomposition for |𝐸 (𝐵)| 1 𝐺 is a macromolecular decomposition ( 𝐴, 𝐵) such that 𝐴 is 𝜀-dense and |𝑉 (𝐺)| ≤ 𝜀 . Say that 𝐺 is 𝜀-macromolecular if 𝐺 admits an 𝜀-macromolecular decomposition. We say that an infinite set H ⊆ F is macromolecular if lim𝐺∈H deg 𝐺 = ∞ and there exists 𝜀 > 0 such that all but finitely many graphs in 𝐻 are 𝜀-macromolecular. The next lemma says that after passing to a subsequence, there is an essentially unique best way to choose the macromolecular decompositions for these graphs. Note that an infinite set of finite transitive graphs is molecular (defined in the introduction) if and only if it is both dense and macromolecular. Given 𝜀 > 0, say that 𝐺 is 𝜀-molecular if 𝐺 is 𝜀-macromolecular and 𝜀-dense, so that an infinite set H of finite transitive graphs is molecular if and only if there exists 𝜀 > 0 such that all but finitely many graphs 𝐺 ∈ H are 𝜀-molecular. Subgraphs Let 𝐺 = (𝑉, 𝐸) be a graph. Given 𝑢, 𝑣 ∈ 𝑉, we write 𝑢𝑣 for the unordered pair {𝑢, 𝑣}, write 𝑢 ∼ 𝑣 to mean that 𝑢𝑣 ∈ 𝐸, and define neigh(𝑢) := {𝑣 ∈ 𝑉 : 𝑢 ∼ 𝑣}. Given a set of vertices 𝑋 and a single vertex 𝑢, we write 𝑢𝑋 := {𝑢𝑣 ∈ 𝐸 : 𝑣 ∈ 𝑋 }. Given 𝑋 that is a subgraph of 𝐺 or a subset of 𝑉, we write 𝜕𝐻 or 𝜕𝐸 𝐻 for the edge boundary 𝜕 𝑋 := {𝑥𝑦 ∈ 𝐸 : 𝑥 ∈ 𝑋 and 𝑦 ∉ 𝑋 }. Given a set of vertex 𝑊 ⊆ 𝑉, we write 𝐺 [𝑊] for the subgraph of 𝐺 induced by 𝑊 and 𝐸 [𝑊] for the edge set 𝐸 (𝐺 [𝑊]). Given two finite subsets 𝑊1 and 𝑊2 of 𝑉 or a finite subsets of 𝑉, we define the density4 of 𝑊1 in 𝑊2 to be |𝑊1 ∩ 𝑊2 | ∥𝑊1 ∥ 𝑊2 := . |𝑊2 | When 𝑊2 = 𝑉, then we simply write ∥𝑊1 ∥ to mean ∥𝑊1 ∥ 𝐺 . We will also apply the same notation to subgraphs of 𝐺, in which case the density of one subgraph in another is defined to be the density of the vertex set of one subgraph in the vertex set of the other. Configurations and clusters Let 𝐺 be a graph and let 𝜔 : 𝐸 → {0, 1} be a configuration. We say that 𝑒 is open or closed according to whether 𝜔(𝑒) = 1 or 𝜔(𝑒) = 0. We think of 𝜔 as encoding the spanning subgraph of 𝐺 with edge set 𝜔−1 (1). So 𝜔 ∪ 𝑋 and 𝜔 ∩ 𝑋 are defined by the above 𝜔 conventions for 𝑋 ⊆ 𝐺, 𝑋 ⊆ 𝑉, or 𝑋 ⊆ 𝐸. Given 𝑢, 𝑣 ∈ 𝑉, we write 𝑢 ↔ 𝑣 or 𝑢 ← → 𝑣 to mean that 3 ‘Non-trivial’ means that there is more than one equivalence class. 4 This definition is unrelated to the notion of dense graphs from earlier. 224 𝑢 and 𝑣 are 𝜔-connected, i.e. there is a path in the graph encoded by 𝜔 from 𝑢 to 𝑣. We also write 𝑣 ↔ ∞ to mean that there are infinitely many vertices 𝑢 satisfying 𝑢 ↔ 𝑣. Given 𝑣 ∈ 𝑉, we define 𝐾𝑣 to be the subgraph of 𝜔 spanned by {𝑢 ∈ 𝑉 : 𝑢 ↔ 𝑣} and call this the cluster at 𝑥. We enumerate the clusters of 𝜔 by 𝐾1 , 𝐾2 , . . . in such a way that |𝐾1 | ≥ |𝐾2 | ≥ . . . . This enumeration is not well-defined when there are multiple clusters of the same volume. In this case, we are happy to break ties arbitrarily. So to avoid this (unimportant) technicality, let us now once and for all fix such a choice of enumeration 𝐾1 , 𝐾2 , . . . with |𝐾1 | ≥ |𝐾2 | ≥ of the clusters in every (countable, locally finite, simple) graph. Now suppose that 𝐺 is finite. In our setting, 𝐾1 will typically be the unique macroscopic cluster. So we will sometimes use the more suggestive notation giant := 𝐾1 . Since 𝜔 ↦→ giant(𝜔) is not an increasing map (with respect to inclusion), for technical reasons we will typically work with the following proxies: for each 𝜀 > 0, define giant𝜀 to be the subgraph of 𝜔 induced by the set of vertices 𝑣 satisfying ∥𝐾𝑣 ∥ ≥ 𝜀. Note that the equation ‘giant = giant𝜀 ’ is a convenient way to say that there exists a unique cluster with density at least 𝜀. Percolation P 𝑝 with varying 𝑝 Let 𝐺 be a graph and let 𝑝 ∈ [0, 1]. We write P 𝑝 = P𝐺𝑝 for the law of a random configuration 𝜔 : 𝐸 → {0, 1} where every 𝜔(𝑒) is iid Bernoulli(𝑝). We will write P = P𝐺 for the law of the canonical montone coupling (𝜔 𝑞 ) 𝑞∈[0,1] of (P𝑞 ) 𝑞∈[0,1] . Let E be an event and suppose that 𝑝 ∈ (0, 1). Then we define P′𝑝 (E) to be the derivative (if it exists) of the map 𝑞 ↦→ P𝑞 (E) evaluated at 𝑞 = 𝑝. Densities and supercriticality Let 𝐺 be a transitive graph and let 𝑝 ∈ [0, 1]. If 𝐺 is infinite, then we set 𝜃 ( 𝑝) = 𝜃 ( 𝑝, 𝐺) := P 𝑝 (𝑜 ↔ ∞). Now suppose that 𝐺 is finite. We define 𝜃 ( 𝑝) = 𝜃 ( 𝑝, 𝐺) := P 𝑝 (𝑜 ∈ giant). Moreover, for each 𝜀 > 0, we define 𝜃 𝜀 ( 𝑝) = 𝜃 𝜀 ( 𝑝, 𝐺) := P 𝑝 (𝑜 ∈ giant𝜀 ), and we say that 𝑝 is 𝜀-supercritical if 𝜃 𝜀 ((1 − 𝜀) 𝑝) ≥ 𝜀. So for an infinite set H of finite transitive graphs and an assignment of parameters 𝑝 : H → [0, 1], the assignment 𝑝 is supercritical (as defined in the introduction) if and only if there exists 𝜀 > 0 such that for all but finitely many graphs 𝐺 ∈ H , the parameter 𝑝(𝐺) is 𝜀-supercritical for 𝐺 or 𝑝 = 1. In addition to Theorem 6.1.1, we use the following technical result from our previous work [EH21a, Proposition 4.1 and Remark 4.2] characterising the so-called sharp density property, which we now recall. For all 𝐺 ∈ F and 𝛽, 𝜀 ∈ (0, 1], we define 𝑝 𝐺 𝑐 (𝛽, 𝜀) to be the unique parameter satisfying P 𝑝 𝐺𝑐 (𝛽,𝜀) (∥𝐾1 ∥ ≥ 𝛽) = 𝜀, which is well-defined by continuity and strict monotonicity of percolation 225 on 𝐺. Now we say that an infinite set H ⊆ F has the sharp-density property if lim sup 𝐺∈H 𝛽∈[𝛼,1] 𝑝𝐺 𝑐 (𝛽, 1 − 𝛿) 𝑝𝐺 𝑐 (𝛽, 𝛿) =1 for all 𝛼 ∈ (0, 1) and 𝛿 ∈ (0, 1/2]. This property in particular implies that the event {∥𝐾1 ∥ ≥ 𝛽} undergoes a sharp-threshold for every fixed 𝛽 > 0 when considering percolation on graphs in H . Theorem 6.2.1. Let H be an infinite set of finite transitive graphs. Then H has the sharp-density property if and only if H does not contain an infinite molecular subset. Percolation thresholds We will use the notation 𝑝 𝑐 both for the usual infinite-cluster percolation threshold 𝑝 𝑐 (𝐺) = inf{𝑝 : 𝜃 ( 𝑝) > 0} when 𝐺 is an infinite transitive graph and for a percolation threshold 𝑝 𝑐 : H → [0, 1] when H is an infinite set of finite transitive graphs (when H admits such a threshold), as defined in [Eas22]. It is well-known that 𝑝 𝑐 (𝐺) ≥ deg 1𝐺−1 for every infinite transitive graph 𝐺. Let us record here the analogous result for finite transitive graphs, which can be proven by the same argument as [EH21a, Lemma 2.8] or [Eas22, Proposition 5]. Lemma 6.2.2. Let H be an infinite set of finite transitive graphs. If 𝑝 : H → [0, 1] is a supercritical sequence of parameters, then lim inf 𝐺∈H (deg 𝐺 − 1) 𝑝(𝐺) ≥ 1. In particular, if 𝑝 𝑐 : H → [0, 1] is a percolation threshold, then lim inf 𝐺∈H (deg 𝐺 − 1) 𝑝 𝑐 (𝐺) ≥ 1. It is also well-known that every infinite transitive graph 𝐺 satisfies the mean-field lower bound 𝜀 for all 𝜀 > 0. This was first proven by Menshikov [Men86] and Aizenman 𝜃 ((1 + 𝜀) 𝑝 𝑐 (𝐺)) ≥ 1+𝜀 and Barsky [AB87b]; various alternative simplified proofs are now available [DT16b; Hut20c; DRT19; Van22a]. For future use, we state a version of this bound applying also to sets of finite graphs without infinite molecular subsets. This was proven in [Eas22, Lemma 10] using an adaptation of the methods of [Van22a] together with the uniqueness theorem of the first paper in this series [EH21a]. Theorem 6.2.3. Let H be an infinite set of finite transitive graphs that does not contain an infinite subset that is molecular, and let 𝑝 𝑐 : H → [0, 1] be a percolation threshold function (which exists 𝜀 by [Eas22]). For all 𝜀 ∈ (0, ∞) and all 𝛿 ∈ (0, 1+𝜀 ), lim P (1+𝜀) 𝑝 𝑐 (∥𝐾1 ∥ ≥ 𝛿) = 1. 𝐺∈H 226 6.3 Applications of Hatami’s theorem to percolation Suppose for each 𝑛 ≥ 1 that 𝐴𝑛 is an increasing event depending on some finite number 𝑁𝑛 of independent random bits, each of which is 1 with probability 𝑝 and 0 with probability 1 − 𝑝. Recall that the sequence of events ( 𝐴𝑛 )𝑛≥1 is said to have a sharp threshold if there exists a sequence 𝑝 1 , 𝑝 2 , . . . of numbers in (0, 1) such that lim P (1+𝜀) 𝑝 𝑛 ( 𝐴𝑛 ) = 1 𝑛 and lim P (1−𝜀) 𝑝 𝑛 ( 𝐴𝑛 ) = 0 𝑛 for every 𝜀 > 0 (in which case this holds with 𝑝 𝑛 = min{𝑝 ∈ [0, 1] : P 𝑝 ( 𝐴𝑛 ) = 1/2}). In this section, we outline a new application of sharp threshold theory to percolation using theorems due to Friedgut [Fri98], Bourgain [Fri99b], and Hatami [Hat12b], which have previously been overlooked by the percolation community. Let us first give some further background and definitions. Russo’s formula [Rus82] states that if 𝐴 is an increasing event depending on finitely many bits then ∑︁ 𝑑 P 𝑝 ( 𝐴) = P 𝑝 (𝑒 is pivotal for 𝐴), 𝑑𝑝 𝑒∈𝐸 where an edge 𝑒 is said to be pivotal for the (increasing) event 𝐴 if 𝜔 \ {𝑒} ∉ 𝐴 and 𝜔 ∪ {𝑒} ∈ 𝐴. The probability that 𝑒 is pivotal for 𝐴 is known as the influence of 𝑒 and is denoted by 𝐼 𝑝 ( 𝐴, 𝑒). Talagrand’s inequality [Tal94] states that   −1 2 1 𝑑 P 𝑝 ( 𝐴) ≥ 𝑐 𝑝(1 − 𝑝) log P 𝑝 ( 𝐴)(1 − P 𝑝 ( 𝐴)) log , 𝑑𝑝 𝑝(1 − 𝑝) 𝑝(1 − 𝑝) max𝑒 𝐼 𝑝 ( 𝐴, 𝑒) so that events with small maximal influence must have large total influence. As explained in [EH21a, Section 3.4], Talagrand’s inequality implies that increasing events that depend on 𝑛 bits in a sufficiently symmetric way automatically have sharp thresholds provided that the threshold occurs at a value of 𝑝 that is subalgebraically small in 𝑛. This condition cannot be removed in general, since the existence of a triangle in the Erdős-Rényi random graph 𝐺 (𝑛, 𝑝) has a coarse threshold at 𝑝 of order 𝜆/𝑛. (Other powers of 𝑛 can be obtained from the event that the Erdős-Rényi graph contains a complete subgraph on 𝑘 vertices.) While this application of Talagrand’s inequality is a standard part of the theory of percolation on bounded-degree and infinite graphs, we would like to bring attention to a related but less well-known result due to Friedgut [Fri98]. Friedgut 𝑑 showed that if 𝑝 is bounded away from 0 and 1, and 𝑑𝑝 P 𝑝 ( 𝐴) is bounded above, then in fact 𝐴 can be approximated arbitrarily well by events determined by a bounded number of edges (where the bound on the number of edges depends on the accuracy of the approximation). Friedgut’s theorem 227 is the key to proving our results about bounded-degree graphs, where parameters of interest are always bounded away from 0 and 1. When 𝑝 is algebraically small in 𝑛, the absence of edges of large influence is no longer sufficient to ensure a sharp threshold, and Friedgut’s theorem does not apply, but one can instead look to Bourgain [Fri99b] for a result of a similar flavour that is applicable. Bourgain showed that the absence of a sharp threshold must imply the existence of sets of edges of bounded size (𝐹𝑛 )𝑛≥1 for which 𝜔 ∪ 𝐹𝑛 is more likely to belong to 𝐴𝑛 than 𝜔 by at least a constant (for infinitely many 𝑛 and appropriate choices of the parameter 𝑝). When the threshold occurs at a subalgebraically small value of 𝑝, the conclusions of Bourgain’s theorem are strictly weaker than those of Talagrand, so that Bourgain’s theorem is not relevant to percolation on bounded degree graphs. In this paper, we will make use not of Bourgain’s theorem but rather a powerful strengthening of this theorem due to Hatami [Hat12b]. Roughly speaking, this theorem allows us to replace the conclusion that 𝜔 ∪ 𝐹𝑛 is more likely to belong to 𝐴𝑛 than 𝜔 by a constant with the conclusion that 𝜔 ∪ 𝐹𝑛 ∈ 𝐴𝑛 with arbitrarily high probability. While this distinction may seem minor at first glance, it is in fact very significant. When 𝑝 is bounded away from 0 and 1, Hatami’s theorem’s is equivalent to Friedgut’s theorem from the point of view of our applications. However, to allow for smaller values of 𝑝, which will be necessary in later sections, we will always refer to Hatami’s theorem. We now state Hatami’s theorem. For each positive integer 𝑛, we write [𝑛] for the set {1, . . . , 𝑛}, and for each parameter 𝑝 ∈ (0, 1) write 𝜇 𝑝 for the law of a random variable 𝑥 = (𝑥𝑖 )𝑖∈[𝑛] where the 𝑥𝑖 ’s are i.i.d. Bernoulli(𝑝). Theorem 6.3.1 (Hatami 2012). Let 𝑛 ∈ N, let 𝑓 : {0, 1} [𝑛] → {0, 1} be non-constant and increasing, and let 𝑝 ∈ (0, 1/2]. For every 𝜀 > 0, there exists a set 𝑆 ⊆ [𝑛] such that ! 2 ⌈𝐽 ( 𝑝)⌉ 𝑓 , 𝜇 𝑝 ( 𝑓 (𝑥) | 𝑥𝑖 = 1 ∀𝑖 ∈ 𝑆) ≥ 1 − 𝜀 and |𝑆| ≤ exp 1012 2 𝜀 𝜇 𝑝 ( 𝑓 (𝑥)) 2 where 𝐽 𝑓 ( 𝑝) := 2𝑝(1 − 𝑝) 𝑑𝜇 𝑝 ( 𝑓 ) 𝑑𝑝 . We will apply this theorem via the following corollary. We write P′𝑝 (E) to denote 𝑑P 𝑝 (E) 𝑑𝑝 . Corollary 6.3.2. Let 𝐺 be a finite graph, let 𝑝 ∈ (0, 1), and let 𝜀1 , 𝜀2 , 𝜀3 ∈ (0, 1). If E ⊆ {0, 1} 𝐸 is a non-trivial increasing event such that P 𝑝 (E) ≥ 𝜀1 and 228 𝑝P′𝑝 (E) ≤ 1 , 𝜀2 then there is a set of edges 𝐹 ⊆ 𝐸 such that  P 𝑝 (𝜔 ∪ 𝐹 ∈ E) ≥ 1 − 𝜀3 and  250 |𝐹 | ≤ exp . (𝜀1 𝜀2 𝜀3 ) 2 Proof of Corollary 6.3.2. The corollary is essentially immediate when 𝑝 ≤ 1/2. To allow for the case that 𝑝 > 1/2, we simulate percolation of parameter 𝑝 on 𝐺 by percolation of a parameter 𝜙( 𝑝) ≤ 1/2 on a multigraph with more edges. This argument does not contain any surprises and can safely be skipped on a first reading. Fix 𝑝 ∈ (1/2, 1) and let 𝐻 = (𝑉, 𝐹) be the multigraph formed from 𝐺 by replacing each edge 𝑒 by 1 ⌉ parallel edges 𝑒 1 , . . . , 𝑒 ℓ with the same endpoints as 𝑒. For each configuration 𝜔 ℓ := ⌈log2 1−𝑝 on 𝐻, let Φ(𝜔) be the configuration on 𝐺 in which an edge 𝑒 is open if and only if there is some 1 ≤ 𝑖 ≤ ℓ for which 𝑒𝑖 is open in 𝜔. For each 𝑞 ∈ (0, 1), define 𝑔(𝑞) := P𝐺 𝑞 (A), ℎ(𝑞) := P𝑞𝐻 (Φ−1 (A)), and 𝜙(𝑞) := 1 − (1 − 𝑞) 1/ℓ , 𝐻 𝐻 so that 𝑔 = ℎ ◦ 𝜙 and the pushforward measure Φ∗ P𝜙(𝑞) satisfies Φ∗ P𝜙(𝑞) = P𝐺 𝑞 . By our choice of ℓ, we know that 𝜙( 𝑝) ∈ (0, 1/2], allowing us to apply Theorem 6.3.1. This guarantees that there is a set of edges 𝑇 ⊆ 𝐹 such that !   2 ⌈𝐼⌉ −1 𝐻 |𝑇 | ≤ exp 1012 2 , (6.3.1) P𝜙( 𝑝) 𝜔 ∪ 𝑇 ∈ Φ (A) ≥ 1 − 𝜀 3 and 𝜀3 (ℎ ◦ 𝜙( 𝑝)) 2 where 𝐼 = 2𝜙( 𝑝)(1 − 𝜙( 𝑝))(ℎ′ ◦ 𝜙( 𝑝)). 𝐻 Since Φ∗ P𝜙( = P𝐺𝑝 , we know that P𝐺𝑝 (𝜔 ∪ Φ(𝑇) ∈ A) ≥ 1 − 𝜀3 . Since 𝑔 = ℎ ◦ 𝜙, we have by the 𝑝) chain rule that ℎ′ ◦ 𝜙 = 𝑔′/𝜙′. We also have by direct calculation that (𝜙′ ( 𝑝)) −1 = ℓ(1 − 𝑝) (ℓ−1)/ℓ , 1 1 ≤ ℓ ≤ 2 log2 1−𝑝 we deduce that and since log2 1−𝑝 1 𝜙′ ( 𝑝) ≤ 4(1 − 𝑝) log2 1 ≤ 4. 1− 𝑝 Together with the simple observations that 𝜙( 𝑝) ≤ 𝑝 and 1 − 𝜙( 𝑝) ≤ 1, this lets us simplify our bound on |𝑇 | from eq. (6.3.1) to ! ′ ( 𝑝)⌉ 2 ⌈4 · 2 · 𝑝 · 𝑔 |𝑇 | ≤ exp 1012 . 𝜀32 · 𝑔( 𝑝) 2 The result now follows since ⌈𝑥⌉ ≤ 2𝑥 for all 𝑥 ≥ 1, |Φ(𝑇)| ≤ |𝑇 | and, by hypothesis, 𝑔( 𝑝) ≥ 𝜀1 and 𝑝 · 𝑔′ ( 𝑝) ≤ 𝜀12 . □ 229 The key advantage of this result over the usual sharp-threshold results used in percolation theory is that we can guarantee that 𝜔 ∪ 𝐹 ∈ E occurs with with arbitrarily high probability. In particular, we can force this event to have a good-probability intersection with any other event B such that B \ E has good probability; usually we will apply this in the case B ⊆ E 𝑐 . (On the other hand, this method will tend to give estimates that are very poor quantitatively.) Given an increasing event E, a set of edges 𝐹, and a configuration 𝜔 ∈ {0, 1} 𝐸 , we say that 𝐹 is an activator for 𝐸 if 𝜔 ∉ E but 𝜔 ∪ 𝐹 ∈ E, and define ActE [𝐹] := {𝜔 ∪ 𝐹 ∈ E} ∩ {𝜔 ∉ E} to be the event that 𝐹 is an activator for E. We also write ActE [𝑒] := ActE [{𝑒}] when 𝑒 ∈ 𝐸 is a single edge, so that ActE [𝑒] is the event that 𝑒 is closed and pivotal for E. Corollary 6.3.3. For all 𝜀 > 0, there exists 𝑁 < ∞ such that the following holds. Let 𝐺 be a finite graph. Let E be an increasing event. Let 𝑝 ∈ (0, 1). Suppose that 𝜀 ≤ P 𝑝 (E) ≤ 1 − 𝜀 and 𝑝P′𝑝 (E) ≤ 1 . 𝜀 Let B be any event with P 𝑝 (B \ E) ≥ 𝜀. Then there is a set of edges 𝐹 ⊆ 𝐸 with |𝐹 | ≤ 𝑁 such that P 𝑝 (B ∩ ActE [𝐹]) ≥ 𝜀 . 2 Proof. This follows from Corollary 6.3.2 applied with 𝜀1 = 𝜀2 = 𝜀 and 𝜀3 = 𝜀/2. Indeed, letting 𝐹 be as in the statement of that corollary, we have that P 𝑝 (B ∩ ActE [𝐹]) ≥ P 𝑝 ((B \ E) ∩ (𝜔 ∪ 𝐹 ∈ E)) ≥ 𝜀 − 𝜀 𝜀 ≥ . 2 2 □ A strategy to show that events have low probability. This corollary leads to the following general strategy for showing that an event B is unlikely under P 𝑝 : 1. Find a suitable increasing event E. 2. Justify that for one (or many) parameters 𝑞 ≤ 𝑝, the expression 𝑞P′𝑞 (E) is bounded above. 3. Show that at such 𝑞, it is unlikely that both ActE [𝐹] and B hold simultaneously, where 𝐹 is a deterministic bounded-size set of edges. 4. Use Corollary 6.3.3 to deduce that at such 𝑞, the event B is unlikely. 230 5. Finally, show that because B is unlikely at 𝑞, it is therefore also unlikely at 𝑝. This is to be interpreted as a loose strategy rather than a detailed recipe: each step will involve arguments that are specific to the event at hand. We implement a strategy in the same spirit in Sections 6.4 and 6.7. We conclude this section by making note of the following elementary lemma, which states that we always have many good parameters 𝑞 where 𝑞P′𝑞 is bounded above. Here L denotes the Lebesgue measure. Lemma 6.3.4. Let 𝐺 = (𝑉, 𝐸) be a graph. Fix 𝑝, 𝛿 ∈ (0, 1) such that the interval 𝐼 := [ 𝑝, (1 + 𝛿) 𝑝] is contained in (0, 1]. If E ⊆ {0, 1} 𝐸 is an increasing event that is determined by the state of finitely many edges then     1 2𝜀 ′ L 𝑝 ∈ 𝐼 : 𝑝P 𝑝 (E) ≤ ≥ 1− L (𝐼) 𝜀 𝛿 for every 𝜀 > 0. Proof. Let 𝐽 = {𝑝 ∈ 𝐼 : 𝑝P′𝑝 (E) ≤ 𝜀 −1 }. Since E is increasing and determined by the state of finitely many edges, the map 𝑞 ↦→ P𝑞 (E) is a polynomial that defines an increasing function [0, 1] → [0, 1]. It follows that ∫ ∫ 1 ′ L (𝐼 \ 𝐽) ≤ P𝑞 (E) 𝑑𝑞 ≤ P′𝑞 (E) 𝑑𝑞 = P (1+𝛿) 𝑝 (E) − P 𝑝 (E) ≤ 1. 𝜀(1 + 𝛿) 𝑝 𝐼 𝐽 We deduce the claim by rearranging, using that L (𝐼) = 𝛿 𝑝 and that 𝛿−1 + 1 ≤ 2𝛿−1 . 6.4 □ The unimodular bounded-degree case In this section we prove our main theorems in the case that all graphs in our family are unimodular and have uniformly bounded degrees. (Finite transitive graphs are always unimodular, so this restriction is only relevant for infinite graphs.) All the theorems in this section are special cases of those established in later sections; we present this case separately since it is much less technical and will be the primary case of interest to many readers. It will also serve as a warm-up to the general case, with several of the ideas used in this section appearing again in a more technical context later in the paper. This section also contains all our results about infinite graphs. For every 𝑑 ≥ 2, let us fix a consistent choice of percolation threshold function 𝑝 𝑐 : F𝑑 → [0, 1], which exists by Theorem 6.1.2. In the following proposition, we write 𝑝 𝑐 (𝐺) to denote the value 231 of this percolation threshold function at 𝐺 when 𝐺 is finite and to denote the usual infinite cluster percolation threshold for 𝐺 when 𝐺 is infinite. Moreover,    E 𝑝 ∥giant∥  𝜃 ( 𝑝, 𝐺) :=   P 𝑝 (𝑜 ↔ ∞)  if 𝐺 is finite (6.4.1) if 𝐺 is infinite. Proposition 6.4.1. Let H be an infinite set of (finite or infinite) transitive graphs converging locally to some infinite transitive graph 𝐺 ∞ . For every constant 𝑝 > lim supH 𝑝 𝑐 (𝐺), lim 𝜃 ( 𝑝, 𝐺) = 𝜃 ( 𝑝, 𝐺 ∞ ). H We now give a brief overview of (non)unimodularity, the mass-transport principle and the modular function, referring the reader to [LP16b; Hut20g] for further background. Let 𝐺 = (𝑉, 𝐸) be a connected, locally finite, transitive graph, and let Aut(𝐺) be the automorphism group of 𝐺. The modular function Δ = Δ𝐺 : 𝑉 2 → (0, ∞) is defined by Δ(𝑢, 𝑣) = | Stab𝑣 𝑢| , | Stab𝑢 𝑣| where Stab𝑣 = {𝛾 ∈ Aut(𝐺) : 𝛾𝑣 = 𝑣} is the stabilizer of 𝑣 and Stab𝑣 𝑢 = {𝛾𝑢 : 𝛾 ∈ Stab𝑣 } is the orbit of 𝑢 under Stab𝑣 . We say that 𝐺 is unimodular if Δ ≡ 1 and that 𝐺 is nonunimodular otherwise. Every finite transitive graph is unimodular, as is every Cayley graph and every amenable transitive graph [SW90]. The mass-transport principle. Let 𝐺 be a connected, locally finite, unimodular transitive graph. The mass-transport principle states that ∑︁ ∑︁ 𝐹 (𝑜, 𝑥) = 𝐹 (𝑥, 𝑜) (6.4.2) 𝑥∈𝑉 𝑥∈𝑉 for every 𝐹 : 𝑉 2 → [0, ∞] that is invariant under the diagonal action of Aut(𝐺) on 𝑉 2 , meaning that 𝐹 (𝛾𝑥, 𝛾𝑦) = 𝐹 (𝑥, 𝑦) for every 𝑥, 𝑦 ∈ 𝑉 and 𝛾 ∈ Aut(𝐺). Note that this identity holds trivially when 𝐺 is finite. Proof of Proposition 6.4.1 In this subsection we use Hatami’s theorem to prove Proposition 6.4.1. We will deduce this proposition from a chain of simple lemmas, several of which do not require the bounded degree assumption. We begin with the following immediate consequence of Corollary 6.3.3. 232 Lemma 6.4.2. For every 𝜀 > 0 there exists 𝑁 < ∞ such that the following holds. Let 𝐺 be a graph, let 𝑜 be a vertex of 𝐺, let 𝑝 ∈ (0, 1), and let 𝑚, 𝑛 be integers with 𝑚 < 𝑛. If P 𝑝 (𝑚 ≤ |𝐾𝑜 | < 𝑛) ≥ 𝜀, P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝜀, and 𝑝P′𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 1 𝜀 then there exists a set of edges 𝐹 ⊆ 𝐸 with |𝐹 | ≤ 𝑁 such that
𝜀 P 𝑝 {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] ≥ . 2 Proof. This follows from Corollary 6.3.3 with E := {|𝐾𝑜 | ≥ 𝑛} and B := {𝑚 ≤ |𝐾𝑜 | < 𝑛} ⊆ E𝑐. □ We next turn this estimate concerning activators into one concerning pivotals at a cost of 𝑝 |𝐹 | /2|𝐹 |. This estimate is much less wasteful in the bounded degree case, where the relevant values of 𝑝 will not be small, than it is in the high degree case. Lemma 6.4.3. Let 𝐺 be a graph, let 𝑜 be a vertex of 𝐺, let 𝑝 ∈ (0, 1), and let 𝑚, 𝑛 be integers with 𝑚 < 𝑛. For each non-empty finite set of edges 𝐹 ⊆ 𝐸 there exists an edge 𝑢𝑣 ∈ 𝐹 such that

𝑝 |𝐹 | P 𝑝 {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣] ≥ P 𝑝 {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] . 2 |𝐹 | Note that the left hand side of the inequality concerns the cluster of 𝑢 while the right hand side concerns the cluster of 𝑜. Proof. Let 𝑒 1 , . . . , 𝑒 |𝐹 | be an enumeration of 𝐹. Consider a configuration 𝜔 ∈ {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] and let 𝑖 < |𝐹 | be the maximum index such that |𝐾𝑜 (𝜔 ∪ {𝑒 1 , . . . , 𝑒𝑖 })| < 𝑛. We can write 𝑒𝑖+1 = 𝑢𝑣 in such a way that 𝜔 ∪ {𝑒 1 , . . . , 𝑒𝑖 } ∈ {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣]. Thus, by the pigeonhole principle, there exists a non-random index 𝑖 with an endpoint labelling 𝑒𝑖+1 = 𝑢𝑣 such that
P 𝑝 𝜔 ∪ {𝑒 1 , . . . , 𝑒𝑖 } ∈ {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣] ≥ 233
1 P 𝑝 {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] . 2 |𝐹 | Since the event {𝜔 ∪ {𝑒 1 , . . . , 𝑒𝑖 } ∈ {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣]} is independent of the restriction of 𝜔 to {𝑒 1 , . . . , 𝑒𝑖 } it follows that P 𝑝 {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣]

≥ P 𝑝 𝜔 ∪ {𝑒 1 , . . . , 𝑒𝑖 } ∈ {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣] P 𝑝 ({𝑒 1 , . . . , 𝑒𝑖 } ∈ 𝜔) ≥
𝑝 |𝐹 | P 𝑝 {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] 2 |𝐹 | as required. □ Lemma 6.4.4. Let 𝐺 be a unimodular transitive graph, and let 𝑚 < 𝑛 be natural numbers. Then P′𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝑚 · P 𝑝 {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣]
for every 𝑝 ∈ (0, 1) and every edge 𝑢𝑣 ∈ 𝐸. Proof. The event that |𝐾𝑜 | ≥ 𝑛 is fully determined by the state of finitely many edges. As such, the map 𝑞 ↦→ P𝑞 (|𝐾𝑜 | ≥ 𝑛) is differentiable at 𝑝, and, by Russo’s formula, has derivative given by ∑︁ P 𝑝 ({|𝐾𝑜 (𝜔\𝑒)| < 𝑛} ∩ {|𝐾𝑜 (𝜔 ∪ 𝑒)| ≥ 𝑛}) P′𝑝 (|𝐾𝑜 | ≥ 𝑛) = 𝑒∈𝐸 1 ∑︁ P 𝑝 ({𝜔(𝑒) = 0} ∩ {|𝐾𝑜 (𝜔)| < 𝑛} ∩ {|𝐾𝑜 (𝜔 ∪ 𝑒)| ≥ 𝑛}) 1 − 𝑝 𝑒∈𝐸
1 ∑︁ ∑︁ = P 𝑝 {𝑜 ↔ 𝑎} ∩ act |𝐾 𝑎 |≥𝑛 [𝑎𝑏] , 1 − 𝑝 𝑎∈𝑉 = 𝑏∈neigh(𝑎) where neigh(𝑎) denotes the set of neighbours of 𝑎. Applying the mass-transport principle to exchange the roles of 𝑜 and 𝑎 in the last line yields that P′𝑝 (|𝐾𝑜 | ≥ 𝑛) = ≥ 1 1− 𝑝 max ∑︁   E 𝑝 |𝐾𝑜 | 1(act |𝐾𝑜 |≥𝑛 [𝑜𝑏]) 𝑏∈neigh(𝑜)   E 𝑝 |𝐾𝑜 | 1(act |𝐾𝑜 |≥𝑛 [𝑜𝑏]) ≥ 𝑏∈neigh(𝑜) max 𝑚 · P 𝑝 (|𝐾𝑜 | ≥ 𝑚, act |𝐾𝑜 |≥𝑛 [𝑜𝑏]), 𝑏∈neigh(𝑜) and the claim follows by transitivity. □ Lemma 6.4.5. For all 𝑑 ∈ N and 𝜀 ∈ (0, 1), there exists 𝑚 = 𝑚(𝑑, 𝜀) < ∞ such that if 𝐺 ∈ U𝑑 , 𝑝 ∈ (0, 1) and 𝑛 ≥ 𝑚 are such that P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝜀 then n o L 𝑞 ∈ [ 𝑝, 1] : P𝑞 (𝑚 ≤ |𝐾𝑜 | < 𝑛) > 𝜀 ≤ 𝜀. 234 Recall that U𝑑 denotes the set of all (finite or infinite) unimodular transitive graphs with vertex degree at most 𝑑. Proof. It suffices to consider the case 𝑛 ≥ 2, the case 𝑛 = 1 being vacuous. Fix 𝐺 ∈ U𝑑 , 𝑝 ∈ (0, 1), and integers 𝑛 ≥ 𝑚 with 𝑛 ≥ 2 and suppose that P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝜀. We will prove the contrapositive of the claim: There exists a constant 𝑀 = 𝑀 (𝑑, 𝜀) ≥ 2 such that 
L 𝑞 ∈ [ 𝑝, 1] : P𝑞 (𝑚 ≤ |𝐾𝑜 | < 𝑛) > 𝜀 > 𝜀 ⇒ (𝑚 ≤ 𝑀) . Since 𝑛 ≥ 2, we know by a union bound that 𝜀 ≤ P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ ∑︁ P 𝑝 (𝑜𝑢 is open) ≤ 𝑝𝑑, 𝑢∼𝑜 and hence 𝑝 ≥ 𝑑𝜀 . We may also assume that 𝑝 ≤ 1 − 𝜀, the claim being trivial otherwise. Take 𝐽 := {𝑞 ∈ [ 𝑝, 1] : 𝑞P′𝑞 (|𝐾𝑜 | ≥ 𝑛) ≤ 2𝜀 −2 }. Since L ([ 𝑝, 1]\𝐽) ≤ 𝜀 by Lemma 6.3.4, it suffices to prove that there exists 𝑀 = 𝑀 (𝑑, 𝜀) < ∞ such that
∃𝑞 ∈ 𝐽 such that P𝑞 (𝑚 ≤ |𝐾𝑜 | < 𝑛) ≥ 𝜀 ⇒ (𝑚 ≤ 𝑀) . (6.4.3) To this end, suppose that 𝑞 ∈ 𝐽 is such that P𝑞 (𝑚 ≤ |𝐾𝑜 | < 𝑛) ≥ 𝜀. Applying Lemma 6.4.2 with E = {|𝐾𝑜 | ≥ 𝑛} and B = {𝑚 ≤ |𝐾𝑜 | < 𝑛} ⊆ E 𝑐 , we deduce that there exists 𝑁 = 𝑁 (𝜀) < ∞ and a set of edges 𝐹 ⊆ 𝐸 with |𝐹 | ≤ 𝑁 such that
𝜀 P𝑞 {|𝐾𝑜 | ≥ 𝑚} ∩ act |𝐾𝑜 |≥𝑛 [𝐹] ≥ . 2 Using Lemma 6.4.3, it follows that there exists an edge 𝑢𝑣 ∈ 𝐹 such that
𝑞 |𝐹 | 𝜀 𝜀 1+𝑁 P𝑞 {|𝐾𝑢 | ≥ 𝑚} ∩ act |𝐾𝑢 |≥𝑛 [𝑢𝑣] ≥ =: 𝑐 · ≥ 2 |𝐹 | 2 4𝑁 𝑑 𝑁 where 𝑐 = 𝑐(𝑑, 𝜀) is a positive constant. Since 𝐺 is unimodular, we may apply Lemma 6.4.4 to deduce that P′𝑞 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝑚𝑐. Contrasting this with the hypothesis that 𝑞 ∈ 𝐽 yields that 𝑚≤ 2 2𝑑 ≤ 3. 2 𝑐 𝑝𝜀 𝑐𝜀 We deduce that the implication (6.4.3) holds with 𝑀 = 𝑀 (𝑑, 𝜀) = 2𝑑/(𝑐𝜀 3 ), completing the proof. □ 235 In the next lemma, we write L for the Lebesgue measure. In the context of finite graphs, we used ‘giant’ to denote an arbitrary choice of largest cluster. Let us extend this definition as follows: if there exists an least one infinite cluster, let ‘giant’ be the union of all infinite clusters. The reader may notice that ‘giant’ is undefined when there are arbitarily large finite clusters but no infinite clusters; we will never use the notation ‘giant’ in such a situation. Lemma 6.4.6. Let 𝑑 ≥ 2 and 𝜀 > 0, there exists 𝑚(𝜀, 𝛿) < ∞ such that for all 𝐺 ∈ U𝑑 , n o
𝐺 L 𝑝 ∈ [ 𝑝 𝑐 (𝐺), 1] : P 𝑝 |𝐾𝑜 | ≥ 𝑚 but 𝑜 ∉ giant ≥ 𝜀 ≤ 𝜀. Proof of Lemma 6.4.6. We start by dealing with the finite graphs in U𝑑 . By the definition of a percolation threshold, there exists 𝛿 > 0 such that P 𝑝 𝑐 +𝜀/2 (|𝐾𝑜 | ≥ 𝛿 |𝑉 |) ≥ 𝛿 for all but finitely many of the finite graphs 𝐺 ∈ U𝑑 . Thus, by Lemma 6.4.5, there exists 𝑚 ∈ N such that n 𝜀o 𝜀 ≥ 1 − 𝑝 𝑐 (𝐺) − 𝜀 (6.4.4) L 𝑝 ∈ [ 𝑝 𝑐 (𝐺) + , 1] : P 𝑝 (𝑚 ≤ |𝐾𝑜 | < 𝛿 |𝑉 |) ≤ 2 2 for all but finitely many (isomorphism classes of) finite graphs 𝐺 ∈ U𝑑 . By increasing 𝑚 if necessary, we may take this estimate to hold for every finite graph 𝐺 ∈ U𝑑 . For each finite graph 𝐺 ∈ U𝑑 , let 𝐽 (𝐺) be the set whose Lebesgue measure is bounded in (6.4.4). By Theorems 6.1.1 and 6.2.1, all but finitely many finite graphs 𝐺 ∈ H have the property that P 𝑝 (|𝐾𝑜 | ≥ 𝛿 |𝑉 | but 𝑜 ∉ giant) ≤ 𝜀 2 for every 𝑝 ≥ 𝑝 𝑐 (𝐺) + 𝜀/2, and hence by a union bound that P 𝑝 (|𝐾𝑜 | ≥ 𝑚 but 𝑜 ∉ giant) ≤ 𝜀 𝜀 + =𝜀 2 2 for every 𝑝 ∈ 𝐽 (𝐺). This proves the claim for the finite graphs in U𝑑 since L ([ 𝑝 𝑐 (𝐺), 1]\𝐽 (𝐺)) ≤ 𝜀 for every 𝐺 ∈ U𝑑 . We now turn to the infinite graphs in U𝑑 . By Theorem 6.2.3, there exists 𝛿 > 0 such that P 𝑝 𝑐 + 2𝜀 (|𝐾𝑜 | ≥ 𝑛) ≥ P 𝑝 𝑐 + 2𝜀 (𝑜 ↔ ∞) ≥ 𝛿 for every infinite 𝐺 ∈ H and every 𝑛 ∈ N. So, by Lemma 6.4.5, there exists 𝑚 = 𝑚(𝑑, 𝜀) such that the set of parameters n h o 𝜀 i 𝐽𝑛 = 𝐽𝑛 (𝐺) = 𝑝 ∈ 𝑝 𝑐 + , 1 : P𝑞 (𝑚 ≤ |𝐾𝑜 | < 𝑛) ≤ 𝜀 2 236  
satisfies L 𝑝 𝑐 + 𝜀2 , 1 \𝐽𝑛 ≤ 𝜀2 for every infinite 𝐺 ∈ H and every 𝑛 ∈ N. Noting that 𝐽1 ⊇ 𝐽2 ⊇ Ñ · · · for every infinite 𝐺 ∈ H , we deduce that the intersection 𝐽 = 𝐽 (𝐺) = 𝑛≥1 𝐽𝑛 satisfies h  𝜀 𝜀 𝜀 𝜀 i L ([ 𝑝 𝑐 , 1] \𝐽) = + lim L 𝑝 𝑐 + , 1 \𝐽𝑛 ≤ + = 𝜀 2 𝑛→∞ 2 2 2 for every infinite 𝐺 ∈ U𝑑 . This implies the claim since P 𝑝 (|𝐾𝑜 | ≥ 𝑚 but 𝑜 ↮ ∞) = lim P 𝑝 (𝑚 ≤ |𝐾𝑜 | < 𝑛) ≤ 𝜀 𝑛→∞ for every infinite 𝐺 ∈ U𝑑 and every 𝑝 ∈ 𝐽 (𝐺). □ Lemma 6.4.7. Let H be an infinite set of (infinite or finite) transitive graphs converging locally to some infinite transitive graph 𝐺 ∞ . For all 𝑝 : H → [0, 1] satisfying lim inf H 𝑝/𝑝 𝑐 > 1, if lim lim sup P 𝑝 (|𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant) = 0, 𝑛→∞ (6.4.5) H then lim |𝜃 ( 𝑝, 𝐺) − 𝜃 ( 𝑝, 𝐺 ∞ )| = 0. H Proof. For all 𝐺 ∈ H and 𝑛 ≥ 1, ♥ := |𝜃 ( 𝑝, 𝐺) − 𝜃 ( 𝑝, 𝐺 ∞ )| ≤ 𝜃 ( 𝑝, 𝐺) − P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) + P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) − P𝐺𝑝 ∞ (|𝐾𝑜 | ≥ 𝑛) + P𝐺𝑝 ∞ (|𝐾𝑜 | ≥ 𝑛) − 𝜃 ( 𝑝, 𝐺 ∞ ) . {z } {z } | | {z } | ♥2𝑛 ♥1𝑛 ♥3𝑛 Since lim inf H 𝑝/𝑝 𝑐 > 1, for all 𝑛, lim P 𝑝 (|giant| ≥ 𝑛) = 1, H and hence, by Theorems 6.1.1 and 6.2.1, lim sup ♥1𝑛 = lim sup P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant) , H (6.4.6) H which tends to zero as 𝑛 tends to infinity if eq. (6.4.5) holds. In particular, if eq. (6.4.5) holds then   3 1 2 lim sup ♥ ≤ lim sup inf ♥𝑛 + ♥𝑛 + ♥𝑛 𝑛≥1 H′ H′   ≤ lim sup lim sup ♥1𝑛 + ♥2𝑛 + ♥3𝑛 𝑛→∞ H′ ≤ lim sup lim sup ♥1𝑛 + sup lim sup ♥2𝑛 + lim sup ♥3𝑛 = 0. 𝑛→∞ | H′ {z =0 by eq. (6.4.5) 𝑛≥1 } H′ 𝑛→∞ | {z } | {z } =0 =0 □ 237 We are now ready to complete the proof of Proposition 6.4.1. To make the proof slightly shorter, we will invoke the following well-known result of Schonmann [Sch99]: for every infinite transitive graph 𝐺, the density 𝜃 (·, 𝐺) is continuous on ( 𝑝 𝑐 (𝐺), 1]. This is not strictly necessary: we could bypass this step by using Lemma 6.5.1 (adapting its proof to all unimodular transitive graphs) and Corollary 6.10.2 (implying that that 𝐺 ∞ is unimodular.) Proof of Proposition 6.4.1. Pick 𝜀 > 0 such that 𝑝 > lim supH 𝑝 𝑐 (𝐺) + 𝜀. By Lemma 6.4.6, we can find sequences 𝑝 1 , 𝑝 2 , 𝑝 3 : H → [0, 1] such that 𝑝 1 → 𝑝 − 𝜀, 𝑝 2 ↑ 𝑝, and 𝑝 3 ↓ 𝑝 as 𝐺 → ∞ with 𝐺 ∈ H , and such that for all 𝑖 ∈ {0, 1, 2}, lim lim sup P 𝑝𝑖 (|𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant) = 0. 𝑛→∞ H By Lemma 6.4.7, for all 𝑖,

lim 𝜃 𝑝𝑖 , 𝐺 − 𝜃 𝑝𝑖 , 𝐺 ∞ = 0. H (6.4.7) By Theorem 6.2.3, we have lim inf H 𝜃 ( 𝑝 1 , 𝐺) > 0. So by eq. (6.4.7), limH 𝑝 1 ≥ 𝑝 𝑐 (𝐺 ∞ ) and hence 𝑝 > 𝑝 𝑐 (𝐺). In particular, 𝜃 (·, 𝐺 ∞ ) is continuous at 𝑝 by [Sch99]. So for both 𝑖 ∈ {1, 2}, lim 𝜃 ( 𝑝𝑖 , 𝐺 ∞ ) = 𝜃 ( 𝑝, 𝐺 ∞ ). H (6.4.8) By combining eqs. (6.4.7) and (6.4.8), for both 𝑖 ∈ {1, 2}, lim 𝜃 ( 𝑝𝑖 , 𝐺) = 𝜃 ( 𝑝, 𝐺 ∞ ), H which yields the desired claim by monotonicity because 𝑝 is sandwiched between 𝑝 1 and 𝑝 2 . 6.5 □ Germs and Holes In this section, let H ⊆ F be an infinite set that does not contain any molecular subsequences, and consider some assignment of positive integers 𝑀 : H → N. Given an assignment of parameters 𝑝 : H → [0, 1], let Germ( 𝑝) = Germ( 𝑝, 𝑀, H ) be the statement that lim lim sup P 𝑝 (|𝐾𝑜 | ≥ 𝑛𝑀 but 𝑜 ∉ giant) = 0, 𝑛→∞ H and let Hole( 𝑝) = Hole( 𝑝, 𝑀, H ) be the statement that
lim lim inf inf P 𝑝 ∥giant∥ 𝐴 ≥ 1/𝑛 = 1. 𝑛→∞ H 𝐴⊆𝑉 (𝐺) | 𝐴|≥𝑛𝑀 In both equations, we use the convention that inf ∅ := 1. 238 Lemma 6.5.1. For every supercritical assignment 𝑝, if Germ( 𝑝) holds then Hole( 𝑝) holds. Lemma 6.5.2. For every supercritical assignment 𝑝 and every constant 𝜀 > 0, if Hole( 𝑝) holds then Germ((1 + 𝜀) 𝑝) holds. Lemma 6.5.3. Let 𝐺 be a graph, let 𝑢, 𝑣 be vertices, and let 𝑝, 𝑞 ∈ [0, 1]. For all 𝑛 ∈ N,  𝜔𝑞 

P 𝑜 ̸← → 𝑢 but 𝜕𝐾𝑢 𝜔 𝑝 ∩ 𝜕𝐾𝑣 𝜔 𝑞 ≥ 𝑛 ≤ 𝑒 −𝑛| 𝑝−𝑞| . We next state the universal tightness theorem [Hut21c, Theorem 2.2], which guarantees that the size of the largest intersection of a cluster with a fixed set of vertices is always of the same order as its mean with high probability. This theorem also holds for percolation on weighted graphs; we state a special case that is adequate for our purposes. Theorem 6.5.4 (Universal Tightness). There exist universal constants 𝐶, 𝑐 > 0 such that the following holds. Let 𝐺 = (𝑉, 𝐸) be a (countable, locally finite) graph, let Λ ⊆ 𝑉 be a finite non-empty set of vertices, and let 𝑝 ∈ [0, 1] be a parameter. Set |𝑀 | := max{|𝐾𝑣 ∩ Λ| : 𝑣 ∈ 𝑉 }. Then

P 𝑝 |𝑀 | ≥ 𝛼E 𝑝 |𝑀 | ≤ 𝐶𝑒 −𝑐𝛼 and P 𝑝 |𝑀 | ≤ 𝜀E 𝑝 |𝑀 | ≤ 𝐶𝜀 for every 𝛼 ≥ 1 and 0 < 𝜀 ≤ 1. Proof of Lemma 6.5.1. For all 𝛿 > 0, 𝐺 ∈ H , and 𝐴 ⊆ 𝑉 (𝐺), by a union bound and by linearity of expectation,    
P 𝑝 ∥giant∥ 𝐴 ≥ 𝛿 ≥ P 𝑝 max ∥𝐾𝑢 ∥ 𝐴 ≥ 𝛿 − P 𝑝 max ∥𝐾𝑢 ∥ 𝐴 ≥ 𝛿 𝑢∈𝐴 𝑢∈𝐴\ giant   1 ≥ P 𝑝 max ∥𝐾𝑢 ∥ 𝐴 ≥ 𝛿 − max P 𝑝 (|𝐾𝑢 | ≥ 𝛿 | 𝐴| but 𝑢 ∉ giant) . 𝑢∈𝐴 𝛿 𝑢∈𝐴 So for all 𝑚, 𝑛 ∈ N, ♥𝑚,𝑛 := lim inf inf P 𝑝 ∥giant∥ 𝐴 ≥ 1/𝑚 H
𝐴⊆𝑉 (𝐺) | 𝐴|≥𝑛𝑀   ≥ lim inf inf P 𝑝 max ∥𝐾𝑢 ∥ 𝐴 ≥ 1/𝑚 − lim sup max 𝑚P 𝑝 (|𝐾𝑢 | ≥ 𝑛𝑀/𝑚 but 𝑢 ∉ giant) . H | 𝐴⊆𝑉 (𝐺) 𝑢∈𝐴 H {z 𝑢∈𝑉 (𝐺) } | ♥1𝑚 {z ♥2𝑚,𝑛 By Theorems 6.1.1 and 6.2.1, since 𝑝 is supercritical,   lim inf inf E 𝑝 max ∥𝐾𝑢 ∥ 𝐴 > 0, H 𝐴⊆𝑉 (𝐺) 𝑢∈𝐴 239 } and hence by Theorem 6.5.4, lim𝑚 ♥1𝑚 = 1. By Theorems 6.1.1 and 6.2.1, for all 𝑚, 𝑛 ≥ 1, ♥2𝑚,𝑛 = lim sup 𝑚P 𝑝 (|𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant) . H So if Germ( 𝑝) holds then lim𝑛 ♥2𝑚,𝑛 = 0 for all 𝑚. Therefore, if Germ( 𝑝) holds then by monotonicitiy of ♥𝑚,𝑛 , lim ♥𝑛,𝑛 = lim lim ♥𝑚,𝑛 ≥ lim ♥1𝑚 − lim lim ♥2𝑚,𝑛 = 1, 𝑛 𝑚 𝑛 𝑚 𝑚 𝑛 which implies that Hole( 𝑝) holds. □ Proof of Lemma 6.5.2. Let 𝑞 := (1 + 𝜀) 𝑝. By Theorems 6.1.1 and 6.2.1, there is a constant 𝜀 > 0 such that lim P 𝑝 (N ) = lim P𝑞 (N ) = 0 (6.5.1) H H where N is the complement of the event that there exists a unique cluster 𝐾 satisfying ∥𝐾 ∥ ≥ 𝜀. Suppose for contradiction that Hole( 𝑝) holds but Germ(q) does not. Then we can find an infinite subset H ′ ⊆ H , an assignment 𝑁 : H ′ → N with limH ′ 𝑁/𝑀 = ∞, and a constant 𝜂 > 0 such that for all 𝐺 ∈ H ′, P𝑞 (𝑁 ≤ |𝐾𝑜 | < 𝜀 |𝑉 |) ≥ 𝜂. (6.5.2) Consider some 𝐺 ∈ H ′. Trivially, 
 𝜈(𝐺) := P𝑞 (N ) = E P 𝜔 𝑞 ∈ N | 𝐾𝑜 (𝜔 𝑞 ) , and hence by Markov’s inequality,

P P 𝜔 𝑞 ∈ N | 𝐾𝑜 (𝜔 𝑞 ) ≥ 2𝜈/𝜂 ≤ 𝜂/2. So by eq. (6.5.2) and a union bound, there is a deterministic set Π of possible outcomes for 𝐾𝑜 such that P𝑞 (𝐾𝑜 ∈ Π) ≥ 𝜂/2, (6.5.3) and every 𝐴 ∈ Π satisfies 𝑁 ≤ | 𝐴| < 𝜀 |𝑉 | and
P 𝜔 𝑞 ∈ N | 𝐾𝑜 (𝜔 𝑞 ) = 𝐴 ≤ 2𝜈/𝜂. (6.5.4) Consider some 𝐴 ∈ Π. Let 𝐴 be the set of all edges having at least one endpoint in 𝐴. On the event
that 𝐾𝑜 𝜔 𝑞 = 𝐴, since | 𝐴| < 𝜀 |𝑉 | and every edge in 𝜕 𝐴 is 𝜔 𝑝 -closed, we have 𝜔 𝑝 ∈ N if and 240 
only if 𝜔 𝑝 \𝐴 ∈ N . This is helpful because 𝜔 𝑝 \𝐴 is independent of the event that 𝐾𝑜 𝜔 𝑞 = 𝐴. So we can rewrite eq. (6.5.4) as   P 𝜔 𝑝 \𝐴 ∈ N ≤ 2𝜈/𝜂. In particular, since 𝐴 and 𝐺 were arbitrary, by eq. (6.5.1), Hole( 𝑝), and a union bound, lim′ sup P 𝑝 ({𝜔\𝐵 ∈ N } ∪ N ∪ {giant ∩𝐵 = ∅}) = 0. H 𝐵∈Π | {z } (6.5.5) E (𝐵) Now let 𝑏( 𝐴) = 𝑏( 𝐴, 𝜔) count the number of edges 𝑢𝑣 with 𝑢 ∈ 𝐴 and 𝑣 ∈ 𝑉\𝐴 such that   𝐾𝑣 𝜔\𝐴 ≥ 𝜀 |𝑉 | . Notice that on the event E ( 𝐴), at least one such edge must be 𝜔 𝑝 -open. So by independence, for all 𝑛 ≥ 1, P 𝑝 (E ( 𝐴) | 𝑏( 𝐴) ≤ 𝑛/𝑝) ≥ (1 − 𝑝) 𝑛/𝑝 ≥ 𝑒 −𝑛 > 0. (6.5.6) By constrasting eqs. (6.5.5) and (6.5.6), we must have for every constant 𝑛, lim′ inf P 𝑝 (𝑏( 𝐴) > 𝑛/𝑝) = 1. (6.5.7) H 𝐴∈Π For all 𝐺 and 𝑛, by eq. (6.5.3) and by independence,





P 𝑏 𝐾𝑜 𝜔 𝑞 , 𝜔 𝑝 > 𝑛/𝑝 ≥ P 𝑝 (𝐾𝑜 ∈ Π) inf P 𝑏 𝐴, 𝜔 𝑝 > 𝑛/𝑝 | 𝐾𝑜 𝜔 𝑞 = 𝐴 𝐴∈Π 𝜂 ≥ · inf P 𝑝 (𝑏( 𝐴) > 𝑛/𝑝) . 2 𝐴∈Π So by eq. (6.5.7), for all 𝑛,


lim inf P 𝑏 𝐾 𝜔 , 𝜔 > 𝑛/𝑝 ≥ 𝜂/2. 𝑜 𝑞 𝑝 ′ H (6.5.8)

For all 𝐺 and 𝑛, whenever 𝑏 𝐾𝑜 𝜔 𝑞 , 𝜔 𝑝 > 𝑛/𝑝, there must be at least 𝜀 |𝑉 | vertices 𝑢 such that the event n 𝜔𝑞 o 

T (𝑢) := 𝑜 ̸← → 𝑢 ∩ 𝜕𝐾𝑜 𝜔 𝑞 ∩ 𝜕𝐾𝑢 𝜔 𝑝 ≥ 𝑛/𝑝 (6.5.9) holds. So by linearity of expectation, for all 𝐺 and 𝑛, ♦𝐺,𝑛 := sup P (T (𝑢)) ≥ 𝑢∈𝑉


1 ∑︁ P (T (𝑢)) ≥ 𝜀P 𝑏 𝐾𝑜 𝜔 𝑞 , 𝜔 𝑝 > 𝑛/𝑝 , |𝑉 | 𝑢∈𝑉 and hence by eq. (6.5.8), inf lim inf ♦𝐺,𝑛 ≥ ′ 𝑛≥1 H 241 𝜀𝜂 . 2 (6.5.10) On the other hand, by Lemma 6.5.3, for all 𝐺 and 𝑛, ♦𝐺,𝑛 ≤ 𝑒 −(𝑞−𝑝)·𝑛/𝑝 = 𝑒 −𝜀𝑛 , and hence, lim sup lim sup ♦𝐺,𝑛 ≤ lim sup 𝑒 −𝜀𝑛 = 0, H′ 𝑛→∞ 𝑛→∞ contradicting eq. (6.5.10). 6.6 □ Characterizing discrete phase transitions We now turn our attention to finite transitive graphs of divergent degree. (In fact most of what we do will also apply in the bounded degree case, but has additional technicalities compared to that case.) Our first goal, which we carry out in this section, is to characterise those graphs that have a particularly degenerate kind of percolation phase transition we call a discrete phase transition. Let H be an infinite set of finite connected transitive graphs. We say that H has a discrete percolation phase transition if limH 𝜃 ( 𝑝, 𝐺) = 1 for every supercritical assignment 𝑝 : H → [0, 1]. (Here, the word ‘discrete’ is used since this kind of phase transition, where the density jumps from 0 to 1 over a window of negligible size, is the extreme opposite of a continuous phase transition.) When this occurs, all of our claims about concentration and continuity in the supercritical phase are trivial. In this section we prove the following proposition, which gives a necessary and sufficient condition for the supercritical phase to be discrete. This proposition will play an important role in the remainder of our analysis, allowing us to focus our attention on the case that 𝑝 𝑐 (𝐺) is of order 1/deg(𝐺). Proposition 6.6.1. An infinite set H ⊆ F has a discrete percolation phase transition if and only if H admits a percolation threshold function 𝑝 𝑐 : H → [0, 1] satisfying 𝑝 𝑐 (𝐺) deg 𝐺 = ∞. H 1 − 𝑝 𝑐 (𝐺) lim (6.6.1) The condition (6.6.1) neatly encapsulates that a family of finite vertex-transitive graphs has a discrete phase transition if it has a percolation threshold function that is always either very close to 1 or much larger than the reciprocal of the degree for all large elements of the family; it includes the case that 𝑝 𝑐 (𝐺) = 1 for all but finitely many 𝐺 ∈ H . For an example where 𝑝 𝑐 → 1, take the sequence of cycles (Z𝑛 ), and for an example where 𝑝 𝑐 remains bounded away from 1 but 𝑝 𝑐 /deg → ∞, take cartesian products of complete graphs and cycles (𝐾𝑛 × Z 𝑓 (𝑛) ) where 𝑓 : N → N with lim 𝑓 = ∞ is any sufficiently slowly growing sequence. Since every set of finite transitive graphs with a discrete 242 percolation phase transition admits a threshold function, the fact that the phase transition is discrete can be phrased in terms of the convergence of 𝜃 to a step function when rescaled by 𝑝 𝑐 (𝐺): An infinite set H ⊆ F has a discrete percolation phase transition if and only if it admits a percolation threshold function 𝑝 𝑐 : H → [0, 1] satisfying lim 𝜃 (𝜆𝑝 𝑐 (𝐺), 𝐺) = 1 (𝜆 ≥ 1) 𝐺∈H for each constant 𝜆 ≠ 1. (The 𝜆 = 1 limit of 𝜃 (𝜆𝑝 𝑐 (𝐺), 𝐺) is sensitive to the precise choice of threshold function 𝑝 𝑐 (𝐺).) Corollary 6.6.2. For every infinite set H ⊆ F and for every supercritical assignment 𝑝 : H → [0, 1], lim 𝑝 deg 𝐺 = ∞ =⇒ lim 𝜃 ( 𝑝, 𝐺) = 1. H H Proof. Assume for contradiction that limH 𝑝 deg 𝐺 = ∞ but that for some infinite subset H ′, we have 𝑠 := supH ′ 𝜃 ( 𝑝, 𝐺) < 1. By passing to an infinite subset if necessary, we may assume without loss of generality that H ′ is dense or sparse. In either case, as explained in [Eas22], it follows from Theorem 6.1.2 that H ′ admits a percolation threshold 𝑝 𝑐 : H ′ → [0, 1]. By passing to a further infinite subset if necessary, we may assume without loss of generality that this percolation threshold satisfies either limH ′ 𝑝 𝑐 deg 𝐺 = ∞ or supH ′ 𝑝 𝑐 deg 𝐺 < ∞. In the first case, Proposition 6.6.1 tells us that H ′ has a discrete percolation phase transition. Since 𝑝 is supercritical, it follows that lim′ 𝜃 ( 𝑝, 𝐺) = 1, H contradicting 𝑠 < 1. In the second case, by Theorem 6.2.3, for every positive constant 𝑥,  𝑥  ∥𝐾 ∥ lim inf 𝑃 ≥ = 1, 1 𝑝 H′ 1+𝑥 and hence lim inf 𝜃 ( 𝑝, 𝐺) ≥ ′ H 𝑥 , 1+𝑥 𝑠 which contradicts 𝑠 < 1 when 𝑥 > 1−𝑠 . □ We start with the easier ‘only if’ direction of Proposition 6.6.1. Lemma 6.6.3. Let H ⊆ F . If H does not have a percolation threshold, or if it has a percolation 𝑝 𝑐 (𝐺) threshold 𝑝 𝑐 : H → [0, 1] satisfying lim inf H 1−𝑝 deg 𝐺 < ∞, then the supercritical phase of 𝑐 (𝐺) H is not discrete. 243 Proof. First suppose that H does not have a percolation threshold. Then by the main theorem of [Eas22], H contains an infinite molecular subset H ′. (In fact it contains an infinite 𝑚-molecular subset for infinitely many 𝑚, but we will not need to use this stronger fact.) Since molecular sets are dense, it follows by the results of [Bol+10b] that there exists 𝛼 < ∞ (in fact 𝛼 = 1) such that 2𝛼 𝐺 ↦→ deg𝛼 𝐺 is a percolation threshold for H ′. Then the assignment 𝑝 : 𝐺 ↦→ deg 𝐺 is supercritical for H ′ and satisfies lim sup 𝜃 ( 𝑝(𝐺), 𝐺) ≤ lim sup P 𝑝 (|𝐾𝑜 | > 1) H′ H′ "   deg 𝐺 # 2𝛼 ≤ lim sup 1 − 1 − < 1. deg 𝐺 H′ (6.6.2) Next suppose that H does have a percolation threshold 𝑝 𝑐 : H → [0, 1] but lim inf H 𝑝 𝑐 (𝐺) deg 𝐺 < ∞. 1 − 𝑝 𝑐 (𝐺) Then there exists an infinite subset H ′ with lim sup H′ 𝑝 𝑐 (𝐺) deg 𝐺 < ∞, 1 − 𝑝 𝑐 (𝐺) and hence lim supH ′ 𝑝 𝑐 (𝐺) < 1 and lim supH ′ 𝑝 𝑐 (𝐺) deg 𝐺 < ∞. Without loss of generality (i.e. by passing to a further infinite subset if necessary), we may assume that there exist constants 𝜀 > 0 and 𝛼 < ∞ such that supH ′ 𝑝 𝑐 (𝐺) < 1−𝜀 and limH ′ 𝑝 𝑐 (𝐺) deg 𝐺 = 𝛼, and either limH ′ deg 𝐺 = ∞ 2𝛼 or supH ′ deg 𝐺 < ∞. If limH ′ deg 𝐺 = ∞, then the assignment 𝑝 : 𝐺 ↦→ deg 𝐺 is supercritical for ′ H and satisfies eq. (6.6.2). If supH ′ deg 𝐺 < ∞, then the assignment 𝑝 : 𝐺 ↦→ 1− 𝜀 is supercritical for H ′ and satisfies lim sup 𝜃 ( 𝑝(𝐺), 𝐺) ≤ lim sup P 𝑝 (|𝐾𝑜 | > 1) H′ H′   ≤ lim sup 1 − 𝜀 deg 𝐺 < 1. H′ We have shown that in all cases, we can find an infinite subset H ′ ⊆ H and a supercritical assignment 𝑝 : H ′ → [0, 1] satisfying lim supH ′ 𝜃 ( 𝑝(𝐺), 𝐺) < 1. Then the assignment H → [0, 1] given by    𝑝(𝐺)  𝐺→ ↦  1  if 𝐺 ∈ H ′ if 𝐺 ∈ H \H ′ is supercritical for H and satisfies lim inf H 𝜃 ( 𝑝(𝐺), 𝐺) < 1. Therefore the phase transition on H is not discrete. □ 244 The proof of Proposition 6.6.4 will apply both the universal tightness theorem Theorem 6.5.4 and the following quantitative insertion tolerance estimate of [EH21a, Proposition 2.6]. Both results will be used again several times later in the paper. Note that for percolation on infinite graphs, “insertion tolerance” usually refers to the fact that 𝜔∪{𝑒} has law absolutely continuous with respect to that of 𝜔, or equivalently that P(𝜔(𝑒) = 1|𝜔| 𝐸\𝑒 ) > 0 almost surely. This statement is much less useful for finite graphs, particularly when 𝑝 is very small as is typical in the high-degree case. The following proposition gives conditions under which we can force an edge to appear within a given (possibly random) set without changing the probability of a given event too badly. Proposition 6.6.4 (Quantitative insertion tolerance). Let 𝐺 = (𝑉, 𝐸) be a finite graph, let 𝑝 ∈ (0, 1), and let 𝐹 ⊆ 𝐸 be a collection of edges. Let 𝐴 ⊆ {0, 1} 𝐸 be an event, let 𝜂 > 0 and suppose that for each configuration 𝜔 ∈ 𝐴 there is a distinguished subset 𝐹 [𝜔] with 𝐹 [𝜔] ⊆ 𝐹 \ 𝜔 and |𝐹 [𝜔] | ≥ 𝜂 |𝐹 |. If we define 𝐴+ := {𝜔 ∪ {𝑒} : 𝜔 ∈ 𝐴 and 𝑒 ∈ 𝐹 [𝜔]} then P 𝑝 ( 𝐴+ ) ≥ 𝑝|𝐹 | 𝜂2 · · P 𝑝 ( 𝐴) 2 . 1 − 𝑝 𝑝|𝐹 | + 1 We now begin to work towards the proof of the converse of Lemma 6.6.3 in earnest. We begin with the following lemma, recalling that 𝜃 𝜀 ( 𝑝) = 𝜃 𝜀 ( 𝑝, 𝐺) := P 𝑝 (∥𝐾𝑜 ∥ ≥ 𝜀). Lemma 6.6.5. For each 𝜀 > 0 and 𝜆 < ∞, there exists 𝛿 > 0, and 𝐶 < ∞ such that the following holds for every 𝐺 = (𝑉, 𝐸) ∈ F satisfying |𝑉 | ≥ 𝐶 and deg 𝐺 ≤ 𝛿 |𝑉 |. If 𝑝 ∈ (0, 1) is 𝜀-supercritical and satisfies 𝑝𝜃 ′𝜀 ( 𝑝) ≤ 𝜆 then either 𝑝 ≤ deg𝐶 𝐺 or 𝜃 𝜀 ( 𝑝) > 1 − 𝜀. Proof of Lemma 6.6.5. Let 𝐺 ∈ F . The claim is equivalent to the statement that for each 𝜀 > 0 and 𝜆 < ∞, there exists 𝐶 = 𝐶 (𝜀, 𝜆) and 𝛿 = 𝛿(𝜀, 𝜆) > 0 such that if 𝑝 ∈ (0, 1) is 𝜀-supercritical and satisfies 𝜃 𝜀 ( 𝑝) ≤ 1 − 𝜀 and 𝑝𝜃 ′𝜀 ( 𝑝) ≤ 𝜆 then either |𝑉 | < 𝐶, deg 𝐺 > 𝛿|𝑉 |, or 𝑝 ≤ 𝐶/deg 𝐺. Fix 𝑝 ∈ (0, 1) that is 𝜀-supercritical and satisfies 𝜃 𝜀 ( 𝑝) ≤ 1−𝜀 and 𝑝𝜃 ′𝜀 ( 𝑝) ≤ 𝜆. By Corollary 6.3.3, there exists 𝐶1 = 𝐶1 (𝜀, 𝜆) ≥ 1 and a set of edges 𝐹 ⊆ 𝐸 with |𝐹 | ≤ 𝐶1 such that
𝜀 P 𝑝 act ∥𝐾𝑜 ∥≥𝜀 [𝐹] ≥ . 2 Applying the pigeonhole principle twice, we deduce that there exists 𝑢𝑣 ∈ 𝐹 such that    𝜀 𝜀 P 𝑝 {𝑢 ↮ 𝑣} ∩ ∥𝐾𝑣 ∥ ≥ ≥ . 2𝐶1 4𝐶1 245 (6.6.3) By Theorems 6.1.1 and 6.2.1, since a sparse family of graphs can never contain an infinite macromolecular subset, there exists 𝐶2 = 𝐶2 (𝜀, 𝜆) < ∞ and 𝛿 = 𝛿(𝜀, 𝜆) > 0 such that if |𝑉 | ≥ 𝐶2 and deg 𝐺 ≤ 𝛿 |𝑉 | then     𝜀 𝜀 𝜀 ≤ ∥𝐾𝑣 ∥ < 𝜀 ≤ P 𝑝 ∥𝐾1 ∥ < 𝜀 or ∥𝐾2 ∥ ≥ . (6.6.4) ≤ P𝑝 2𝐶1 2𝐶1 16𝐶1 Let 𝑁 be an 𝜔-connected subset of neigh(𝑢) of maximal volume (breaking ties using some total order on the set of subsets of neigh(𝑢) that is chosen in advance). We have that E 𝑝 |𝑁 | ≥ E 𝑝 |𝐾1 ∩ 𝑁 | ≥ 𝜀 deg(𝐺), and hence by the universal tightness theorem (Theorem 6.5.4) that there exists a constant 𝛿1 = 𝛿1 (𝜀, 𝜆) > 0 such that 𝜀 P 𝑝 (|𝑁 | ≤ 𝛿1 deg) ≤ . 16𝐶1 It follows from this, by a union bound, that the event   𝜀 E := {𝑢 ↮ 𝑣} ∩ {∥𝐾𝑣 ∥ ≥ 𝜀} ∩ {|𝑁 | ≥ 𝛿1 deg} ∩ ∥𝐾2 ∥ < 2𝐶1 satisfies P 𝑝 (E) ≥ 𝜀/(8𝐶1 ) whenever |𝑉 | ≥ 𝐶2 and deg 𝐺 ≤ 𝛿. To conclude, it suffices to prove that there exists a constant 𝐶3 = 𝐶3 (𝜀, 𝜆) < ∞ such that if |𝑉 | ≥ 𝐶2 and deg 𝐺 ≤ 𝛿 |𝑉 | then 𝑝 ≤ 𝐶3 /deg 𝐺. Let
(∗) := P 𝑝 act ∥𝐾𝑢 ∥≥𝜀 [𝑢𝑁] | E , where 𝑢𝑁 denotes the set of edges with one endpoint equal to 𝑢 and the other in 𝑁. We split into two cases according to whether (∗) ≥ 21 (Case 1) or (∗) < 12 (Case 2). Case 1 Since all the vertices of 𝑁 are connected in 𝜔, if act ∥𝐾𝑢 ∥≥𝜀 [𝑢𝑁] occurs then every edge in 𝑢𝑁 is a closed pivotal for the event {∥𝐾𝑢 ∥ ≥ 𝜀}. As such, by the proof of Lemma 6.4.4 (i.e. a simple mass-transport argument), 1 ∑︁ 𝜃 ′𝜀 ( 𝑝) = E 𝑝 [|𝐾𝑜 | 1 (act |𝐾𝑜 |≥𝜀|𝑉 | [𝑜𝑏])] 1 − 𝑝 𝑏∼𝑜 𝛿1 deg 𝐺 P 𝑝 (E, act ∥𝐾𝑢 ∥≥𝜀 [𝑢𝑁]) 2 𝜀𝛿1 ≥ deg 𝐺, 16𝐶1 ≥ and, since 𝑝𝜃 ′𝜀 ( 𝑝) ≤ 𝜆, it follows that 𝑝≤ 16𝐶1𝜆 . 𝜀𝛿1 deg 𝐺 246 Case 2 Notice that if we start with a configuration belonging to the event E\ act ∥𝐾𝑢 ∥≥𝜀 [𝑁] and then open an edge in 𝑢𝑁, we obtain a configuration in which 𝛿1 deg 𝐺 ≤ |𝐾𝑢 | < 𝜀 |𝑉 | . So, by the quantitative insertion-tolerance estimate of Proposition 6.6.4, there exists a constant 𝛿2 = 𝛿2 (𝜀, 𝜆) > 0 such that P 𝑝 ({𝛿1 deg 𝐺 ≤ |𝐾𝑢 | < 𝜀 |𝑉 |} ∩ {∥𝐾𝑣 ∥ ≥ 𝜀}) ≥ 𝛿2 . It follows from Lemma 6.4.4 that 𝜃 ′𝜀 ( 𝑝) ≥ 𝛿1 𝛿2 deg 𝐺, and hence that 𝑝≤ 𝜆 . 𝛿1 𝛿2 deg 𝐺 □ Proof of Proposition 6.6.1. The ‘only if’ direction is Lemma 6.6.3. Suppose for contradiction that the ‘if’ direction is false, so we can find an infinite set H of finite connected transitive graphs with a percolation threshold 𝑝 𝑐 satisfying 𝑝 𝑐 (𝐺) deg 𝐺 = ∞ H 1 − 𝑝 𝑐 (𝐺) lim (6.6.5) and a supercritical sequence 𝑝 satisfying lim inf 𝜃 ( 𝑝(𝐺), 𝐺) < 1. H Let H ′ be an infinite subset of H such that supH ′ 𝜃 ( 𝑝(𝐺), 𝐺) < 1. If limH ′ 𝑝 𝑐 (𝐺) = 1, then the only supercritical sequence for H ′ (up to changing finitely many terms in the sequence) would be the constant assignment 𝐺 ↦→ 1, which trivially satisfies limH ′ 𝜃 (1, 𝐺) = 1. So by passing to a further infinite subset if necessary, we may assume without loss of generality that supH ′ 𝑝 𝑐 (𝐺) < 1. Since eq. (6.6.5) holds but 𝑝 𝑐 (𝐺) is bounded away from 1 on H ′, we know that lim′ 𝑝 𝑐 (𝐺) deg 𝐺 = ∞. H (6.6.6) By Lemma 6.3.4, we can find a constant 𝜀 > 0 and an 𝜀-supercritical sequence 𝑞 for H ′ such that for all but finitely many 𝐺 ∈ H ′, we have 𝜃 𝜀 (𝑞) ≤ 1 − 𝜀 and 𝑞𝜃 ′𝜀 (𝑞) ≤ 𝜀1 . By eq. (6.6.6), we know that H ′ does not contain an infinite subset of graphs with edge densities |𝐸 2| uniformly bounded |𝑉 | away from zero [Bol+10b] or vertex degrees uniformly bounded above (trivially, since 𝑝 𝑐 ≤ 1). Lemma 6.6.5 therefore implies that lim′ 𝑞𝜃 ′𝜀 (𝑞) = ∞, H which yields the required contradiction. □ 247 6.7 Negligibility of large holes The goal of this section is to prove the following proposition, which states that sets of size significantly larger than the degree have large intersection with the giant with high probability. Recall that when 𝑊 and 𝐴 are two finite sets of vertices we write ∥𝑊 ∥ 𝐴 = |𝑊 ∩ 𝐴|/| 𝐴| for the density of 𝑊 in 𝐴, and that giant𝜀 = {𝑣 : ∥𝐾𝑣 ∥ ≥ 𝜀} denotes the set of vertices whose clusters have density at least 𝜀. Recall that F denotes the set of all finite transitive graphs. Proposition 6.7.1. For every supercritical assignment 𝑝 : F → [0, 1], lim lim inf 𝑛→∞ F inf 𝐴⊆𝑉 (𝐺) | 𝐴|≥𝑛 deg 𝐺
P 𝑝 ∥giant∥ 𝐴 ≥ 1/𝑛 = 1, where inf ∅ := 1. At the core of our proof is an induction argument that is similar to our proof of [EH21a, Proposition 4.1] in the previous paper of this series. We will start by proving Proposition 6.7.1 conditionally on a technical lemma whose proof is deferred to Section 6.7. Before giving the proof of Proposition 6.7.1,  let us give an informal overview of the strategy. Notice that the original event giant𝜀 𝐴 ≥ 𝛿 is increasing. So, if we suppose for contradiction that the proposition fails at some 𝜀-supercritical 𝑝, then the conclusion of the proposition also fails to hold for all 𝑞 ≤ 𝑝. By Lemma 6.5.2, this implies that at every (1 − 𝜀) 𝑝 ≤ 𝑞 ≤ 𝑝, the cluster at 𝑜 is mesoscopic with good probability. Thus, we must have a whole interval 𝐼 of parameters such that we reach a contradiction if we can show that there is some 𝑞 ∗ ∈ 𝐼 where 𝐾𝑜 is unlikely to be a mesoscopic under P𝑞∗ . To find such a 𝑞 ∗ ∈ 𝐼, we will use the following lemma to pick a “good” sequence ( 𝑝 𝑛 )𝑛≥1 within this interval 𝐼. We will then sprinkle repeatedly, moving from P 𝑝 𝑛1 to P 𝑝 𝑛2 to P 𝑝 𝑛3 and so forth, where ( 𝑝 𝑛𝑖 )𝑖≥1 is a well-chosen subsequence of ( 𝑝 𝑛 )𝑛≥1 . As we do so, we will deduce a sequence of increasingly strong estimates about measure P 𝑝 𝑛 , eventually finding an 𝑛 where we can prove that 𝑞 ∗ = 𝑝 𝑛 has the desired properties needed for us to obtain a contradiction. Lemma 6.7.2. For every 0 < 𝜀 < 1 there exists 𝛿(𝜀) > 0 such that the following holds. Let 𝐺 = (𝑉, 𝐸) ∈ F satisfy |𝑉 | ≥ 𝛿−1 . For every 𝜀-supercritical parameter 𝑝 ∈ (0, 1), there exists

an increasing sequence of parameters ( 𝑝 𝑛 )𝑛≥1 in 1 − 𝜀2 𝑝, 𝑝 such that the following inequalities hold for every 𝑛 ≥ 1: 1. 𝑝 𝑛 𝜃 ′𝜀 ( 𝑝 𝑛 ) ≤ 8𝜀 −1 ; 248 2. 𝑝 𝑛+1 − 𝑝 𝑛 ≥ 𝜀3−𝑛−3 (deg 𝐺) −1 ; 3. 𝜃 𝜀 ( 𝑝 𝑛+1 ) − 𝜃 𝜀 ( 𝑝 𝑛 ) ≤ 2−𝑛 . Proof of Lemma 6.7.2. Fix 𝐺 = (𝑉, 𝐸) ∈ F and an 𝜀-supercritical parameter 𝑝 ∈ (0, 1). By Lemma 6.3.4, the set 𝐽 = {𝑞 ∈ ((1 − 𝜀/2) 𝑝, 𝑝) : 𝑞𝜃 ′𝜀 (𝑞) ≤ 8𝜀 −1 } has Lebesgue measure L (𝐽) ≥ 1 𝜀 1 4 𝜀 𝑝. By Lemma 6.2.2, there exists 𝛿 > 0 such that if |𝑉 | ≥ 1/𝛿 then 𝑝 ≥ 2 deg , so that L (𝐽) ≥ 8 deg . Applying the same argument as in the proof of [EH21a, Lemma 4.7] but with ‘𝜃 𝜀 (·)’ used in place of ‘P· (∥𝐾1 ∥ ≥ 𝛽)’ yields a sequence ( 𝑝 𝑛 )𝑛≥1 in 𝐽 such that 𝑝 𝑛+1 −𝑝 𝑛 ≥ 3−𝑛−1 L (𝐽) ≥ 𝜀3−𝑛−3 (deg 𝐺) −1 and 𝜃 𝜀 ( 𝑝 𝑛+1 ) − 𝜃 𝜀 ( 𝑝 𝑛 ) ≤ 2−𝑛 for every 𝑛 ≥ 1 as required. (Indeed, this argument works for any non-decreasing function taking values in [0, 1] as explained in the proof of [EH21a, Lemma 4.7].) □ Let us now introduce some notation that will be useful in the rest of the section. Let 𝐺 ∈ F and let 𝜀, 𝛿 > 0. For each vertex 𝑣 of 𝐺, define the event   deg 𝛿,𝜀 ≤ |𝐾𝑣 | < 𝜀 |𝑉 | . meso𝑣 := 𝛿 To lighten notation, we will also adopt the shorthand act𝑥𝜀 [𝐹] := act ∥𝐾 𝑥 ∥≥𝜀 [𝐹] for the event that a set of edges 𝐹 is an activator for the event {∥𝐾𝑥 ∥ ≥ 𝜀}. The next lemma is the inductive step for our repeated sprinkling argument. It tells us that by sprinkling along our good sequence ( 𝑝 𝑛 )𝑛≥1 , we can iteratively shrink the size of a certain set of edges 𝐹 while maintaining a good probability 𝜀 that meso𝛿,𝜀 𝑜 ∩ act𝑜 [𝐹] occurs. Once 𝐹 becomes a singleton, we will have found our parameter 𝑞 ∗ = 𝑝 𝑛 with which we can obtain a contradiction. Lemma 6.7.3 (Shrinking by sprinkling). Let 𝑝, 𝜀, 𝛿 ∈ (0, 1) and 𝐺 = (𝑉, 𝐸) ∈ F . Suppose that
𝑝 is 𝜀-supercritical and that ( 𝑝 𝑛 )𝑛≥1 is a sequence in (1 − 𝜀2 ) 𝑝, 𝑝 satisfying conditions 2 and 3 from Lemma 6.7.2. For each (𝑛, 𝑘, 𝛼), let 𝐴𝑛 (𝑘, 𝛼) be the statement that there exists a set of edges 𝐹 with |𝐹 | = 𝑘 satisfying   𝜀 [𝐹] P 𝑝 𝑛 meso𝛿,𝜀 ∩ act ≥ 𝛼. 𝑜 𝑜 Given any 𝑘 ≥ 2 and 𝛼 > 0, we can find 𝑁 (𝜀, 𝑘, 𝛼) < ∞ such that following holds if |𝑉 | ≥ 𝑁 deg 𝐺: For each 𝑛 ≥ 𝑁, there exists 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 such that for all 𝑚 > 𝑛, 𝐴𝑛 (𝑘, 𝛼) =⇒ 𝐴𝑚 (𝑘 − 1, 𝛽) We stress that the constants 𝛽 and 𝑁 appearing in Lemma 6.7.3 are independent of the choices of 𝐺, 𝑝, ( 𝑝 𝑛 )𝑛≥1 , and 𝛿. The proof of Lemma 6.7.3 is highly technical, and is deferred to the next subsection. By repeatedly applying Lemma 6.7.3, we obtain the following. 249 Lemma 6.7.4. Let H ⊆ F be an infinite family of graphs that is sparse. Let 𝜀 > 0 be a constant, and suppose that 𝑝 : H → (0, 1) is an assignment such that for every 𝐺, the parameter 𝑝(𝐺) is 𝜀-supercritical for 𝐺. Then   lim lim sup inf P𝑞 meso1/𝑚,𝜀 = 0. 𝑜 𝑚→∞ H 𝑞∈[(1−𝜀) 𝑝,𝑝] Proof. Suppose for contradiction that this is not the case. Then there exists a constant 𝜂 > 0 such that for all 𝑚 ≥ 1, there is an infinite subset H𝑚 ⊆ H such that for all 𝐺 ∈ H𝑚 , for all 𝑞 ∈ [(1 − 𝜀) 𝑝, 𝑝],   1/𝑚,𝜀 P𝑞 meso𝑜 ≥ 𝜂. Since H is sparse, we can moreover assume that for all 𝑚 ≥ 1 and for all 𝐺 ∈ H𝑚 , |𝐺 | ≥ 𝑚 deg 𝐺. Consider some 𝑚 ≥ 1 and some 𝐺 = (𝑉, 𝐸) ∈ H𝑚 . We will show that if 𝑚 is sufficiently large with respect to 𝜂 and 𝜀, then we can force a contradiction. Let 𝛿1 (𝜀) > 0 be the constant from Lemma 6.7.2. Suppose that 𝑚 ≥ 1/𝛿1 , and let ( 𝑝 𝑛 )𝑛≥1 be the sequence in ((1 − 𝜀/2) 𝑝, 𝑝) that is thereby guaranteed to exist. By Corollary 6.3.3, there is a constant 𝑘 (𝜀, 𝜂) < ∞ such that for every 𝑚 ≥ 1, there is a set of edges 𝐹 ⊆ 𝐸 (which may depend on 𝑚) such that |𝐹 | ≤ 𝑘 and
  min 𝜀 2 , 𝜂, 𝜀/8 1/𝑚,𝜀 𝜀 ∩ act𝑜 [𝐹] ≥ 𝛽0 := P 𝑝 𝑛 meso𝑜 . 2 Equivalently, in the language of Lemma 6.7.3 (where the constant “𝛿” is 1/𝑚), 𝐴𝑛 (𝑘, 𝛽0 ) holds for all 𝑛. Let 𝑁0 be the constant “𝑁 (𝜀, 𝑘, 𝛽0 )” from Lemma 6.7.3. Recursively define the sequence (𝑁0 , 𝛽0 ), (𝑁1 , 𝛽1 ), . . . , (𝑁 𝑘−1 , 𝛽 𝑘−1 ), starting with (𝑁0 , 𝛽0 ) as already defined, as follows: Suppose that we have defined 𝑁𝑖−1 and 𝛽𝑖−1 for some 𝑖 ∈ {1, . . . , 𝑘 − 1}. Then set 𝛽𝑖 to be the constant “𝛽(𝜀, 𝑘 − 𝑖 + 1, 𝛽𝑖−1 , 𝑁𝑖−1 )” from Lemma 6.7.3 and set 𝑁𝑖 := 𝑁 ∧ 𝑁𝑖−1 + 1 where 𝑁 is the constant “𝑁 (𝜀, 𝑘 − 𝑖, 𝛽𝑖 )” from Lemma 6.7.3. Note that 𝑀 := 𝑁 𝑘−1 and 𝛿2 := 𝛽 𝑘−1 are constants that are entirely determined by 𝜀 and 𝜂. In particular, we may assume that 𝑚 ≥ 𝑀, so that for all 𝑖 ∈ {1, . . . , 𝑘 − 1}, |𝐺 | ≥ 𝑁𝑖 deg 𝐺. (6.7.1) Claim 6.7.5. 𝐴 𝑝 𝑀 (1, 𝛿2 ) holds. Proof of claim. We will use induction on 𝑖 to prove more generally that 𝐴 𝑁𝑖 (𝑘 − 𝑖, 𝛽𝑖 ) holds for every 𝑖 ∈ {0, . . . , 𝑘 − 1}, which yields the claim as the case 𝑖 = 𝑘 − 1. The base case 𝑖 = 0 holds 250 because 𝐴𝑛 (𝑘, 𝛽0 ) holds for every 𝑛 ≥ 1, and therefore in particular for 𝑛 = 𝑁0 . For the inductive step, suppose that 𝐴 𝑁𝑖 (𝑘 − 𝑖, 𝛽𝑖 ) holds for some 𝑖 ∈ {0, . . . , 𝑘 − 2}. Since 𝑁𝑖 ≥ 𝑁 where 𝑁 is the constant “𝑁 (𝜀, 𝑘 − 𝑖, 𝛽𝑖 )” from Lemma 6.7.3, and since eq. (6.7.1) holds, Lemma 6.7.3 implies that 𝐴𝑛 (𝑘 − (𝑖 + 1), 𝛽𝑖+1 ) holds for every 𝑛 > 𝑁𝑖 , and in particular, for 𝑛 = 𝑁𝑖+1 , as required. □ Pick an edge 𝑢𝑣 ∈ 𝐸 witnessing the fact that 𝐴 𝑝 𝑀 (1, 𝛿2 ) holds, i.e. such that   𝜀 P 𝑝 𝑀 meso1/𝑚,𝜀 ∩ act [𝑢𝑣] ≥ 𝛿2 . 𝑜 𝑜 By a union bound, we may assume without loss of generality that the endpoint 𝑢 ∈ 𝑢𝑣 satisfies   𝛿 2 1/𝑚,𝜀 𝜀 P 𝑝 𝑀 meso𝑢 ∩ act𝑢 [𝑢𝑣] ≥ . 2 Then by Lemma 6.4.4, 𝛿2 𝑚 deg 𝐺 . 2 1 By Lemma 6.2.2, when 𝑚 is large, we must have 𝑝 𝑀 ≥ 2 deg 𝐺 . Therefore, 𝜃 ′𝜀 ( 𝑝 𝑀 ) ≥ 𝑝 𝑀 𝜃 ′𝜀 ( 𝑝 𝑀 ) ≥ 𝛿2 𝑚 , 4 which contradicts the first condition enumerated in Lemma 6.7.2 when 𝑚 is sufficiently large with respect to 𝜀 and 𝜂. □ We will now deduce Proposition 6.7.1 from Lemma 6.7.4. Note that this proof remains conditional on Lemma 6.7.3 (through our use of Lemma 6.7.4), which we will prove in the next subsection. Proof of Proposition 6.7.1. Suppose for contradiction the claimed equation does not hold. For every fixed 𝐺 = (𝑉, 𝐸) ∈ F , ♦𝑛,𝐺 := inf 𝐴⊆𝑉 | 𝐴|≥𝑛 deg 𝐺 P 𝑝 ∥giant∥ 𝐴 ≥ 1/𝑛
is trivially non-decreasing with respect to 𝑛. So there must exist a constant 𝛾 > 0 such that for all 𝑛 ≥ 1 there is an infinite set H𝑛 ⊆ F such that ♦𝑛,𝐺 ≤ 1 − 𝛾 for all 𝐺 ∈ H𝑛 . By picking a distinct element from each H𝑛 , we can build an infinite set H ⊆ F such that for all 𝑛 ≥ 1, lim sup ♦𝑛,𝐺 ≤ 1 − 𝛾. H In particular, in the language of Section 6.5, the set H does not have property Hole( 𝑝, 𝑑) where 𝑑 (𝐺) := deg 𝐺. By passing to an infinite subset of H , we may assume without loss of generality 251 that there is a constant 𝜀 > 0 such that for every 𝐺 ∈ H , 𝑝(𝐺) ∈ (0, 1) and 𝑝(𝐺) is 2𝜀-supercritical for 𝐺.   For each 𝐺 ∈ H , pick a parameter 𝑞(𝐺) ∈ [(1 − 𝜀) 𝑝, 𝑝] that minimises P𝑞 meso1/𝑚,𝜀 , which 𝑜 exists by continuity. Note that 𝑞 : H → [0, 1] is supercritical because (1 − 𝜀) 𝑝 is supercritical because 𝑝(𝐺) is always 2𝜀-supercritical for 𝐺. By Lemma 6.7.4,   1/𝑚,𝜀 lim lim sup P𝑞 meso𝑜 = 0. 𝑛→∞ H In particular, by Theorems 6.1.1 and 6.2.1 and since sparse families of graphs cannot contain molecular subsequences, lim lim sup P 𝑝 (|𝐾𝑜 | ≥ 𝑚𝑑 but 𝑜 ∉ giant) = 0, 𝑛→∞ H i.e. H has the Germ(𝑞, 𝑑) property. By Lemma 6.5.1, it follows that H has the Hole(𝑞, 𝑑) property. So by monotonicity, H also has the Hole( 𝑝, 𝑑) property, a contradiction. □ Proof of Lemma 6.7.3 Let 𝐺 = (𝑉, 𝐸) ∈ F , and write 𝑑 for the vertex degree of 𝐺. Let 𝑝, 𝜀, 𝛿 ∈ (0, 1), and suppose that 𝑝
is 𝜀-supercritical for 𝐺. Suppose that ( 𝑝 𝑛 )𝑛≥1 is a sequence in (1 − 𝜀2 ) 𝑝, 𝑝 satisfying conditions 2 and 3 from Lemma 6.7.2. Fix 𝑘 ≥ 2 and 𝛼 > 0. Let 𝑛 < 𝑚 be arbitrary positive integers. Assume that 𝐴𝑛 (𝑘, 𝛼) holds, and let 𝐹 be a set of edges witnessing this. Our goal is to find 𝑁 (𝜀, 𝑘, 𝛼) < ∞ and 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 such that if |𝑉 | ≥ 𝑁 𝑑 and 𝑛 ≥ 𝑁, then 𝐴𝑚 (𝑘 − 1, 𝛽) must hold. Notice that 𝐴𝑚 (𝑟, 𝛽) =⇒ 𝐴𝑚 (𝑘 − 1, 𝛽) for all 𝑟 ∈ {1, . . . , 𝑘 − 1}, since we can always extend a set of edges witnessing 𝐴𝑚 (𝑟, 𝛽) to a set of edges witnessing 𝐴𝑚 (𝑘 − 1, 𝛽) by adding (𝑘 − 1) − 𝑟 many arbitrary edges. In the following claims, we will write 𝑐 1 , 𝑐 2 , · · · ∈ (0, 1) for small positive constants that are determined by the triple (𝜀, 𝑘, 𝛼). When a claim involves a new constant 𝑐𝑖 that has not previously appeared, we are asserting the existence of a constant 𝑐𝑖 (𝜀, 𝑘, 𝛼) ∈ (0, 1) with 𝑐𝑖 ≤ min(𝑐 1 , . . . , 𝑐𝑖−1 ) that would make the claim true. The constant 𝑐 1 is introduced in Claim 1, the constant 𝑐 2 in Claim 2, and so forth. Claim 6.7.6. If |𝑉 | ≥ 𝑐−1 1 𝑑, then we can find vertices 𝑥 and 𝑦 with 𝑥𝑦 ∈ 𝐹 such that P 𝑝 𝑛 (E1 ) ≥ 𝑐 1 where 𝜀 E1 := meso𝛿,𝜀 𝑜 ∩ act𝑜 [𝐹] ∩ {𝑦 ∈ giant𝜀 }. 252 Proof. Similarly to in the proof of Lemma 6.6.5, by applying the pigeonhole principle twice, there are vertices 𝑥 and 𝑦 with 𝑥𝑦 ∈ 𝐹 such that   𝛼 𝜀 𝜀 } ≥ . P 𝑝 𝑛 meso𝛿,𝜀 ∩ act [𝐹] ∩ {𝑦 ∈ giant 𝑜 𝑜 2𝑘 2𝑘 By Theorem 6.1.1 and Theorem 6.2.1, there exists 𝑁 (𝜀, 𝛼, 𝑘) < ∞ such that if |𝑉 | ≥ 𝑁 𝑑 then  𝛼 𝜀  P 𝑝 𝑛 ∥𝐾1 ∥ ≥ 𝜀 and ∥𝐾2 ∥ < ≥ 1− . 2𝑘 4𝑘 𝛼 The conclusion follows by a union bound, with 𝑐 1 := 𝑁 −1 ∧ 4𝑘 . □ We will use 𝑥 and 𝑦 to denote the vertices whose existence is guaranteed by this claim throughout the rest of the proof. Given a configuration 𝜔 and a vertex 𝑢, let 𝑆(𝑢) = 𝑆(𝑢, 𝜔) be the largest 𝜔-connected subset of neigh(𝑢), breaking any ties according to an arbitrary deterministic rule. Claim 6.7.7. If |𝑉 | ≥ 𝑐−1 1 𝑑, then P 𝑝 𝑛 (E2 ) ≥ 𝑐 2 where  E2 := E1 ∩ ∥𝑆(𝑥)∥ neigh(𝑥) ≥ 𝑐 2 . Proof. Consider any vertex 𝑢. Since 𝑝 𝑛 ≥ (1−𝜀) 𝑝 and 𝑝 is 𝜀-supercritical, E 𝑝 𝑛 giant𝜀 neigh(𝑢) ≥ 𝜀. So by Markov’s inequality,  𝜀 𝜀 P 𝑝 𝑛 giant𝜀 neigh(𝑢) ≥ ≥ . 2 2 There can never be more than 1/𝜀 clusters that each contains at least 𝜀 |𝑉 | vertices. Therefore E 𝑝 𝑛 ∥𝑆(𝑢)∥ neigh(𝑢) ≥ 𝜀3 . 4 So by Theorem 6.5.4, there exists a universal constant 𝐶 ∈ (1, ∞) such that   𝜀3 𝑐1 𝑐1 P 𝑝 𝑛 ∥𝑆(𝑢) ∥ neigh(𝑢) ≤ · ≤ . 4 2𝐶 2 3 𝑐1 The conclusion follows by a union bound, with 𝑐 2 := 𝜀4 · 2𝐶 . −1 Claim 6.7.8. If |𝑉 | ≥ 𝑐−1 1 𝑑 and 𝑛 ≥ 𝑐 3 , then either 𝐴𝑚 (1, 𝑐 3 ) holds or P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 where E3 := E2 \ Act𝜀𝑜 [𝑥𝑆(𝑥)]. 253 □ 𝑐2 2 Proof. Supose that |𝑉 | ≥ 𝑐−1 1 𝑑 and P 𝑝 𝑛 (E3 ) < 𝑐 := 4 . Our goal is to show that 𝐴𝑚 (1, 𝑐) holds if we assume that 𝑛 ≥ 𝑁 for a sufficiently large choice of 𝑁 (𝜀, 𝛼, 𝑘). Indeed, this would establish the claim with 𝑐 3 := 𝑁 −1 ∧ 𝑐. By a union bound, P 𝑝 𝑛 (E2 \E3 ) ≥ 𝑐 2 − 𝑐22 𝑐2 ≥ . 4 2 On the event E2 \E3 , we always have  𝑢 ∈ neigh(𝑥) : act𝜀𝑜 [𝑥𝑢] neigh(𝑥) ≥ ∥𝑆(𝑥) ∥ neigh(𝑥) ≥ 𝑐 2 . So there exists 𝑢 ∈ neigh(𝑥) such that   𝑐 2 𝜀 P 𝑝 𝑛 meso𝛿,𝜀 ∩ act [𝑥𝑢] ≥ · 𝑐 2 = 2𝑐. 𝑜 𝑜 2 Let 𝑁 be the smallest positive integer such that 2 𝑁1−1 ≤ 𝑐, and assume that 𝑛 ≥ 𝑁. By the third condition enumerated in Lemma 6.7.2, 𝜃𝜀 ( 𝑝𝑚 ) − 𝜃𝜀 ( 𝑝𝑛) ≤ 𝑚−1 ∑︁ 1 1 ≤ ≤ 𝑐. 𝑘 𝑁−1 2 2 𝑘=𝑛 Therefore, by a union bound and the monotone coupling of P 𝑝 𝑛 and P 𝑝 𝑚 ,     𝜀 𝛿,𝜀 𝜀 P 𝑝 𝑚 meso𝛿,𝜀 ∩ act [𝑥𝑢] ≥ P meso ∩ act [𝑥𝑢] − (𝜃 𝜀 ( 𝑝 𝑚 ) − 𝜃 𝜀 ( 𝑝 𝑛 )) ≥ 2𝑐 − 𝑐 = 𝑐, 𝑝 𝑜 𝑜 𝑜 𝑜 𝑛 and hence 𝐴𝑚 (1, 𝑐) holds. □ Claim 6.7.9. If |𝑉 | ≥ 𝑐−1 4 𝑑 and P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 , then P 𝑝 𝑛 (E) ≥ 𝑐 4 where E := E1 ∩ {|𝐾𝑥 | ≥ 𝑐 2 𝑑} . 1 Proof. By Lemma 6.2.2, there exists 𝑁 (𝜀, 𝑘, 𝛼) with 𝑁 ≥ 𝑐−1 1 such that if |𝑉 | ≥ 𝑁 then 𝑝 𝑛 ≥ 2𝑑 . Assume that |𝑉 | ≥ 𝑁 𝑑, which implies that |𝑉 | ≥ 𝑁 and |𝑉 | ≥ 𝑐−1 1 𝑑, and assume that P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 . Our goal is to find 𝑐(𝜀, 𝑘, 𝛼) ∈ (0, 𝑐 3 ] such that P 𝑝 𝑛 (E) ≥ 𝑐. Indeed, then the claim holds with 𝑐 4 := 𝑁 −1 ∧ 𝑐. On the event E3 , we know that |𝑆(𝑥)| ≥ 𝑐 2 𝑑. So we are trivially done, with 𝑐 := 𝑐23 , if P 𝑝 𝑛 (E3 ∩ {𝜔 ∩ 𝑥𝑆(𝑥) ≠ ∅}) ≥ 𝑐23 . So by a union bound, we may instead assume without loss of generality that 𝑐3 P 𝑝 𝑛 (E3 ∩ {𝜔 ∩ 𝑥𝑆(𝑥) = ∅}) ≥ . 2 254 Notice that if we start with a configuration in E3 ∩ {𝜔 ∩ 𝑥𝑆(𝑥) = ∅}, then open any edge in 𝑥𝑆(𝑥), we obtain a configuration in E. Therefore by Proposition 6.6.4 with 𝐹 := 𝑥 neigh(𝑥), 𝐴 := E3 ∩ {𝜔 ∩ 𝑥𝑆(𝑥) = ∅}, 𝜂 := 𝑐 2 , and 𝐹 [𝜔] := 𝑥𝑆(𝑥), P 𝑝 𝑛 (E) ≥  𝑐 2  𝑐 2 𝑐22 𝑝𝑛 𝑑 1/2 3 3 · · · ≥ 𝑐22 · =: 𝑐. 1 − 𝑝𝑛 𝑝𝑛 𝑑 + 1 2 1/2 + 1 2 □ For each vertex 𝑧, pick a map 𝜙 𝑧 ∈ Aut 𝐺 with 𝜙 𝑧 (𝑥) = 𝑧, which exists by transitivity, and define E 𝑧 := {𝜔 ◦ 𝜙 𝑧 ∈ E}. Claim 6.7.10. If |𝑉 | ≥ 𝑐−1 4 𝑑 and P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 , then P 𝑝 𝑛 (F1 ) ≥ 𝑐 5 where F1 := {∥{𝑧 ∈ 𝑉 : E 𝑧 holds}∥ ≥ 𝑐 5 } Proof. Assume that |𝑉 | ≥ 𝑐−1 4 𝑑 and P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 . By the previous claim and since every 𝜙 𝑧 is an automorphism, E 𝑝 𝑛 ∥{𝑧 ∈ 𝑉 : E 𝑧 holds}∥ = P 𝑝 𝑛 (E) ≥ 𝑐 4 . Therefore the claim follows from Markov’s inequality, with 𝑐 5 := 𝑐24 . □ For the remaining claims, we will (explicitly) assume that 𝑑 is sufficiently large in order to help with certain rounding errors. When 𝑑 is not this large, we will anyway be able to easily conclude directly. −1 Claim 6.7.11. If |𝑉 | ≥ 𝑐−1 4 𝑑 and 𝑑 ≥ 𝑐 6 , then for every 𝜔 ∈ F1 , there exists a collection of disjoint sets of vertices (B𝑖 : 𝑖 ∈ 𝐼) such that all of the following hold, where B := ∪𝑖∈𝐼 B𝑖 : 1. 𝜔 ∈ Ð 𝑧∈B E 𝑧 . 2. For each 𝑖 ∈ 𝐼, the set B𝑖 is 𝜔-connected. 3. {𝜙 𝑧 (𝑦) : 𝑧 ∈ B} is 𝜔-connected (where 𝑦 is the vertex from Claim 6.7.6). 4. 𝑐 6 ≤ |𝐼|𝑉|𝑑| ≤ 4𝑐 2 . 5. 𝑐 6 ≤ |𝐵𝑑𝑖 | ≤ 𝑐 2 for each 𝑖 ∈ 𝐼. 255 −1 Proof. Define 𝑐 6 := 𝑐2 𝑐65 𝜀 . Assume that |𝑉 | ≥ 𝑐−1 4 𝑑 and 𝑑 ≥ 𝑐 6 , and consider 𝜔 ∈ F1 . For every 𝑧 ∈ 𝑉, if 𝜔 ∈ E 𝑧 , then 𝜙 𝑧 (𝑦) ∈ giant𝜀 (𝜔). There cannot be more than 1/𝜀 clusters that each contains at least 𝜀 |𝑉 | vertices. So there exists a set of vertices 𝑌 with ∥𝑌 ∥ ≥ 𝑐 5 𝜀 such that Ð 𝜔 ∈ 𝑧∈𝑌 E 𝑧 and {𝜙 𝑧 (𝑦) : 𝑧 ∈ 𝑌 } is 𝜔-connected. Consider a cluster 𝐾 of 𝜔 with |𝐾 | ≥ 𝑐 2 𝑑. We claim that 𝐾 can be partitioned into sets of vertices 𝐴1 ⊔ · · · ⊔ 𝐴 𝑘 such that 𝑐23𝑑 ≤ | 𝐴𝑖 | ≤ 𝑐 2 𝑑 for every 𝑖. Indeed, it suffices to establish this when 𝐾 is the set of integers {1, . . . , 𝑁 } for some 𝑁 ≥ 𝑐 2 𝑑. Let 𝑞 := ⌊𝑐 2 𝑑/2⌋, and write 𝑁 = 𝑏𝑞 + 𝑟 where 𝑏 ∈ N and 𝑟 ∈ {1, . . . , 𝑞 − 1}. Now take the partition into the consecutive intervals {1, . . . , 𝑞}, . . . , {(𝑏 − 2)𝑞 + 1, . . . , (𝑏 − 1)𝑞} and {(𝑏 − 1)𝑞 + 1, . . . , 𝑁 }. Each interval has size 𝑞 or 𝑞 + 𝑟. Notice that thanks to our choice of 𝑐 6 , and since 𝑑 ≥ 𝑐−1 6 , we have 𝑐 2 𝑑 ≥ 6, and 𝑐2 𝑑 hence ⌊𝑐 2 𝑑/2⌋ ≥ 3 . Therefore, 𝑐2 𝑑 ≤ 𝑞 ≤ 𝑞 + 𝑟 ≤ 2𝑞 ≤ 𝑐 2 𝑑. 3 For every 𝑧 ∈ 𝑌 , since 𝜔 ∈ E 𝑧 , we know that |𝐾 𝑧 (𝜔)| ≥ 𝑐 2 𝑑. Thanks to the previous paragraph, by splitting large clusters, there therefore exists an equivalence relation ∼ on 𝑉 that is a refinement of 𝜔 ← → such that for every 𝑧 ∈ 𝑌 , the equivalence class of 𝑧 under ∼, say [𝑧], satisfies 𝑐2 𝑑 ≤ |[𝑧] | ≤ 𝑐 2 𝑑. 3 ∥𝑌 ∥ ≥ 𝑐 5 𝜀 In particular, using the bound 𝑑 ≥ 𝑐−1 6 for the second inequality, and using the bound for the third inequality, 𝑐2 𝑑 2 2 |𝑉 | min | [𝑧] | ≥ ≥ ≥ . 𝑧∈𝑌 |𝑌 | 3 𝑐5 𝜀 This allows us apply [EH21a, Lemma 4.12] where (𝑋, 𝑌 , ∼) := (𝑉, 𝑌 , ∼). This implies the existence of a collection of disjoint subsets (𝐵𝑖 : 𝑖 ∈ 𝐼) of 𝑌 such that |𝐼 | ≥ |𝑉 | |𝑉 | ≥ , 4 min𝑧∈𝑌 | [𝑧]| 4𝑐 2 𝑑 and for every 𝑖 ∈ 𝐼, the set 𝐵𝑖 is entirely contained in some equivalence class of ∼ (which may depend on 𝑖) and satisfies |𝐵𝑖 | ≥ |𝑌 | 𝑐5 𝜀 𝑐2 𝑑 min | [𝑧]| ≥ · = 𝑐 6 𝑑. 2 |𝑉 | 𝑧∈𝑌 2 3 Since every 𝐵𝑖 is entirely contained in some equivalence class of ∼, we also know that |𝐵𝑖 | ≤ 𝑐 2 𝑑, and therefore |𝐵𝑖 | 𝑐6 ≤ ≤ 𝑐2 . 𝑑 256 Since 𝐵𝑖 ∩ 𝐵 𝑗 = ∅ for all 𝑖 ≠ 𝑗, and every 𝐵𝑖 satisfies |𝐵𝑖 | ≥ 𝑐 6 𝑑, we know that |𝐼 | · 𝑐 6 𝑑 ≤ |𝑉 |, and therefore |𝑉 | 𝑐6 ≤ ≤ 4𝑐 2 . |𝐼 | 𝑑 □ −1 −1 Claim 6.7.12. There exists 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 such that if |𝑉 | ≥ 𝑐−1 4 𝑑, 𝑛 ≥ 𝑐 7 , 𝑑 ≥ 𝑐 6 , and P 𝑝 𝑛 (F1 ) ≥ 𝑐 5 , then 𝐴𝑚 (𝑘 − 1, 𝛽) holds. −1 Proof. Assume that |𝑉 | ≥ 𝑐−1 4 𝑑, 𝑑 ≥ 𝑐 6 , and P 𝑝 𝑛 (F1 ) ≥ 𝑐 5 . Recall that P denotes the standard monotone coupling of percolation measures. Our goal is to find 𝑁 (𝜀, 𝑘, 𝛼) < ∞ and 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 such that if 𝑛 ≥ 𝑁 then 𝐴𝑚 (𝑘 − 1, 𝛽) holds. Indeed, then the claim follows with 𝑐 7 := 𝑁 −1 ∧ 𝑐 6 . On the event F1 , 𝑐6 𝑐6 𝑐6 𝑑 ∥B ∥ ≥ |𝐼 | · min ∥B𝑖 ∥ ≥ |𝐼 | · ≥ . ≥ 𝑖∈𝐼 |𝑉 | 4𝑐 2 4 So there exists a fixed vertex 𝑧 such that the event E4 := F1 ∩ {𝑧 ∈ B} satisfies P 𝑝 𝑛 (E4 ) ≥ 𝑐54𝑐6 . Let 𝑁 (𝜀, 𝑘, 𝛼) be the smallest positive integer such that 2 𝑁1−1 ≤ 𝑐58𝑐6 , and assume that 𝑛 ≥ 𝑁. Then as in the proof of Claim 6.7.8, by the third condition enumerated in Lemma 6.7.2, 𝜃 𝜀 ( 𝑝 𝑚 )−𝜃 𝜀 ( 𝑝 𝑛 ) ≤ 𝑐58𝑐6 .
So by a union bound, since 𝜔 𝑝 𝑛 ∈ E4 implies that 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑛 , P  

𝑐5 𝑐6 𝜔 𝑝 𝑛 ∈ E4 ∩ 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑚 ≥ . 8 (6.7.2)
When 𝜔 𝑝 𝑛 ∈ E4 and 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑚 , then by monotonicity of the coupling, 𝜔 𝑝 𝑚 ∈ meso𝛿,𝜀 . 𝜙 𝑧 (𝑜) Therefore, using the fact that 𝜙 𝑧 is an automorphism for the first equality,     𝛿,𝜀 𝜀 𝜀 [𝑥𝑦] [𝜙 = P meso ∩ act (𝑥𝑦)] P 𝑝 𝑚 meso𝛿,𝜀 ∩ act 𝑝𝑚 𝑧 𝑜 𝑜 𝜙 𝑧 (𝑜) 𝜙 𝑧 (𝑜)  n o 
𝜀 ≥ P 𝜔 𝑝 𝑛 ∈ E4 ∩ 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑚 ∩ 𝜔 𝑝 𝑚 ∈ act𝜙 𝑧 (𝑜) [𝜙 𝑧 (𝑥𝑦)] . In particular, if o 𝑐 𝑐  n 
5 6 P 𝜔 𝑝 𝑛 ∈ E4 ∩ 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑚 ∩ 𝜔 𝑝 𝑚 ∈ act𝜀𝜙 𝑧 (𝑜) [𝜙 𝑧 (𝑥𝑦)] ≥ , 16 (6.7.3) 5 𝑐6 then we are done because 𝐴𝑚 (1, 𝛽) holds, and hence 𝐴𝑚 (𝑘 − 1, 𝛽) holds, with 𝛽 := 𝑐16 . So we may assume that eq. (6.7.3) is false, and therefore by eq. (6.7.2) and a union bound, that the event o n 
A1 := 𝜙 𝑧 (𝑜) ∉ giant𝜀 𝜔 𝑝 𝑚 ∩ 𝜔 𝑝 𝑚 ∉ act𝜀𝜙 𝑧 (𝑜) [𝜙 𝑧 (𝑥𝑦)] 257 satisfies P 
𝑐5 𝑐6 . 𝜔 𝑝 𝑛 ∈ E4 ∩ A1 ≥ 16 On the event that 𝜔 𝑝 𝑛 ∈ E4 , let B∗ denote the class B𝑖 from the collection (B𝑖 : 𝑖 ∈ 𝐼) (defined with respect to 𝜔 𝑝 𝑛 ) that contains 𝑧, and define the event  A2 := 𝜔 𝑝 𝑚 ∩ {𝜙𝑢 (𝑥𝑦) : 𝑢 ∈ B∗ } ≠ ∅ . Suppose we condition on 𝜔 𝑝 𝑛 and find that 𝜔 𝑝 𝑛 ∈ E4 , then condition on the restriction of 𝜔 𝑝 𝑚 to 𝐸\ {𝜙𝑢 (𝑥𝑦) : 𝑢 ∈ 𝐵∗ }. Then the events A1 and A2 are both increasing events of the unrevealed edges, i.e. the restriction of 𝜔 𝑝 𝑚 to {𝜙𝑢 (𝑥𝑦) : 𝑢 ∈ 𝐵∗ }. The conditional law of this restriction of 𝜔 𝑝 𝑚 is still a product measure, and in particular almost surely satisfies the FKG inequality. Since there are at least |B2∗ | ≥ 𝑐62𝑑 unrevealed edges, the conditional probability that A2 will occur is almost surely at least  𝑝𝑚 − 𝑝𝑛 1− 1− 1 − 𝑝𝑛  | B2∗ |  ≥ 1− 1− 𝜀 3𝑛+3 𝑑  𝑐62𝑑 − 𝑐6 𝜀 ≥ 1 − 𝑒 2·3𝑛+3 =: 𝛽1 , where the first inequality used the second condition enumerated in Lemma 6.7.2. Notice that if A1 and A2 both occur after revealing the remaining edges, then (using that 𝜔 𝑝 𝑛 ∈ E4 ) 𝜔 𝑝 𝑚 ∈ meso𝛿,𝜀 ∩ act𝜀𝜙 𝑧 (𝑜) [𝜙 𝑧 (𝐹\{𝑥𝑦})] . 𝜙 (𝑜) 𝑧 Therefore, using that 𝜙 𝑧 is an automorphism in the first line and applying the conditional FKG inequality to obtain the final inequality,     𝛿,𝜀 𝜀 𝛿,𝜀 𝜀 P 𝑝 𝑚 meso𝑜 ∩ act𝑜 [𝐹\{𝑥𝑦}] = P 𝑝 𝑚 meso𝜙 (𝑜) ∩ act𝜙 𝑧 (𝑜) [𝜙 (𝐹\{𝑥𝑦})] 𝑧 
≥ P 𝜔 𝑝 𝑛 ∈ E4 ∩ A1 ∩ A2 
𝛽1 𝑐 5 𝑐 6 ≥ 𝛽1 · P 𝜔 𝑝 𝑛 ∈ E 4 ∩ A 1 ≥ =: 𝛽, 16 and hence 𝐴𝑚 (𝑘 − 1, 𝛽) holds. □ We are now ready to complete the proof of the lemma. Recall that our goal is to find 𝑁 (𝜀, 𝑘, 𝛼) < ∞ and 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 such that if |𝑉 | ≥ 𝑁 𝑑 and 𝑛 ≥ 𝑁, then 𝐴𝑚 (𝑘 − 1, 𝛽) must hold. We will split our argument into two cases according to whether or not 𝑑 ≥ 𝑐−1 6 . (The ultimate choices of 𝑁 and 𝛽 are obtained by taking the maximum/minimum of the constants in each case.) 258 |𝑉 | ≥ 𝑁1 First suppose that 𝑑 < 𝑐−1 6 . By Lemma 6.2.2, there exists 𝑁1 (𝜀, 𝑘, 𝛼) < ∞ such that if 1 ≥ 𝑐26 . By Lemma 6.4.3, if 𝑝 𝑛 ≥ 𝑐26 then there exists an edge 𝑢𝑣 ∈ 𝐹 such that then 𝑝 𝑛 ≥ 2𝑑     𝑝 |𝐹 | 𝜀 P 𝑝 𝑛 meso𝑢𝛿,𝜀 ∩ act𝑢𝜀 [𝑢𝑣] ≥ 𝑛 P 𝑝 𝑛 meso𝛿,𝜀 ∩ act [𝐹] 𝑜 𝑜 2 |𝐹 | 𝑐6
𝑘 ≥ 2 2𝑘 · 𝛼 =: 𝑐 8 (𝜀, 𝑘, 𝛼). As argued in the proof of Claim 6.7.8, by the third condition of Lemma 6.7.2, there exists 𝑁2 (𝜀, 𝑘, 𝛼) < ∞ such that if 𝑛 ≥ 𝑁2 then 𝜃 𝜀 ( 𝑝 𝑚 ) − 𝜃 𝜀 ( 𝑝 𝑛 ) ≤ 𝑐28 . Set 𝑁 := 𝑁1 ∨ 𝑁2 , and assume that |𝑉 | ≥ 𝑁 𝑑 (hence |𝑉 | ≥ 𝑁) and 𝑛 ≥ 𝑁. Then by a union bound,     𝑐8 𝑐8 P 𝑝 𝑚 meso𝑢𝛿,𝜀 ∩ act𝑢𝜀 [𝑢𝑣] ≥ P 𝑝 𝑛 meso𝑢𝛿,𝜀 ∩ act𝑢𝜀 [𝑢𝑣] − (𝜃 𝜀 ( 𝑝 𝑚 ) − 𝜃 𝜀 ( 𝑝 𝑛 )) ≥ 𝑐 8 − = , 2 2 and hence, by transitivity, 𝐴𝑚 (1, 𝛽) holds (and hence 𝐴𝑚 (𝑘 − 1, 𝛽) holds) with 𝛽 := 𝑐28 . Now suppose that 𝑑 ≥ 𝑐−1 6 . Let 𝛽(𝜀, 𝑘, 𝛼, 𝑛) > 0 be the constant from Claim 6.7.12, which we may assume satisfies 𝛽 ≤ 𝑐 7 . Set 𝑁 := 𝑐−1 7 , and assume that |𝑉 | ≥ 𝑁 𝑑 and 𝑛 ≥ 𝑁. By Claim 6.7.8, either 𝐴𝑚 (1, 𝑐 3 ) holds or P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 . If 𝐴𝑚 (1, 𝑐 3 ) holds then 𝐴𝑚 (1, 𝛽) holds (and hence 𝐴𝑚 (𝑘 − 1, 𝛽) holds), so we may instead assume that P 𝑝 𝑛 (E3 ) ≥ 𝑐 3 . So by Claim 6.7.10, P 𝑝 𝑛 (F1 ) ≥ 𝑐 5 . Therefore by Claim 6.7.12, 𝐴𝑚 (𝑘 − 1, 𝛽) holds, as required. 6.8 The non-macromolecular case The goal of this section is to prove the following proposition, which says that for supercritical percolation on a high-degree non-macromolecular finite transitive graph, 𝑜 is unlikely to belong to a mesoscopic cluster. Proposition 6.8.1. Let H ⊆ F be an infinite set of (isomorphism classes of) finite, connected, vertex transitive graphs with lim𝐺∈H deg 𝐺 = ∞ that contains no infinite macromolecular subsets. Then for every supercritical 𝑝 : H → [0, 1], lim lim sup P 𝑝 (|𝐾𝑜 | ≥ 𝑛, 𝑜 ∉ giant) = 0. 𝑛→∞ (6.8.1) 𝐺∈H Equivalently, we must show that if the convergence claimed in eq. (6.8.1) fails, then we can construct suitable macromolecular decompositions for infinitely many of the graphs in H . In Section 6.8, we show that we can construct these macromolecular decompositions if the giant cluster does not behave as it should in the neighbourhood of the origin. In Section 6.8, we will then show that if the giant cluster does behave as it should in the neighbourhood of the origin, then we 259 can deduce eq. (6.8.1). Together, these yield Proposition 6.8.1. In Section 6.11, we will show that Proposition 6.8.1 implies our main theorems in the case of high-degree non-macromolecular graphs. Macromolecular graphs and the neighbourhood of the origin The goal of this subsection is to prove the following proposition. Informally, this proposition states that if we consider supercritical percolation on a high-degree finite transitive graph that is not macromolecular, then the giant cluster includes a positive proportion of the neighbours of the origin with high probability. (This is not the case for macromolecular graphs, since the “local giants” we see near the origin may fail to be included in the global giant.) Proposition 6.8.2. Let H ⊆ F be an infinite set of (isomorphism classes of) finite, connected, vertex transitive graphs with lim𝐺∈H deg 𝐺 = ∞ that contains no infinite macromolecular subsets. Then   lim lim inf P 𝑝 ∥giant∥ neigh(𝑜) ≥ 𝛼 = 1 𝛼↓0 H for every supercritical 𝑝 : H → [0, 1]. When the graphs in H are dense, then the result follows easily from supercritical uniqueness Theorem 6.1.1. So for most of the proof of this proposition, we may assume that the graphs are sparse. The proof of this proposition will rely on the following definition. Let 𝐺 = (𝑉, 𝐸) ∈ F , let 𝑥 be a vertex of 𝐺, and let 𝛼, 𝛽 ∈ (0, 1). Given a configuration 𝜔 ∈ {0, 1} 𝐸 , we say that 𝑥 is (𝛼, 𝛽)-happy if there is an 𝜔-connected subset 𝑋 ⊆ neigh(𝑥) with ∥ 𝑋 ∥ neigh(𝑥) ≥ 𝛼 such that 𝑋 ⊆ giant 𝛽 , and say that 𝑥 is (𝛼, 𝛽)-sad if there is an 𝜔-connected subset 𝑋 ⊆ neigh(𝑥) with ∥ 𝑋 ∥ neigh(𝑥) ≥ 𝛼 such that 𝑋 ⊈ giant 𝛽 . (A vertex can be both happy and sad, or neither happy nor sad.) Given 𝑥, 𝑦 ∈ 𝑉 and parameters 𝛼, 𝛽, 𝛾, 𝑝 ∈ (0, 1), we say that 𝑥 and 𝑦 are (𝛼, 𝛽, 𝛾, 𝑝)-friends if P 𝑝 ( [{𝑥 is (𝛼, 𝛽)-happy} ∩ {𝑦 is (𝛼, 𝛽)-sad}] ∪ [{𝑦 is (𝛼, 𝛽)-happy} ∩ {𝑥 is (𝛼, 𝛽)-sad}]) ≤ 𝛾. That is, 𝑥 and 𝑦 are friends if it is unlikely that one is happy while the other is sad. We will eventually see that the subgraph spanned by edges between friends can be identified, in an appropriate sense, with the macromolecular structure of 𝐺. Our first lemma gives conditions under which a vertex has at most 𝑂 (1) neighbours that it is not friends with. Lemma 6.8.3. For all 𝛼, 𝛽, 𝛾, 𝛿 ∈ (0, 1), there exists 𝜀 > 0 and 𝑁 < ∞ such that the following holds. Let 𝐺 be a finite connected transitive graph with deg ≥ 𝜀 −1 that is not 𝜀-molecular. Suppose 260 that 𝑝 ∈ (0, 1) satisfies 𝜃 𝛽 ((1 − 𝛿) 𝑝) ≥ 𝛿 and 𝑝𝜃 ′𝛽 ( 𝑝) ≤ 1𝛿 . Then |{𝑥 ∈ neigh(𝑜) : 𝑜 and 𝑥 are not (𝛼, 𝛽, 𝛾, 𝑝)-friends}| ≤ 𝑁. Proof. Let 𝐺 ∈ F . Fix 𝛼, 𝛽, 𝛾, 𝛿, 𝑝 ∈ (0, 1) such that 𝜃 𝛽 ((1 − 𝛿) 𝑝) ≥ 𝛿 and 𝑝𝜃 ′𝛽 ( 𝑝) ≤ 1/𝛿, and suppose 𝜀 > 0 is such that deg ≥ 𝜀 −1 and that 𝐺 is not 𝜀-molecular. By Lemma 6.2.2, there exists 1 . We will write that two vertices are friends to 𝜀 0 = 𝜀0 (𝛿, 𝛽) > 0 such that if 𝜀 ≤ 𝜀0 then 𝑝 ≥ 2 deg mean that they are (𝛼, 𝛽, 𝛾, 𝑝)-friends. Fix an edge 𝑥𝑦 ∈ 𝐸 and assume that 𝑥 and 𝑦 are not friends. By a union bound, we may assume without loss of generality (swapping 𝑥 and 𝑦 if necessary) that the event E that 𝑥 is (𝛼, 𝛽)-happy but 𝑦 is (𝛼, 𝛽)-sad satisfies P 𝑝 (E) ≥ 𝛾/2. On this event, let 𝑋 be an 𝜔-connected subset of neigh(𝑥) with ∥ 𝑋 ∥ neigh(𝑥) ≥ 𝛼 that is contained in giant 𝛽 , and let 𝑌 be an 𝜔-connected subset of neigh(𝑦) with ∥𝑌 ∥ neigh(𝑦) ≥ 𝛼 that is not contained in giant 𝛽 . By Theorem 6.1.1, there exists 𝜀1 = 𝜀 1 (𝛽, 𝛾, 𝛿) ∈ (0, 𝜀0 ) such that if 𝜀 ≤ 𝜀1 then P 𝑝 (∥𝐾2 ∥ ≥ 𝛽/2) ≤ 𝛾/4, so that if 𝜀 ≤ 𝜀1 then by a union bound P 𝑝 (E ∩ {∥𝐾2 ∥ < 𝛽/2}) ≥ 𝛾/4. If E occurs and 𝑦 ∈ giant 𝛽 , then opening any of the edges between 𝑦 and 𝑌 causes giant 𝛽 to increase in size by at least 𝛼 deg. Thus, using transitivity, Russo’s formula implies that  1  𝜃 ′𝛽 ( 𝑝) ≥ P 𝑝 E ∩ {𝑦 ∈ giant 𝛽 } (𝛼 deg) 2 , 2 1 and since 𝑝 ≥ 2 deg and 𝑝𝜃 ′𝛽 ( 𝑝) ≤ 𝛿−1 it follows that P 𝑝 (E ∩ {𝑦 ∈ giant 𝛽 }) ≤ 4 𝛼2 𝛿 deg ≤ 4𝜀 . 𝛼2 𝛿 Thus, if 𝜀 ≤ 𝜀2 = min{𝜀1 , 𝛼2 𝛾𝛿/32} then P 𝑝 (E ∩ {𝑦 ∈ giant 𝛽 }) ≤ 𝛾/8 and hence by a union bound,    P 𝑝 E ∩ 𝑦 ∉ giant 𝛽 ∩ {∥𝐾2 ∥ < 𝛽/2} ≥ 𝛾/8. By applying quantitative insertion-tolerance (Proposition 6.6.4) to open an edge in 𝑦𝑌 , we deduce that there is a constant 𝑐 1 = 𝑐 1 (𝛼, 𝛾) > 0 such that if 𝜀 ≤ 𝜀2 then the event E1 := {𝑥 is (𝛼, 𝛽)-happy} ∩ {𝑦 ∉ giant 𝛽 } ∩ { 𝐾 𝑦 neigh(𝑦) ≥ 𝛼} satisfies P 𝑝 (E1 ) ≥ 𝑐 1 . If the event E1 ∩ {𝑥 ↔ 𝑦} holds, then 𝑥 is not connected to 𝑋, and opening any edge between 𝑥 and 𝑋 causes the size of giant 𝛽 to increase by at least 𝛼 deg. Thus, as above, Russo’s formula together with transitivity imply that 𝑝𝜃 ′𝛽 ( 𝑝) ≥ 𝛼2 deg P 𝑝 (E1 ∩ {𝑥 ↔ 𝑦}) , 4 261 so that P 𝑝 (E1 ∩ {𝑥 ↔ 𝑦}) ≤ 4𝜀 . 𝛼2 𝛿 Thus, if we define 𝜀3 = min{𝜀2 , 𝑐 1 𝛼2 𝛿/8} then by a union bound P 𝑝 (E1 ∩ {𝑥 ↮ 𝑦}) ≥ 𝑐1 2 whenever 𝜀 ≤ 𝜀 3 . Applying quantitative insertion-tolerance to open an edge between 𝑥 and 𝑋, we deduce that there is a constant 𝑐 2 = 𝑐 2 (𝛼, 𝛾) > 0 such that the event E2 := {∥𝐾𝑥 ∥ neigh(𝑥) ≥ 𝛼} ∩ {𝑥 ∈ giant 𝛽 } ∩ { 𝐾 𝑦 neigh(𝑦) ≥ 𝛼} ∩ {𝑦 ∉ giant 𝛽 } satisfies P 𝑝 (E2 ) ≥ 𝑐 2 whenever 𝜀 ≤ 𝜀3 . In particular, the unordered pair 𝑥𝑦 = {𝑥, 𝑦} satisfies h i   𝑀 (𝑥𝑦) := E 𝑝 𝐾 𝑦 1act ∥ 𝐾 𝑦 ∥ ≥𝛽 [𝑥𝑦] + E 𝑝 |𝐾𝑥 | 1act ∥ 𝐾 𝑥 ∥ ≥𝛽 [𝑥𝑦] ≥ 𝑐 2 𝛼 deg whenever 𝜀 ≤ 𝜀3 . If we define E𝑥 = {𝑦 ∈ neigh(𝑥) : 𝑥 and 𝑦 are not friends} then this estimate holds for every 𝑥 ∈ 𝑉 and 𝑦 ∈ E𝑥 provided that 𝜀 ≤ 𝜀3 . Using transitivity and summing over this estimate, we obtain that if 𝜀 ≤ 𝜀3 then 𝜃 ′𝛽 ( 𝑝) ≥   1 ∑︁ ∑︁ 𝑐2 𝛼 1 ∑︁ ∑︁ E 𝑝 |𝐾𝑥 | 1act ∥ 𝐾 𝑥 ∥ ≥𝛽 [𝑥𝑦] = deg |E𝑜 |. 𝑀 (𝑥𝑦) ≥ |𝑉 | 𝑥∈𝑉 2 |𝑉 | 𝑥∈𝑉 2 𝑦∈E 𝑥 𝑦∈E 𝑥 Since 𝑝 ≥ 1/(2 deg) and 𝑝𝜃 ′𝛽 ( 𝑝) ≤ 𝛿−1 , it follows that if 𝜀 ≤ 𝜀3 then |E𝑜 | ≤ 4(𝑐 2 𝛼𝛿) −1 . This completes the proof. □ We now investigate the subgraph spanned by edges between friends. We say that 𝑥 and 𝑦 are (𝛼, 𝛽, 𝛾, 𝑝)−linked if they are connected in the subgraph of 𝐺 spanned by those edges of 𝐺 whose endpoints are (𝛼, 𝛽, 𝛾, 𝑝)-friends, and define pop(𝛼, 𝛽, 𝛾, 𝑝) by pop(𝛼, 𝛽, 𝛾, 𝑝) := 1 |{𝑥 ∈ 𝑉 : 𝑜 and 𝑥 are (𝛼, 𝛽, 𝛾, 𝑝)-linked}| . deg Note that this quantity can be larger than 1. Lemma 6.8.4. For all 𝛼, 𝛽, 𝛾, 𝛿 ∈ (0, 1), there exists 𝜀 > 0 such that the following holds. Let 𝐺 be a finite connected transitive graph with deg 𝐺 ≥ 𝜀 −1 that is not 𝜀-dense. Suppose that 𝑝 ∈ (0, 1) satisfies 𝜃 𝛽 ((1−𝛿) 𝑝) ≥ 𝛿, 𝑝𝜃 ′𝛽 ( 𝑝) ≤ 𝛿−1 , and pop(𝛼, 𝛽, 𝛾, 𝑝) ≤ 𝛿−1 . Then 𝐺 is 𝜀-macromolecular. Proof. Let 𝐺 and 𝑝 satisfy the given hypotheses, for some constant 𝜀 > 0 to be determined. Write pop for pop(𝛼, 𝛽, 𝛾, 𝑝). Say that vertices 𝑥 and 𝑦 are linked to mean that they are (𝛼, 𝛽, 𝛾, 𝛿)-linked, and note that this induces an Aut 𝐺-invariant equivalence relation on 𝑉 (𝐺). First suppose that this 262 equivalence relation is trivial, i.e. every vertex is linked to every other vertex. By definition of pop, every equivalence class has size pop · deg 𝐺. Therefore, |𝐸 (𝐺)| |𝐺 | 2 = deg 𝐺 𝛿 deg 𝐺 = ≥ , 2 |𝐺 | 2 pop · deg 𝐺 2 i.e. 𝐺 is 𝛿/2-dense. If 𝜀 < 𝜀 0 := 𝛿/2, then this is impossible (since 𝐺 is not 𝜀-dense), and hence the equivalence relation of being linked must be non-trivial. Let ( 𝐴, 𝐵) be the corresponding macromolecular decomposition. By Lemma 6.8.3, there exists 𝜀1 = 𝜀 1 (𝛼, 𝛽, 𝛾, 𝛿) ∈ (0, 𝜀0 ) and 𝑁 (𝛼, 𝛽, 𝛾, 𝛿) < ∞ such that if 𝜀 < 𝜀1 then (since 𝐺 being not 𝜀-dense implies that 𝐺 is not 𝜀-molecular) |{𝑥 ∈ neigh(𝑜) : 𝑜 and 𝑥 are not (𝛼, 𝛽, 𝛾, 𝑝)-friends}| ≤ 𝑁. In particular, by transitivity, since every pair of friends is trivially linked, |𝐸 (𝐵)| 𝑁 ≤ . |𝐺 | 2 1 Similarly, deg 𝐴 ≥ deg 𝐺 − 𝑁. Hence if 𝜀 ≤ 2𝑁 , which implies that deg 𝐺 ≥ 2𝑁, then   |𝐸 ( 𝐴)| deg 𝐴 deg 𝐺 − 𝑁 𝛿 𝑁 𝛿 = ≥ = 1− ≥ . 2 2 pop · deg 𝐺 (2/𝛿) deg 𝐺 2 deg 𝐺 4 | 𝐴|   1 𝛿 Therefore, if 𝜀 < 𝜀2 := min 𝜀1 , 𝑁2 , 2𝑁 , 4 , then ( 𝐴, 𝐵) is an 𝜀-macromolecular decomposition. □ Lemma 6.8.5. Let 𝛼, 𝛽, 𝛾, 𝑝 ∈ (0, 1) and let 𝑚 ∈ N. Let 𝐺 be a finite connected graph and suppose that 𝑣 1 , . . . , 𝑣 𝑚 is a sequence of vertices such that 𝑣 𝑖 and 𝑣 𝑖+1 are (𝛼, 𝛽, 𝛾, 𝑝)-friends for all 𝑖 ∈ {1, . . . , 𝑚 − 1}. Then ! 𝑚 Ø P 𝑝 {𝑣 1 is (𝛼, 𝛽)-sad} ∩ {𝑣 𝑖 is (𝛼, 𝛽)-happy} ≤ 𝑚𝛾. 𝑖=2 Proof. We have by a union bound that 𝑚  𝑚−1 Ø ∑︁ P 𝑝 {𝑣 1 is (𝛼, 𝛽)-sad}∩ {𝑣 𝑖 is (𝛼, 𝛽)-happy} ≤ P 𝑝 (𝑣 𝑖 is (𝛼, 𝛽)-sad, 𝑣 𝑖+1 is (𝛼, 𝛽)-happy) ≤ 𝑚𝛾  𝑖=2 𝑖=1 as claimed, where the second inequality follows from the definition of (𝛼, 𝛽, 𝛾, 𝑝)-friends. □ Lemma 6.8.6. For all 𝛽, 𝛿, 𝜂 ∈ (0, 1), there exists 𝛼 > 0 such that for every finite connected transitive graph 𝐺, for every parameter 𝑝 ∈ (0, 1), if 𝜃 𝛽 ( 𝑝) ≥ 𝛿 then P 𝑝 (𝑜 is neither (𝛼, 𝛽)-happy nor (𝛼, 𝛽)-sad) ≤ 𝜂. 263 Proof. Suppose that 𝜃 𝛽 ( 𝑝) ≥ 𝛿. Then by transitivity, E 𝑝 giant 𝛽 neigh(𝑜) ≥ 𝛿, and hence by   Markov’s inequality, P 𝑝 giant 𝛽 neigh(𝑜) ≥ 2𝛿 ≥ 2𝛿 . Now there can never be more than (1/𝛽)many clusters each having density at least 𝛽. Thus, 𝜆 := max ∥𝐾𝑢 ∥ neigh(𝑜) 𝑢∈𝑉 (𝐺)   𝛽𝛿2 𝛿 satisfies P 𝑝 𝜆 ≥ 𝛽𝛿 ≥ , and in particular, E [𝜆] ≥ 𝑝 2 2 4 . So by the universal tightness theorem (Theorem 6.5.4), there is a universal constant 𝐶 < ∞ such that for all 𝛼 > 0, P 𝑝 (𝜆 ≤ 𝛼) ≤ 𝐶 · 4𝛼 . 𝛽𝛿2 2 Now the conclusion follows by choosing 𝛼 := 𝜂𝛽𝛿 4𝐶 , noting that 𝑜 is neither (𝛼, 𝛽)-happy nor (𝛼, 𝛽)-sad if and only if 𝜆 < 𝛼. □ We next prove the following deterministic graph theory lemma, via a probabilistic argument. Lemma 6.8.7. For all 𝑘 ∈ N there exist 𝜀 > 0 and 𝑚 ∈ N such that if 𝐺 is a finite, connected, regular graph that is not 𝜀-dense and has at least one edge, then 𝐺 contains a (possibly self-intersecting) path of vertices 𝑣 1 , . . . , 𝑣 𝑚 satisfying 𝑚 Ø neigh(𝑣 𝑖 ) ≥ 𝑘 deg . 𝑖=1 Proof. We claim that the result holds with 𝑚 := 6𝑘 2 and 𝜀 := (2𝑚) −1 . Let 𝐺 = (𝑉, 𝐸) be a finite connected regular graph that is not 𝜀-dense and has at least one edge. We will prove the claim by case analysis according to whether diam 𝐺 ≥ 3𝑘. First suppose that diam 𝐺 ≥ 3𝑘. Let 𝑣 1 , . . . , 𝑣 3𝑘 be the first 3𝑘 vertices in a geodesic path between two points of maximal distance in 𝐺. Extend the path 𝑣 1 , . . . , 𝑣 3𝑘 in an arbitrary way to a path 𝑣 1 , . . . , 𝑣 𝑚 , which has the correct length. (For example, repeatedly cross the edge 𝑣 3𝑘−1 𝑣 3𝑘 .) Since 𝑣 1 , . . . , 𝑣 3𝑘 is a geodesic, we must have that neigh(𝑣 𝑖 ) ∩ neigh(𝑣 𝑗 ) = ∅ for all 1 ≤ 𝑖, 𝑗 ≤ 3𝑘 with |𝑖 − 𝑗 | ≥ 3, so 𝑘 𝑚 ∑︁ Ø neigh(𝑣 𝑖 ) ≥ neigh(𝑣 3 𝑗 ) = 𝑘 deg 𝑖=1 𝑗=1 as required. 264 Now suppose that diam 𝐺 ≤ 3𝑘. Let P be the law of an independent sequence 𝑢 1 , . . . , 𝑢 2𝑘 of vertices in 𝑉 each chosen uniformly at random, so that 1 ∑︁ deg 2 |𝐸 | 1 1𝑣∈neigh(𝑢) = = ≤ 2𝜀 = 2 |𝑉 | 𝑢∈𝑉 |𝑉 | 𝑚 |𝑉 | P (𝑣 ∈ neigh(𝑢𝑖 )) = for each 𝑣 ∈ 𝑉 and 1 ≤ 𝑖 ≤ 2𝑘. It follows by independence and linearity of expectation that  deg E neigh(𝑢𝑖 ) ∩ neigh(𝑢 𝑗 ) = P (𝑣 ∈ neigh(𝑢𝑖 )) P 𝑣 ∈ neigh(𝑢 𝑗 ) = |𝑉 | |𝑉 | 𝑣∈𝑉 ∑︁
2 ≤ deg 𝑚 for every 𝑖 ≠ 𝑗 and hence by inclusion-exclusion that E 2𝑘 Ø neigh(𝑢𝑖 ) ≥ 2𝑘 deg − ∑︁ E neigh(𝑢𝑖 ) ∩ neigh(𝑢 𝑗 ) ≥ 2𝑘 deg −(2𝑘) 2 deg ≥ 𝑘 deg . 𝑚 𝑖≠ 𝑗 𝑖=1 Ð2𝑘 In particular, there is a deterministic sequence of vertices 𝑢˜ 1 , . . . , 𝑢˜ 2𝑘 such that | 𝑖=1 neigh( 𝑢˜𝑖 )| ≥ 𝑘 deg. By picking a geodesic (which necessarily has length ≤ 3𝑘) from 𝑢˜𝑖 to 𝑢˜𝑖+1 for each 𝑖, then extending arbitrarily to obtain the correct length, we can find a path 𝑣 1 , . . . , 𝑣 𝑚 that contains 𝑢˜ 1 , . . . , 𝑢˜ 2𝑘 as a subsequence and hence satisfies 𝑚 Ø neigh(𝑣 𝑖 ) ≥ 𝑖=1 2𝑘 Ø neigh( 𝑢˜𝑖 ) ≥ 𝑘 deg 𝑖=1 as desired. □ We are now ready to complete the proof of Proposition 6.8.2. In addition to the lemmas from the present subsection, our proof will also apply Proposition 6.7.1 from the previous section, stating that sets of larger than degree order have large intersection with the giant with high probability. Proof of Proposition 6.8.2. Let 𝑞 : H → [0, 1] be a supercritical assignment of parameters. Our goal is to prove that   lim lim inf P𝑞 ∥giant∥ neigh(𝑜) ≥ 𝛼 = 1. (6.8.2) 𝛼↓0 H We may assume without loss of generality that 𝑞(𝐺) < 1 for all 𝐺 ∈ H . In particular, we may assume that there exists a constant 𝛿0 > 0 such that for every 𝐺 ∈ H , 𝑞(𝐺) is 𝛿0 -supercritical for 𝐺. By Lemma 6.3.4 applied to the interval 𝐼 = 𝐼 (𝐺) := [(1 − 𝛿0 /2)𝑞, (1 + 𝛿0 /2)(1 − 𝛿0 /2)𝑞],  n o  2𝜀 ′ L (𝐼) for all 𝐺 ∈ H and all 𝜀 > 0, L 𝑝 ∈ 𝐼 : 𝑝𝜃 𝛿0 /8 ( 𝑝) ≤ 1/𝜀 ≥ 1 − 𝛿0 /2 265 where L denotes the Lebesgue measure. In particular, for all 𝐺 ∈ H there exists 𝑝 = 𝑝(𝐺) ∈ 𝐼 such that 𝑝𝜃 ′𝛿0 /8 ( 𝑝) ≤ 1/𝛿 where 𝛿 := 𝛿0 /8. Note that 𝑝 is always 𝛿-supercritical because 𝑞 is 𝛿0 -supercritical and 𝑝 ≥ (1 − 𝛿0 /2)𝑞. By Theorem 6.1.1 and Theorem 6.2.1,
lim P𝑞 giant = giant𝛿 = 1. H So by monotonicity, since 𝑝(𝐺) ≤ 𝑞(𝐺) for all 𝐺 ∈ H , it suffices to establish that   lim lim inf P 𝑝 giant𝛿 neigh(𝑜) ≥ 𝛼 = 1. 𝛼↓0 H To this end, fix an arbitrary constant 𝜂 > 0, and note that the event that giant𝛿 neigh(𝑜) ≥ 𝛼 is the same as the event that 𝑜 is (𝛼, 𝛿)-happy, for all 𝛼 > 0. We will find a constant 𝛼 > 0 such that for all but finitely many 𝐺 ∈ H , P 𝑝 (𝑜 is (𝛼, 𝛿)-happy) ≥ 1 − 𝜂. (6.8.3) By Proposition 6.7.1, there exists 𝜀0 > 0 such that for all 𝐺 ∈ H , for every set of vertices 𝐴 ⊆ 𝑉 (𝐺) with | 𝐴| ≥ 𝜀0−1 deg 𝐺,   𝜂 P 𝑝 giant𝛿 𝐴 ≥ 𝜀0 ≥ 1 − . 3 By Lemma 6.8.7, there exists 𝜀1 > 0 and 𝑚 ∈ N such that every finite, connected, regular graph 𝐺 that is not 𝜀1 -dense and has at least one edge must contain a path of vertices 𝑣 1 , . . . , 𝑣 𝑚 satisfying 𝑚 Ø neigh(𝑣 𝑖 ) ≥ 𝑖=1 2 deg 𝐺. 𝜀0 By Lemma 6.8.6, there exists 𝛼0 > 0 such that for all 𝐺 ∈ H , 𝜂 P 𝑝 (𝑜 is (𝛼0 , 𝛿)-happy or (𝛼0 , 𝛿)-sad) ≥ 1 − , 3 (6.8.4) and by monotonicity, this inequality also holds with any 𝛼 ∈ (0, 𝛼0 ) in place of 𝛼0 . Define 𝜂 𝛼 := min(𝛼0 , 1/𝑚) and 𝛾 := 3𝑚 . Now by Lemma 6.8.4, there exists 𝜀2 > 0 such that for all but finitely many 𝐺 ∈ H (namely, every 𝐺 ∈ H with deg 𝐺 ≥ 1/𝜀2 that is not 𝜀2 -macromolecular), (Case A) 𝐺 is 𝜀2 -dense or (Case B) 𝐺 satisfies pop(𝛼, 𝛿, 𝛾, 𝑝) ≥ 1/𝜀 1 . We will establish eq. (6.8.3) in each of these two cases in turn. Case A Note that when 𝑜 is (𝛼, 𝛿)-sad, there exists a cluster 𝐾 ⊈ giant𝛿 satisfying ∥𝐾 ∥ ≥ 𝛼 deg 𝐺 2𝛼 |𝐸 (𝐺)| . = |𝐺 | |𝐺 | 2 266 In particular, if 𝐺 is 𝜀 2 -dense and 𝑜 is (𝛼, 𝛿)-sad then giant𝛿 ≠ giant2𝛼𝜀2 . By Theorem 6.1.1 and Theorem 6.2.1, for all but finitely many 𝐺 ∈ H ,   2𝜂 P 𝑝 giant2𝛼𝜀2 ≠ giant𝛿 ≤ . 3 So for all but finitely many 𝐺 ∈ H , if 𝐺 is 𝜀2 -dense then P 𝑝 (𝑜 is (𝛼, 𝛿)-happy) ≥ P 𝑝 (𝑜 is (𝛼, 𝛿)-happy or (𝛼, 𝛿)-sad) − P 𝑝 (𝑜 is (𝛼, 𝛿)-sad) 𝜂 2𝜂 ≥ 1− − = 1 − 𝜂. 3 3 Case B By Lemma 6.8.3, there exists 𝑁 < ∞ such that for all but finitely many 𝐺 ∈ H (namely, for a suitable constant 𝜀3 > 0, every 𝐺 ∈ H with deg 𝐺 ≥ 1/𝜀3 that is not 𝜀3 -macromolecular), |{𝑥 ∈ neigh(𝑜) : 𝑜 and 𝑥 are not (𝛼, 𝛿, 𝛾, 𝑝)-friends}| ≤ 𝑁. In particular, for all but finitely many 𝐺 ∈ H (those graphs that additionally satisfy deg 𝐺 ≥ 2𝑁), 1 1 |{𝑥 ∈ neigh(𝑜) : 𝑜 and 𝑥 are (𝛼, 𝛿, 𝛾, 𝑝)-friends}| ≥ . deg 𝐺 2 (6.8.5) Now suppose that some given 𝐺 ∈ H satisfies pop(𝛼, 𝛿, 𝛾, 𝑝) ≥ 1/𝜀1 , eq. (6.8.5), and (a trivially harmless hypothesis) deg 𝐺 > 0. Consider the spanning subgraph of 𝐺 containing only the edges 𝑢𝑣 ∈ 𝐸 (𝐺) such that 𝑢 and 𝑣 are (𝛼, 𝛿, 𝛾, 𝑝)-friends, and let 𝐴 denote the connected component of this graph containing 𝑜. Note that |𝐸 ( 𝐴)| | 𝐴| 2 = 𝜀1 1 deg 𝐴 deg 𝐺 ≤ < 𝜀1 , ≤ = 2 | 𝐴| 2 | 𝐴| 2 pop(𝛼, 𝛿, 𝛾, 𝑝) 2 and the graph 𝐴 contains at least one edge because deg 𝐴 ≥ 12 deg 𝐺 > 0. So by definition of 𝜀 1 , the graph 𝐴 must contain a path of vertices 𝑣 1 , . . . , 𝑣 𝑚 satisfying 2 1 |𝑇 | ≥ deg 𝐴 ≥ deg 𝐺 𝜀0 𝜀0 where 𝑇 := 𝑚 Ø neigh(𝑣 𝑖 ), 𝑖=1 and without loss of generality, since 𝐴 is transitive, we may choose this path such that 𝑣 1 = 𝑜. Therefore, by definition of 𝜀0 ,   P 𝑝 giant𝛿 𝑇 ≥ 𝜀0 ≥ 1 − 𝜂/3. 267 We always have 𝑚 ∑︁ 𝑖=1 𝑚 1 ∑︁ giant𝛿 neigh(𝑣 𝑖 ) = giant𝛿 ∩ neigh(𝑣 𝑖 ) deg 𝐺 𝑖=1 1 giant𝛿 ∩𝑇 deg 𝐺 |𝑇 | 1 giant𝛿 𝑇 ≥ giant𝛿 𝑇 . = deg 𝐺 𝜀0 ≥ So if giant𝛿 𝑇 ≥ 𝜀0 , then some 𝑣 𝑖 with 𝑖 ∈ {1, . . . , 𝑚} must be (1/𝑚, 𝛿)-happy and hence (𝛼, 𝛿)-happy. In particular, we deduce that ! 𝑚 Ø 𝜂 {𝑣 𝑖 is (𝛼, 𝛿)-happy} ≥ 1 − . P𝑝 3 𝑖=1 Now by combining this estimate with eq. (6.8.4) and the estimate appearing in Lemma 6.8.5 via a union bound, P 𝑝 (𝑜 is (𝛼, 𝛽)-happy) ≥ 1 − P 𝑝 (𝑜 is neither (𝛼, 𝛿)-happy nor (𝛼, 𝛿)-sad) ! 𝑚 Ù − P𝑝 {𝑣 𝑖 is not (𝛼, 𝛿)-happy} 𝑖=1 − P 𝑝 {𝑣 1 is (𝛼, 𝛿)-sad} ∩ 𝑚 Ø ! {𝑣 𝑖 is (𝛼, 𝛿)-happy} 𝑖=2 ≥ 1 − 𝜂/3 − 𝜂/3 − 𝑚𝛾 = 1 − 𝜂. □ Negligibility of mesoscopic clusters In this subsection, we complete the proof of Proposition 6.8.1. Given our work in the previous subsection, it suffices to show that for supercritical percolation on a high-degree finite transitive graph, if the giant cluster includes a positive proportion of the neighbourhood of the origin with high probability, then eq. (6.8.1) holds. Proof of Proposition 6.8.1. Let H ⊆ F be an infinite set with lim𝐺∈H deg 𝐺 = ∞ that does not contain any infinite macromolecular subsets. Let 𝑝 : H → [0, 1] be supercritical. We may assume without loss of generality that 𝑝(𝐺) < 1 for all 𝐺 ∈ H . In particular, we may assume that there exists a constant 𝛿 > 0 such that for every 𝐺 ∈ H , the parameter 𝑝(𝐺) is 𝛿-supercritical for 𝐺. Fix an arbitrary constant 𝜂 > 0. Our goal is to find a constant 𝑘 < ∞ such that for all but finitely many graphs 𝐺 ∈ H , P 𝑝 (|𝐾𝑜 | ≥ 𝑘, 𝐾𝑜 ≠ giant) ≤ 𝜂. (6.8.6) 268 By Theorems 6.1.1 and 6.2.1, for all but finitely many 𝐺 ∈ H ,
P 𝑝 giant𝛿 ≠ giant ≤ 𝜂/2. So it suffices to find 𝑘 < ∞ such that for all but finitely many 𝐺 ∈ H , P 𝑝 (𝑜 ∈ M) ≤ 𝜂/2 where M := {𝑢 ∈ 𝑉 (𝐺) : 𝑘 ≤ |𝐾𝑢 | < 𝛿 |𝐺 |}. (6.8.7) Define 𝑞(𝐺) := (1 − 𝛿/2) 𝑝(𝐺) for all 𝐺 ∈ H . By Proposition 6.8.2, there exists a constant 𝛼 > 0 such that for all but finitely many 𝐺 ∈ H ,   P𝑞 ∥giant∥ neigh(o) ≥ 𝛼 ≥ 1 − 𝜂2 . 200 (6.8.8) By Theorems 6.1.1 and 6.2.1 again, for all but finitely many 𝐺 ∈ H ,
𝜂2 P𝑞 giant𝛿 ≠ giant ≤ . 300 (6.8.9) By Lemma 6.2.2, for all but finitely many 𝐺 ∈ H , 𝑝(𝐺) ≥ 1 . 2 deg 𝐺 (6.8.10) We will show that if a graph in H satisfies eqs. (6.8.8) to (6.8.10) then it must also satisfy eq. (6.8.7) with 𝑘 := 1000𝛼−1 𝜂−2 𝛿−1 log(300𝜂−2 𝛿−1 ). Indeed, suppose for contradiction that some particular graph 𝐺 = (𝑉, 𝐸) ∈ F satisfies eqs. (6.8.8) to (6.8.10) but not eq. (6.8.7). By the negation of eq. (6.8.7) and Markov’s inequality, (and by transitivity and linearity of expectation)  𝜂 𝜂 P 𝑝 ∥M ∥ ≥ ≥ . 4 4 By eqs. (6.8.8) and (6.8.9), the set S := {𝑢 ∈ 𝑉 : giant𝛿 neigh(𝑢) ≥ 𝛼} satisfies P𝑞 (𝑜 ∈ S) ≥ 1 − 𝜂2 𝜂2 𝜂2 − ≥ 1− . 200 300 64 So by Markov’s inequality,  𝜂 𝜂 P𝑞 ∥S∥ ≥ 1 − ≥ 1− . 8 8 Recall that (𝜔𝑡 : 𝑡 ∈ [0, 1]) ∼ P denotes the standard monotone coupling of percolation measures on 𝐺. By a union bound,  𝜂 𝜂 𝜂 𝜂 𝜂 P M (𝜔 𝑝 ) ≥ and S(𝜔 𝑞 ) ≥ 1 − ≥ − = . (6.8.11) 4 8 4 8 8 269 Note that on the event being estimated in eq. (6.8.11), M (𝜔 𝑝 ) ∩ S(𝜔 𝑞 ) ≥ S(𝜔 𝑞 ) + M (𝜔 𝑝 ) − 1 ≥ 𝜂 . 8 Moreover, we can rewrite this intersection density as ∑︁ 1 1S(𝜔𝑞 ) (𝑢) M (𝜔 𝑝 ) ∩ S(𝜔 𝑞 ) = |𝑉 | 𝑢∈M (𝜔 𝑝 ) ∑︁ ∑︁ 1 1 = 1S(𝜔𝑞 ) (𝑢) |𝑉 | 𝐾𝑢 (𝜔 𝑝 ) 𝑢∈M (𝜔 𝑝 ) 𝑣∈𝐾𝑢 (𝜔 𝑝 ) ∑︁ ∑︁ 1 1 = 1S(𝜔𝑞 ) (𝑢) |𝑉 | 𝐾 (𝜔 ) 𝑣 𝑝 𝑣∈M (𝜔 𝑝 ) 𝑢∈𝐾 𝑣 (𝜔 𝑝 ) ∑︁ 1 S(𝜔 𝑞 ) 𝐾𝑣 (𝜔 𝑝 ) . = |𝑉 | 𝑣∈M (𝜔 𝑝 ) Therefore, by transitivity and linearity of expectation, h E   ∑︁  1  𝜂 𝜂 𝜂2 S(𝜔 𝑞 ) 𝐾𝑜 (𝜔 𝑝 ) 1M (𝜔 𝑝 ) (𝑜) = E  S(𝜔 𝑞 ) 𝐾𝑣 (𝜔 𝑝 )  ≥ · = ,  |𝑉 | 𝑣∈M (𝜔 𝑝 )  8 8 64   i and thus by Markov’s inequality,     𝜂2 𝜂2 . ≥ P 𝑜 ∈ M (𝜔 𝑝 ) ∩ S(𝜔 𝑞 ) 𝐾𝑜 (𝜔 𝑝 ) ≥ 128 128 So by eq. (6.8.9) and a union bound,       𝜂2 𝜂2 𝜂2 𝜂2 P 𝑜 ∈ M (𝜔 𝑝 ) ∩ S(𝜔 𝑞 ) 𝐾𝑜 (𝜔 𝑝 ) ≥ − ≥ . ∩ giant(𝜔 𝑞 ) = giant𝛿 (𝜔 𝑞 ) ≥ 128 128 300 300 (6.8.12) On the event being estimated in eq. (6.8.12), every vertex 𝑣 in the unique 𝜔 𝑞 -cluster of density ≥ 𝛿 satisfies 𝜕𝐾𝑜 (𝜔 𝑝 ) ∩ 𝜕𝐾𝑣 (𝜔 𝑞 ) ≥ S(𝜔 𝑞 ) ∩ 𝐾𝑜 (𝜔 𝑝 ) · 𝛼 deg 𝐺 = 𝐾𝑜 (𝜔 𝑝 ) · S(𝜔 𝑞 ) 𝐾𝑜 (𝜔 𝑝 ) · 𝛼 deg 𝐺 ≥𝑘· 𝜂2 · 𝛼 deg 𝐺. 128 Therefore, there must exist a deterministic vertex 𝑣 satisfying   𝜔𝑝 𝛼𝜂2 𝑘 𝜂2 𝛿 (∗) := P 𝜕𝐾𝑜 (𝜔 𝑝 ) ∩ 𝜕𝐾𝑣 (𝜔 𝑞 ) ≥ deg 𝐺 and 𝑜 ← ̸ →𝑣 ≥ . 128 300 270 On the other hand, by Lemma 6.5.3,       𝛼𝜂2 𝑘 𝛿 1 𝛼𝜂2 𝑘 𝛼𝜂2 𝛿𝑘 (∗) ≤ exp −( 𝑝 − 𝑞) deg 𝐺 ≤ exp − · · deg 𝐺 = exp − . 128 2 2 deg 𝐺 128 512 We now have a contradiction because our choice of 𝑘 was large enough to ensure that   𝛼𝜂2 𝛿𝑘 𝜂2 𝛿 exp − < . 512 300 □ 6.9 The macromolecular case In this section, we describe the asymptotic behaviour of the density of the giant cluster for supercritical percolation on high-degree finite transitive graphs that are macromolecular. Recall that H ⊆ F is said to be macromolecular if lim𝐻 deg 𝐺 = ∞ and there exists a constant 𝜀 > 0 such that all but finitely many 𝐺 ∈ H admit an 𝜀-macromolecular decomposition ( 𝐴(𝐺), 𝐵(𝐺)). We will often abbreviate ( 𝐴, 𝐵) := (( 𝐴(𝐺), 𝐵(𝐺)) : 𝐺 ∈ H ), noting that at most finitely many of these pairs ( 𝐴(𝐺), 𝐵(𝐺)) might be undefined. Let us say that H is irreducibly macromolecular if additionally, these 𝜀-macromolecular decompositions ( 𝐴(𝐺), 𝐵(𝐺)) can be chosen in such a way that there does not exist an infinite subset H ′ ⊆ H such that {𝐴(𝐺) : 𝐺 ∈ H } is macromolecular. In this case, let us also call ( 𝐴, 𝐵) an irreducibly macromolecular decompositon. In this section, we will focus on irreducibly macromolecular families. We will later show that every macromolecular family contains an infinite family that is irreducibly macromolecular (and in fact, the corresponding macromolecular decompositions are essentially uniquely determined). As such, general macromolecular families will inherit the results of this section. Our main goal is to prove the following proposition, which establishes that this asymptotic density is determined by the local geometry of the quotient graphs 𝐵. Proposition 6.9.1. Let 𝐻 ⊆ F be an infinite set of (isomorphism classes of) finite, connected, vertex transitive graphs with lim𝐺∈H deg 𝐺 = ∞. Suppose that 𝑝 : H → [0, 1] is a supercritical assignment satisfying supH 𝑝 deg 𝐺 < ∞. For each 𝐺 ∈ H , define 𝜓(𝐺) := mf ( 𝑝(𝐺) deg 𝐺) and 𝑝 ∗ (𝐺) := 𝑝(𝐺)𝜓(𝐺) 2 . If H is sparse and admits an irreducible macromolecular decomposition ( 𝐴, 𝐵), then lim lim sup 𝜃 ( 𝑝, 𝐺) − 𝜓 · P𝐵𝑝∗ (|𝐾𝑜 | ≥ 𝑛) = 0. 𝑛→∞ H 271 (6.9.1) In the first subsection, we will use the result of Section 6.7 to show that in the setting of this proposition, under P𝐺𝑝 , 𝑜 is unlikely to belong to a mesoscopic cluster whose volume is much larger then deg 𝐺. In the second subsection, we prove Proposition 6.9.1. In the third subsection, we will explain how Proposition 6.9.1 implies our main theorems concerning concentration (Theorem 6.1.3) and equicontinuity (Theorem 6.1.11) for graphs that admit irreducible macromolecular decompositions. Warning: In this section, we will apply our main results about non-macromolecular graphs in order to analyse the family of graphs 𝐴 coming from an irreducibly macromolecular decomposition ( 𝐴, 𝐵). More precisely, we will apply both Proposition 6.9.1, establishing a uniform tail on the distribution of non-giant clusters, as well as the consequences of this proposition that are derived in Section 6.11, namely the results in column 4 of the table in that section, establishing that the supercritical giant cluster density is concentrated and local. The proofs of these results about non-macromolecular graphs do not rely on the analysis of macromolecular graphs appearing in this section, so there is no danger of circular reasoning. In fact, the analysis in this section only relies on these results for non-macromolecular families of graphs that are also dense, since the family of graphs 𝐴 coming from a macromolecular decomposition is always dense by definition. For this special case of dense non-macromolecular graphs, all of our results could be deduced much more easily and directly from Theorem 6.1.1 and a refinement of Lemma 6.11.3, without having to go through the work of Section 6.8. Proof of Proposition 6.9.1 Our goal is to establish eq. (6.9.1). It suffices to establish that every infinite subset of H contains a further infinite subset for which these equations hold. As such, throughout this proof we may without loss of generality replace H by an arbitrary infinite subset of H . We would like to apply our results from Section 6.8 concerning high-degree non-macromolecular graphs to A := {𝐴(𝐺) : 𝐺 ∈ H }. It follows easily from the definition of irreducible macromolecular decomposition that limH deg 𝐴 = ∞ (because deg 𝐺 − deg 𝐴 is uniformly bounded above) and A does not contain an infinite subset that is macromolecular. What is less obvious is that 𝑝 is supercritical for A. Strictly speaking, it does not make sense to ask whether 𝑝 is supercritical for A because graphs in A do not lie in the domain of 𝑝. To avoid this technicality, by passing to an infinite subset of H if necessary, let us assume that the map H → A sending 𝐺 ↦→ 𝐴(𝐺) is injective and hence a bijection. This lets us identify our given assignment of parameters 𝑝 : H → [0, 1] with an assignment 𝑝 : A → [0, 1]. 272 Claim 6.9.2. The assignment 𝑝 : A → [0, 1] is supercritical for A. Proof. Since A is a family of dense graphs, the assignment 𝑝 𝑐 := 1/deg 𝐴 defines a percolation threshold for A [Bol+10b]. (See discusson in Section 6.1). So our goal is to show that lim inf 𝑝 deg 𝐴 > 1. H (6.9.2) By Lemma 6.2.2, since 𝑝 is supercritical for H , lim inf 𝑝 deg 𝐺 > 1. H (6.9.3) Since deg 𝐺 − deg 𝐴 is uniformly bounded above and limH deg 𝐺 = ∞, deg 𝐴 = 1. H deg 𝐺 lim Combining eqs. (6.9.3) and (6.9.4) yields eq. (6.9.2). (6.9.4) □ We are now able to invoke Proposition 6.8.2 from Section 6.7 to analyse P 𝑝𝐴 . We will also invoke the main result of Section 6.8, concerning a uniform tail for non-macromolecular graphs, and its consequences explained in Section 6.11, specifically, the validity of the mean-field approximation and the concentration of the giant cluster cluster density for non-macromolecular graphs. Our ∗ next step is to leverage this control of percolation on 𝐴 to build a suitable coupling of P 𝑝𝐴 and P𝐵𝑝∗ . Informally, we want this coupling to confirm the picture that mesoscopic clusters in P𝐺𝑝 with volume of order | 𝐴| behave like microscopic clusters with volume of order 1 in P𝐵𝑝∗ . The idea is that these mesocopic clusters in P𝐺𝑝 must be essentially built from joining the “local” giant clusters from a bounded number of the copies of 𝐴 in 𝐺, and these connections between “local” giant clusters behave like P𝐵𝑝∗ . Let us now introduce the notation required to make this precise. For each 𝐺 ∈ H , let 𝑢 ↦→ [𝑢] denote the class function for an Aut 𝐺-invariant equivalence relation on 𝑉 that induces the macromolecular decomposition ( 𝐴, 𝐵). We naturally identify the set of classes {[𝑢] : 𝑢 ∈ 𝑉 } with the vertex set 𝑉 (𝐵) of 𝐵. Let 𝐹 be the set of all edges 𝑥𝑦 ∈ 𝐸 such that [𝑥] ≠ [𝑦]. For each [𝑢] ∈ 𝑉 (𝐵), make the following definitions: Let 𝐸 [𝑢] be the set of all edges 𝑥𝑦 ∈ 𝐸 such that 𝑥, 𝑦 ∈ [𝑢]. Let 𝜔[𝑢] := 𝜔| 𝐸 [𝑢] , and for each 𝑣 ∈ [𝑢], let 𝐾𝑣 [𝑢] := 𝐾𝑣 (𝜔[𝑢]). → − Let 𝐹 be the set of all ordered pairs (𝑢, 𝑣) such that the corresponding unordered pairs 𝑢𝑣 = {𝑢, 𝑣} → − belongs to 𝐹. Let Q = Q𝐺 be the law of a random vector 𝛽 ∈ {0, 1} 𝐹 whose entries are iid Bernoulli(𝜓). Thanks to Claim 6.9.2, after passing to an infinite subset of H if necessary, we may assume that there exists a constant 𝜀 > 0 such that for all 𝐴 ∈ A, the parameter 𝑝( 𝐴) is 𝜀-supercritical for 𝐴. 273 Fix some choice of 𝜀 for the rest of this proof. Define 𝑚 := | 𝐴| = |[𝑜]|. For each [𝑢] ∈ 𝑉 (𝐵), define 𝑟 [𝑢] := {𝑣 ∈ [𝑢] : |𝐾𝑣 [𝑢] | ≥ 𝜀𝑚}, → − and given vectors 𝜔 ∈ {0, 1} 𝐸 and 𝛽 ∈ {0, 1} 𝐹 , let E [𝑢] be the assertion that for every pair → − (𝑥, 𝑦) ∈ 𝐹 that satisfies 𝑥 ∈ [𝑢] and 𝜔(𝑥, 𝑦) = 1, 𝛽(𝑥, 𝑦) = 1 if and only if 𝑥 ∈ 𝑟 [𝑢]. Claim 6.9.3. For each 𝐺 ∈ H , there exists a choice of coupling P = P𝐺 of 𝜔 ∼ P𝐺𝑝 and 𝛽 ∼ Q𝐺 in which 𝜔| 𝐹 and 𝛽 are independent of each other such that lim min P (E [𝑢]) = 1. (6.9.5) H [𝑢]∈𝑉 (𝐵) Proof. Fix 𝜂 > 0 and let 𝐺 ∈ H . We will show that for all but finitely many choices for 𝐺, we can build suitable coupling that satisfies P (E [𝑢]) ≥ 1 − 𝜂 for every [𝑢]. Let 𝑛 be a large positive integer to be determined. Independently sample the following families of random variables, and write P for their joint law: 1. Sample 𝜔| 𝐹 ∼ Ë 𝐹 Bern( 𝑝). → − 2. Independently for each (𝑢, 𝑣) ∈ 𝐹 , sample Ì 𝜈 (𝑢,𝑣) ∼ Bern( 𝑝) and 𝛽(𝑢, 𝑣) ∼ Bern(𝜓) 𝐸 [𝑜] that are coupled so that ! Δ := P 𝑝 (|𝐾𝑜 [𝑜]| ≥ 𝑛) − 𝜓 = P  𝐾𝑜 𝜈 (𝑢,𝑣) |
≥ 𝑛 △ {𝛽(𝑢, 𝑣) = 1} . {z } E (𝑢,𝑣) → − 3. Independently for each (𝑢, 𝑣) ∈ 𝐹 , sample 𝜙 (𝑢,𝑣) ∼ Unif (Γ𝑢 ) where Γ𝑢 is the set of all graph automorphisms of 𝐺 that map 𝑜 ↦→ 𝑢. 274 4. Sample 𝜈 ∼ Ë 𝐸 Bern( 𝑝). → − For each (𝑢, 𝑣) ∈ 𝐹 , let 𝜈ˆ (𝑢,𝑣) : 𝐸 [𝑜] ⇀ {0, 1} be the partial function encoding the edges revealed in an exploration of the cluster at 𝑜 from inside (with respect to an arbitrary deterministic ordering of 𝐸 [𝑜]) that is halted as soon as the event that 𝐾𝑜 (𝜈 (𝑢,𝑣) ) ≥ 𝑛 is determined by the states of the revealed edges. (See [Eas24, Section 2.1] for more discussion of these partial functions and for a warm-up to the argument below.) Given partial functions 𝑓 and 𝑔, we write 𝑓 ⊔ 𝑔 for the override, i.e. the partial function with domain dom( 𝑓 ⊔ 𝑔) = dom( 𝑓 ) ∪ dom(𝑔) that coincides with 𝑓 on dom( 𝑓 )\ dom(𝑔) and coincides with 𝑔 → − on dom(𝑔)\ dom( 𝑓 ). Let 𝑥 1 , . . . , 𝑥𝑟 be an enumeration of the set of all pairs (𝑢, 𝑣) ∈ 𝐹 satisfying → − 𝜔(𝑢𝑣) = 1, listed according to an arbitrary but deterministic total order on 𝐹 that is fixed ahead of time. Now set

𝜔 := 𝜔 𝐹 ⊔ 𝜈ˆ𝑥1 ◦ 𝜙𝑥1 ⊔ · · · ⊔ 𝜈ˆ𝑥𝑟 ◦ 𝜙𝑥𝑟 ⊔ 𝜈. Observe that 𝜔 and 𝛽 have the required marginals and independence properties, so what remains is to establish the required control of E [𝑢] for every [𝑢]. We will focus on E [𝑜], but the same → − arguments work for all [𝑢]. Let F1 be the event that E (𝑢, 𝑣) holds for some (𝑢, 𝑣) ∈ 𝐹 with → − 𝑣 ∈ [𝑜], and let F2 be the event that there exist (𝑢, 𝑣), (𝑢′, 𝑣′) ∈ 𝐹 such that 𝑣, 𝑣′ ∈ [𝑜] and 𝜔(𝑢𝑣) = 𝜔(𝑢′𝑣 ′) = 1 but 𝜈ˆ (𝑢,𝑣) ◦ 𝜙 (𝑢,𝑣) and 𝜈ˆ (𝑢′ ,𝑣 ′ ) ◦ 𝜙 (𝑢′ ,𝑣 ′ ) are incompatible as partial functions, i.e. they disagree on some portion of the intersection of their domains. Note that E [𝑜] ⊆ F1 ∪ F2 . So it suffices to show that P(F1 ), P(F2 ) ≤ 𝜂2 . Control of F1 almost surely, Let 𝑧 denote the number of edges in 𝜕 [𝑜] that are open in 𝜔. By a union bound, P (F1 | 𝑧) ≤ 𝑧Δ. So by another union bound and by Markov’s inequality, for all 𝜆 > 0, E[𝑧] + 𝜆Δ. 𝜆 Note that for some constant 𝐶 < ∞ independent of 𝐺, we have E[𝑧] = 𝑝 deg 𝐵 < ∞ because deg 𝐵 supH 𝑝 deg 𝐺 < ∞ by hypothesis, and supH deg 𝐺 < ∞ by definition of macromolecular decompoP (F1 ) ≤ 𝜂 sition. Fix 𝜆 := 4𝐶/𝜂 so that E[𝑧] 𝜆 ≤ 4 . By continuity of mf and by eq. (6.9.4), lim |mf ( 𝑝 deg 𝐺) − mf ( 𝑝 deg 𝐴)| = 0. H 275 So by Theorems 6.1.1 and 6.2.1 and Proposition 6.8.1 and our results about non-macromolecular graphs (specifically, (𝑇, 4) and (𝐿, 4) in Section 6.11), lim lim sup P 𝑝 (|𝐾𝑜 [𝑜] | ≥ 𝑙) − 𝜓 = 0. 𝑙→∞ H 𝜂 for It follows that by picking our constant 𝑛 to be sufficiently large, we can guarantee that Δ ≤ 4𝜆 all but finitely many 𝐺. Fix such a choice of 𝑛 for the rest of the proof. By combining our bounds, we now have 𝜂 𝜂 𝜂 P (F1 ) ≤ + = . 4 4 2 Control of F2 → − For each (𝑢, 𝑣) ∈ 𝐹 , consider the cluster
𝐶 (𝑢,𝑣) := 𝐾𝑣 𝜈ˆ (𝑢,𝑣) ◦ 𝜙 (𝑢,𝑣) , → − where any edges with undefined state are treated as closed. Notice that for all 𝑥, 𝑦 ∈ 𝐹 , the clusters 𝐶𝑥 and 𝐶𝑦 are independent of 𝜔| 𝐹 , and if 𝜈ˆ𝑥 ◦ 𝜙𝑥 and 𝜈ˆ 𝑦 ◦ 𝜙 𝑦 are incompatible then 𝐶𝑥 ∩ 𝐶𝑦 ≠ ∅. So by a union bound and independence, ∑︁
𝑝 2 P 𝐶𝑥 ∩ 𝐶𝑦 ≠ ∅ P (F2 ) ≤ 𝑥,𝑦 → − → − where the sum is over all 𝑥 = (𝑢, 𝑣) ∈ 𝐹 and 𝑦 = (𝑢′, 𝑣′) ∈ 𝐹 such that 𝑣, 𝑣′ ∈ [𝑜]. Consider some 𝑥 = (𝑢, 𝑣) and 𝑦 = (𝑢′, 𝑣′). Let K be the set of all possible outcomes for 𝐶𝑥 for any (equivalently → − every) 𝑥 ∈ 𝐹 . By independence, we can expand ∑︁

|{( 𝑓 , 𝑔) ∈ Γ𝑢 × Γ𝑢′ : 𝑓 (𝑋) ∩ 𝑔(𝑌 ) ≠ ∅}| . P 𝐶𝑥 ∩ 𝐶𝑦 ≠ ∅ = P (𝐶𝑥 = 𝑋) P 𝐶𝑦 = 𝑌 · |Γ𝑢 | |Γ𝑢′ | 𝑋,𝑌 ∈K Now by summing over all choices for 𝑥 and 𝑦 and exchanging the order of summation,  2 2 deg 𝐵 P (F2 ) ≤ ( 𝑝 | 𝐴|) min P ( 𝑓 (𝑋) ∩ 𝑔(𝑋) ≠ ∅) | 𝐴| 𝑋,𝑌 ∈K where P is the law of two independent uniformly random graph automorphisms 𝑓 and 𝑔 of 𝐺 [𝑜]. Recall that 𝑝 deg 𝐵 ≤ 𝐶. Consider some 𝑋, 𝑌 ∈ K. By a union bound and independence, ∑︁ P ( 𝑓 (𝑋) ∩ 𝑔(𝑋) ≠ ∅) ≤ P (𝑢 ∈ 𝑓 (𝑋)) P (𝑢 ∈ 𝑔(𝑌 )) . 𝑢∈[𝑜] Notice that for all 𝑢 ∈ [𝑜], we have 𝑢 ∈ 𝑓 (𝑋) if and only if 𝑓 −1 (𝑢) ∈ 𝑋, the law of 𝑓 −1 is the same as the law of 𝑓 , and the law of 𝑓 (𝑢) is uniform over [𝑜]. In particular, for all 𝑢 ∈ [𝑜], P (𝑢 ∈ 𝑓 (𝑋)) = 276 |𝑋 | 𝑛 ≤ , | 𝐴| | 𝐴| and the same is true of 𝑔(𝑌 ). So P ( 𝑓 (𝑋) ∩ 𝑔(𝑋) ≠ ∅) ≤ 𝑛2 , | 𝐴| and since 𝑋 and 𝑌 were arbitrary, 𝐶 2 𝑛2 , | 𝐴| which is smaller than 𝜂2 for all but finitely many 𝐺 because limH deg 𝐴 = ∞. P (F2 ) ≤ □ Fix such a family of couplings (P𝐺 : 𝐺 ∈ H ) for the rest of the proof. Given some 𝐺 ∈ H , we can naturally identify 𝐹 with the edge set of 𝐵. Now our coupling P induces the following pair of percolation configurations 𝜔, ˜ 𝜔ˆ ∈ {0, 1} 𝐹 on 𝐵: for all 𝑢𝑣 ∈ 𝐹, 𝜔(𝑢𝑣) ˆ := 𝜔(𝑢𝑣)1𝑟 [𝑢] (𝑢)1𝑟 [𝑣] (𝑣) and 𝜔(𝑢𝑣) ˜ := 𝜔(𝑢𝑣) 𝛽(𝑢, 𝑣) 𝛽(𝑣, 𝑢). Note that 𝜔ˆ only depends on 𝜔, and by construction, the law of 𝜔(𝑢, ˜ 𝑣) is exactly P𝐵𝑝∗ . For each [𝑢] ∈ 𝑉 (𝐵), let Π[𝑢] is the set of all edges 𝑥𝑦 ∈ 𝐹 such that 𝑥 ∈ [𝑢] and 𝜔(𝑥𝑦) = 1, let 𝑠[𝑢] be the set of all vertices 𝑣 ∈ 𝑉\[𝑢] such that 𝑣 ∉ 𝑟 [𝑣] and there exists some 𝑥𝑦 ∈ 𝐹 with 𝑥 ∈ [𝑢] and 𝑦 ∈ 𝐾𝑣 [𝑣] satisfying 𝜔(𝑥𝑦) = 1, and let 𝑡 [𝑢] be the set of all vertices 𝑣 ∈ 𝑠[𝑢] such that there exists 𝑥 ∈ 𝑉\( [𝑣] ∪ 𝑟 [𝑢]) with 𝑣𝑥 ∈ 𝐸 satisfying 𝜔(𝑣𝑥) = 1. Claim 6.9.4. We have lim lim sup P 𝑝 (|Π[𝑜] | > 𝑖) = 0; 𝑖→∞ H lim lim sup P 𝑝 (|𝑠[𝑜] | > 𝑖) = 0; 𝑖→∞ H lim P 𝑝 (𝑡 [𝑜] ≠ ∅) = 0. H Proof. We will prove each of the three equations in turn. First equation For all 𝐺, we have E 𝑝 |Π[𝑜]| = 𝑝 deg 𝐵. As noted in the proof of the previous claim, it follows easily from the definition of macromolecular decomposition and from supH 𝑝 deg 𝐺 < ∞ that sup 𝑝 deg 𝐵 < ∞. H So by Markov’s inequality, lim sup lim sup P 𝑝 (|Π [𝑜]| > 𝑖) ≤ lim sup 𝑖→∞ H 𝑖→∞ 277 1 sup 𝑝 deg 𝐺 = 0. 𝑖 H (6.9.6) Second equation Let 𝐺 ∈ H and 𝑖 ≥ 1. Let 𝑋 be the set of all vertices 𝑦 ∈ 𝑉\[𝑜] such that there exists 𝑥 ∈ [𝑜] satisfying 𝑥𝑦 ∈ 𝐹 and 𝜔(𝑥𝑦) = 1. The event that |𝑠[𝑜] | > 𝑖 is contained in the union F1 (𝑖) ∪ F2 (𝑖) √ where F1 (𝑖) is the event that |𝑋 | ≥ 𝑖, and F2 (𝑖) is the event that there exists 𝑥𝑦 ∈ Π[𝑜] with √ 𝑥 ∈ [𝑜] and 𝑦 ∉ 𝑟 [𝑦] such that 𝐾 𝑦 [𝑦] ≥ 𝑖. Since |𝑋 | ≤ |Π [𝑜] | always holds, it follows from our analysis above of the first equation that sup E 𝑝 |𝑋 | < ∞ (6.9.7) H and lim lim sup P 𝑝 (F1 ( 𝑗)) = 0. 𝑗→∞ H By a union bound, independence, and transitivity, # " ∑︁ P 𝑝 (F2 (𝑖)) ≤ E 𝑝 1𝑥∈𝑋 1|𝐾 𝑥 [𝑥]|≥√𝑖 1𝑥∉𝑟 [𝑥] 𝑥∈𝑉   √ ≤ E 𝑝 |𝑋 | · P 𝑝 |𝐾𝑜 [𝑜] | ≥ 𝑖 and 𝑜 ∉ 𝑟 [𝑜] . By Theorems 6.1.1 and 6.2.1 and Proposition 6.8.1,   √ lim P 𝑝 |𝐾𝑜 [𝑜] | ≥ 𝑖 and 𝑜 ∉ 𝑟 [𝑜] = 0, 𝑗→∞ and hence by eq. (6.9.7), lim P 𝑝 (F1 (𝑖)) = 0. 𝑗→∞ The conclusion now follows by a union bound. Third equation Let 𝐼 be the set of all edges 𝑥𝑦 ∈ 𝐸 such that [𝑥] = [𝑦] or {𝑥, 𝑦} ∩ 𝑟 [𝑜] ≠ ∅. Let 𝑌 be the set of all vertices 𝑦 ∈ 𝑉\[𝑜] such that there exists 𝑥 ∈ 𝑉\ ( [𝑦] ∪ 𝑟 [𝑜]) satisfying 𝜔(𝑥𝑦) = 1. Note that 𝑡 [𝑜] ≠ ∅ if and only if 𝑌 ∩ 𝑠[𝑜] ≠ ∅. For all 𝑦 ∈ 𝑉, by a union bound we almost surely have P 𝑝 (𝑦 ∈ 𝑌 | 𝜔| 𝐼 ) ≤ 𝑝 (deg 𝐺 − deg 𝐴) =: Λ 278 because after having revealed the states of all edges in 𝐼, in order for 𝑦 to ultimately belong to 𝑌 , at least one of the at most (deg 𝐺 − deg 𝐴)-many unrevealed edges 𝑒 ∈ 𝐹 with 𝑦 ∈ 𝑒 must turn out to be open. So by a union bound and since 𝑠[𝑜] is determined by 𝜔| 𝐼 , we almost surely have " # ∑︁ P 𝑝 (𝑡 [𝑜] ≠ ∅ | 𝜔| 𝐼 ) ≤ E 𝑝 1𝑦∈𝑌 1𝑦∈𝑠[𝑜] | 𝜔| 𝐼 𝑦∈𝑉 = ∑︁ P 𝑝 (𝑦 ∈ 𝑌 | 𝜔| 𝐼 ) ≤ Λ |𝑠[𝑜] | . 𝑦∈𝑠[𝑜] So by another union bound,  P 𝑝 (𝑡 [𝑜] ≠ ∅) ≤ P 𝑝  √ 1 |𝑠[𝑜]| ≥ √ + Λ. Λ (6.9.8) By definition of macromolecular decomposition, deg 𝐺 − deg 𝐴 is bounded above uniformly in 𝐺. Moreover, limH 𝑝 = 0 because limH deg 𝐺 = ∞ and supH 𝑝 deg 𝐺 < ∞. Therefore, lim Λ = 0. (6.9.9) H In particular, thanks to our analysis above of the second equation,   1 lim P 𝑝 |𝑠[𝑜] | ≥ √ = 0. H Λ (6.9.10) The conclusion now follow by combining eqs. (6.9.8) to (6.9.10). □ We will now use these simple properties and a mass-transport argument to relate the distribution of microscopic clusters in P𝐵𝑝∗ to the distribution of mesoscopic clusters in P𝐺𝑝 . Let 1 Ø 𝑘 := 𝐾𝑣 (𝜔) , 𝑚𝜓 𝑣∈𝑟 [𝑜] and let Round(𝑘) be the integer closest 𝑘, rounding up in case of a tie. Claim 6.9.5. For every positive integer 𝑛, lim P𝐺𝑝 (Round(𝑘) = 𝑛) − P𝐵𝑝∗ ( 𝐾 [𝑜] = 𝑛) = 0. H Proof. Let 𝑛 be a positive integer, let 𝜂 > 0, and let 𝐺 ∈ H . Note that Δ := P𝐺𝑝 (Round(𝑘) = 𝑛) − P𝐵𝑝∗ ( 𝐾 [𝑜] = 𝑛) ≤ P {Round(𝑘) = 𝑛}△{ 𝑘˜ = 𝑛} | {z } F 279
(6.9.11) where 𝑘˜ denotes the size of the cluster of [𝑜] in 𝜔, ˜ which we view as a configuration on 𝐵. Our goal is to show that Δ ≤ 𝜂 for all but finitely many choices of 𝐺. Given 𝑖 ≥ 1, say that [𝑢] is 𝑖-good if all of the following events hold: • 𝜔[𝑢] has a unique cluster 𝐾 satisfying |𝐾 | ≥ 𝜀𝑚. 𝜓 . • |𝑟 [𝑢] − 𝜓𝑚| ≤ 16𝑛 • |Π[𝑢] | ≤ 𝑖, |𝑠[𝑢] | ≤ 𝑖, and 𝑡 [𝑢] = ∅. By Theorems 6.1.1 and 6.2.1, our main results about non-macromolecular graphs (row 4 in Section 6.11), and Claim 6.9.4, there exists a constant 𝑖 ∈ N such that P 𝑝 ([𝑜] is 𝑖-good) ≤ 𝜂 2𝑛 (6.9.12) for all but finitely many choices for 𝐺. Fix such a choice of 𝑖 for the rest of the proof, and simply say that if [𝑢] is good to mean that 𝑢 is 𝑖-good. Let 𝑋 be the cluster of [𝑜] in 𝜔˜ ∩ 𝜔, ˆ and let 𝑌 be the set of all edges 𝑥𝑦 ∈ 𝐹 such that [𝑥] ∈ 𝑋 or [𝑦] ∈ 𝑋. Say that an edge 𝑥𝑦 ∈ 𝐹 is intact if E [𝑥] holds and E [𝑦] holds, and note that this implies that 𝜔(𝑥𝑦) ˆ = 𝜔(𝑥𝑦). ˜ In particular, if every edge in 𝑌 is intact, then ˜ 𝑘ˆ = 𝑘, (6.9.13) where 𝑘ˆ denotes the size of the cluster of [𝑜] in 𝜔. ˆ We claim that if additionally every element of 𝑋 is good then F cannot hold. Indeed, if every element of 𝑋 is good, then 𝑘𝑚𝜓, which by definition of 𝑘 equals the number of vertices 𝜔-connected to [𝑜], satisfies ∑︁ ∑︁ |𝑟 [𝑣] | ≤ 𝑘𝑚𝜓 ≤ (|𝑟 [𝑣]| + |𝑠[𝑣]|) , [𝑣]∈𝐾 [𝑜] ( 𝜔) ˆ and moreover, [𝑣]∈𝐾 [𝑜] ( 𝜔) ˆ     1 1 ˆ 1− 𝜓𝑚 𝑘ˆ ≤ 𝑘𝑚𝜓 ≤ 1 + 𝜓𝑚 𝑘ˆ + 𝑖 𝑘. 16𝑛 16𝑛 So by eq. (6.9.13) and our hypothesis that limH 𝑚 = ∞, for all but finitely many choices of 𝐺,   𝑘˜ 1 𝑖 ˜ ˜ 𝑘−𝑘 ≤ 𝑘 + ≤ . (6.9.14) 16𝑛 𝑚𝜓 8𝑛 By manipulating eq. (6.9.14), we have 𝑘˜ ≤ 2𝑘. So by applying eq. (6.9.14) again, 𝑘 − 𝑘˜ ≤ 280 𝑘 . 4𝑛 (6.9.15) ˜ Similarly, if Notice that if 𝑘˜ = 𝑛, then by eq. (6.9.14), we have 𝑘 − 𝑘˜ < 12 , and hence 𝑘 = 𝑘. Round(𝑘) = 𝑛, then 𝑘 ≤ 𝑛 + 1 and hence 𝑘 = 𝑘˜ by eq. (6.9.15). So F cannot hold, as claimed. Let F1 be the event that F holds but not every element of 𝑋 is good, and let F2 := F \F2 . By our work above, on the event F2 , not every edge in 𝑌 is intact. For each [𝑢] ∈ 𝑉 (𝐵), let F [𝑢], F1 [𝑢], F2 [𝑢], 𝑋 [𝑢], and 𝑌 [𝑢] be the analogues of F , F1 , etc. defined with [𝑢] in place of [𝑜]. Given classes [𝑢], [𝑣], let 𝑓1 ([𝑢], [𝑣]) ∈ {0, 1} be the indicator for the event that F1 [𝑢] holds, [𝑣] ∈ 𝑋 [𝑢], and [𝑣] is not good. Given a class [𝑢] and an edge 𝑒 ∈ 𝐹, let 𝑓2 ([𝑢], 𝑒) ∈ {0, 1} be the indicator for the event that F2 [𝑢] holds, 𝑒 ∈ 𝑌 [𝑢], and 𝑒 is not intact. We know that for all [𝑢] ∈ 𝑉 (𝐵), if F [𝑢] holds then 𝑓𝑖 ([𝑢], 𝑥) = 1 for some 𝑥 and 𝑖. So by eq. (6.9.11) and transitivity,      1  ∑︁  1  ∑︁   E E 𝑓1 ([𝑢], [𝑣])  + 𝑓2 ([𝑢], 𝑒)  ≥ Δ, |𝐵|   |𝐵|  [𝑢],𝑒   [𝑢],[𝑣]    | {z } | {z } Γ1 (6.9.16) Γ2 where the first sum is over all [𝑢], [𝑣] ∈ 𝑉 (𝐵) and the second sum is over all [𝑢] ∈ 𝑉 (𝐵) and 𝑒 ∈ 𝐹. (Recall that |𝐵| is the number of vertices in 𝐵.) So it suffices to show that Γ1 , Γ2 ≤ 𝜂2 . For every [𝑢], on the event F [𝑢], we have |𝑋 [𝑢] | ≤ 𝑛. So for every class [𝑣], there are never more than 𝑛 classes [𝑢] satisfying 𝑓1 ([𝑢], [𝑣]) = 1, and of course there are no such classes [𝑢] when [𝑣] is good. So for all [𝑢], ∑︁    𝜂 E  𝑓1 ([𝑢], [𝑣])  ≤ 𝑛P ( [𝑢] is not good) ≤ , 2  [𝑢]    (6.9.17) where the second inequality comes from eq. (6.9.12). In particular, Γ1 ≤ 𝜂2 . Similarly, for every [𝑢], since on the event F2 [𝑢] we have ∑︁ |𝑌 [𝑢]| ≤ |Π[𝑣]| ≤ 𝑛𝑖 [𝑣]∈𝑋 [𝑢] because every every element of 𝑋 [𝑢] is good, it follows that for all 𝑒 ∈ 𝐹, ∑︁    E  𝑓2 ([𝑢], 𝑒)  ≤ 2𝑛𝑖,  [𝑢]    (6.9.18) the factor of ‘2’ arising from the fact that the classes of the endpoints of the edge 𝑒 ∈ 𝐹 could belong to two distinct 𝜔ˆ ∩ 𝜔-clusters. ˜ In particular, Γ2 ≤ 𝜂2 , as required. □ 281 We are now ready to conclude the proof of eq. (6.9.1). Fix 𝜂 > 0. By Proposition 6.7.1 and Lemma 6.5.2 (as well as Theorems 6.1.1 and 6.2.1), there exists a sufficiently large positive integer 𝑁0 such that for every integer 𝑁 ≥ 𝑁0 , for all but finitely many G ∈ H , 𝜃 ( 𝑝, 𝐺) − P𝐺𝑝 (|𝐾𝑜 | ≥ 𝐿𝑚) ≤  𝑁 + 12 𝜂 , 4  where 𝐿 = 𝐿 (𝑁, 𝐺) := 𝜓 . By Theorems 6.1.1 and 6.2.1 and Proposition 6.8.1, for all but finitely many 𝐺, 𝜂 P𝐺𝑝 (|𝐾𝑜 | ≥ 𝐿) − P𝐺𝑝 (|𝐾𝑜 | ≥ 𝐿 and 𝑜 ∈ 𝑟 [𝑜]) ≤ . 4 Note that |𝐾𝑜 | ≥ 𝐿 and 𝑜 ∈ 𝑟 [𝑜] if and only if 𝑘 ≥ 𝑁 and 𝑜 ∈ 𝑟 [𝑜]. By transitivity,   1 ∑︁ 𝐺 𝐺 𝐺 |𝑟 [𝑜] | P 𝑝 (𝑘 ≥ 𝑁 and 𝑜 ∈ 𝑟 [𝑜]) = P 𝑝 (𝑘 ≥ 𝑁 and 𝑢 ∈ 𝑟 [𝑜]) = E 𝑝 1 𝑘 ≥𝑁 . 𝑚 𝑚 𝑢∈[𝑜] By our main results for non-macromolecular graphs (row 4 in Section 6.11), for all but finitely many 𝐺,   |𝑟 [𝑜]| 𝜂 𝐺 |𝑟 [𝑜]| 1 𝑘 ≥𝑁 − 𝜓 · P𝐺𝑝 (𝑘 ≥ 𝑁) ≤ E𝐺𝑝 −𝜓 ≤ . E𝑝 𝑚 𝑚 4 By Claim 6.9.5 and the fact that 𝜓 is always bounded above by 1, for all but finitely many 𝐺, 𝜓 · P𝐺𝑝 (𝑘 ≥ 𝑁) − 𝜓 · P𝐵𝑝∗ (|𝐾𝑜 | ≥ 𝑁) ≤ 𝜂 . 4 By combining our bounds, we find that for all but finitely many 𝐺, 𝜃 ( 𝑝, 𝐺) − P𝐵𝑝∗ (|𝐾𝑜 | ≥ 𝑁) ≤ 𝜂, as required. 6.10 The nonunimodular case In this section we complete the proof of the bounded-degree case of our results by treating the nonunimodular case. Perhaps surprisingly, following [Hut20g], percolation is much better understood on nonunimodular transitive graphs than on unimodular transitive graphs. More specifically, we will briefly explain how very strong quantitative forms of all our main results can be deduced in the nonunimodular case from the results of [Hut20h], which establish sharp, quantitative tail bounds on the distribution of finite clusters under the 𝐿 2 boundedness condition, and [Hut20g], which imply that this condition holds in the nonunimodular case; our main task in this section will be to outline how uniform versions of these results can be deduced by a compactness argument. Nonunimodular 282 transitive graphs cannot arise as limits of finite transitive graphs by Corollary 6.10.2, and we will keep this section brief since it is tangential to the main focus of the paper. For nonunimodular 𝐺, the tilted mass-transport principle states that ∑︁ ∑︁ 𝐹 (𝑜, 𝑥) = 𝐹 (𝑥, 𝑜)Δ(𝑜, 𝑥) 𝑥∈𝑉 (6.10.1) 𝑥∈𝑉 for every 𝐹 : 𝑉 2 → [0, ∞] that is invariant under the diagonal action of Aut(𝐺) on 𝑉 2 . The following proposition, established in [Hut20b, Corollary 5.4], tells us that the modular function is determined by the local geometry of the graph. This proposition is proven using a probabilistic interpretation of the modular function as a Radon-Nikodym cocycle (see [Hut20g; BC12]) and is not at all obvious from the algebraic definition given above! Proposition 6.10.1. Let (𝐺 𝑛 )𝑛≥1 be a sequence of connected, locally finite transitive graphs converging locally to some locally finite transitive graph 𝐺, and let (𝑜 𝑛 )𝑛≥1 and 𝑜 be vertices of (𝐺 𝑛 )𝑛≥1 and 𝐺 respectively. For each 𝑟 ≥ 1 there exists 𝑁 < ∞ such that for every 𝑛 ≥ 𝑁 there exists an isomorphism 𝜙𝑛 from the ball of radius 𝑟 around 𝑜 𝑛 in 𝐺 𝑛 to the ball of radius 𝑟 around 𝑜 in 𝐺 satisfying Δ𝐺 𝑛 (𝑢, 𝑣) = Δ𝐺 (𝜙(𝑢), 𝜙(𝑣)) for every 𝑢, 𝑣 in the ball of radius 𝑟 around 𝑜 𝑛 in 𝐺 𝑛 . Together with the cocycle identity [Hut20g, Lemma 2.3], which states that Δ(𝑥, 𝑧) = Δ(𝑥, 𝑦)Δ(𝑦, 𝑧) for every 𝑥, 𝑦, 𝑧 ∈ 𝑉 (and hence that 𝐺 is unimodular if and only if Δ(𝑜, 𝑥) = 1 for every neighbour of the origin), this proposition has the following immediate corollary. Corollary 6.10.2. For each 𝑑 ≥ 1, let G𝑑 be the space of all isomorphism classes of connected, transitive graphs of degree at most 𝑑, and let U𝑑 and N𝑑 = G𝑑 \ U𝑑 be the sets of unimodular and nonunimodular elements of G𝑑 respectively. The sets U𝑑 and N𝑑 are both closed (and hence both open) in G𝑑 with respect to the local topology. Note that the family U𝑑 contains both finite and infinite graphs. The following theorem is a “uniform in 𝐺” version of a theorem whose non-uniform version follows from the results of [Hut20g; Hut20h]. The bound on the volume tail of finite clusters it yields is sharp for 𝑝 not too close to 0 or 1 as explained in detail in [Hut20h]. 283 Theorem 6.10.3. Let 𝑑 ≥ 1 and let N𝑑 be the set of (isomorphism classes of) nonunimodular transitive graphs of degree at most 𝑑. There exist positive constants 𝑐 = 𝑐(𝑑) > 0 and 𝐶 = 𝐶 (𝑑) < ∞ such that   P𝐺𝑝 (𝑛 ≤ |𝐾𝑜 | < ∞) ≤ 𝐶𝑛−1/2 exp −𝑐| 𝑝 − 𝑝 𝑐 | 2 𝑛 for every 𝑛 ≥ 1, 𝐺 ∈ N𝑑 , and 𝑝 ∈ [0, 1]. After sketching the proof of this theorem we will use it to deduce the following corollary, which is much stronger than the equicontinuity statements proven in the other parts of the paper. Corollary 6.10.4. Let 𝑑 ≥ 1 and let N𝑑 be the set of (isomorphism classes of) nonunimodular transitive graphs of degree at most 𝑑. For each 𝐺 ∈ N𝑑 , 𝜃 ( 𝑝, 𝐺) is an analytic function of 𝑝 on ( 𝑝 𝑐 (𝐺), 1] and there exists a constant 𝐶 = 𝐶 (𝑑) such that 𝐶 𝑑 𝜃 ( 𝑝, 𝐺) ≤ 𝑑𝑝 1− 𝑝 and 𝜃 ( 𝑝, 𝐺) ≥ 1 − 𝐶 |1 − 𝑝|𝐶 for every 𝐺 ∈ N𝑑 and 𝑝 ∈ ( 𝑝 𝑐 , 1]. We now introduce some relevant machinery from [Hut20g]. Let 𝐺 = (𝑉, 𝐸) be a connected, locally finite, nonunimodular (vertex-)transitive graph, and let Δ = Δ𝐺 : 𝑉 2 → (0, ∞) be the modular function of 𝐺. For each 𝜆 ∈ R, the tilted susceptibility is defined to be ∑︁ 𝜒 𝑝,𝜆 = 𝜒𝐺 = P 𝑝 (𝑜 ↔ 𝑥)Δ𝜆 (𝑜, 𝑥), 𝑝,𝜆 𝑥∈𝑉 so that 𝜒 𝑝,0 = E 𝑝 |𝐾𝑜 | is the ordinary susceptibility. It is a consequence of the tilted mass-transport principle that 𝜒 𝑝,𝜆 = 𝜒 𝑝,1−𝜆 , and since 𝜒 𝑝,𝜆 is also a convex function of 𝜆 ∈ R this leads to a special role for the critically tilted susceptibility 𝜒 𝑝,1/2 = min𝜆 𝜒 𝑝,𝜆 . The tiltability threshold is defined to be 𝑝 𝑡 (𝐺) = sup{𝑝 ∈ [0, 1] : 𝜒 𝑝,1/2 < ∞}. The following is a special case of the main theorem of [Hut20g]. Theorem 6.10.5. Let 𝐺 = (𝑉, 𝐸) be a connected, locally finite, nonunimodular (vertex-)transitive graph. Then 𝑝 𝑐 (𝐺) < 𝑝 𝑡 (𝐺). In order to prove Theorem 6.10.3, we will argue that Theorem 6.10.5 implies the following “uniform in 𝐺” version of the same theorem. 284 Corollary 6.10.6. For each 𝑑 ≥ 1 there exist positive constants 𝑐 = 𝑐(𝑑) and 𝐶 = 𝐶 (𝑑) such that 𝜒𝐺 𝑝 𝑐 ,1/2 ≤ 𝐶 𝑝 𝑡 (𝐺) − 𝑝 𝑐 (𝐺) ≥ 𝑐 and for every 𝐺 ∈ N𝑑 . To deduce Corollary 6.10.6 from Theorem 6.10.5, we will also apply the following theorem of [Hut20b], which is a consequence of the results of [Hut20g]. Theorem 6.10.7 ([Hut20b], Theorem 5.6). The critical probability 𝑝 𝑐 defines a continuous function on N𝑑 . That is, if 𝐺 𝑛 is a sequence in N𝑑 converging to some 𝐺 ∈ N𝑑 then 𝑝 𝑐 (𝐺 𝑛 ) → 𝑝 𝑐 (𝐺). We will also require the following lemma. Lemma 6.10.8. For each fixed 𝑝 ∈ [0, 1] and 𝜆 ∈ R, the tilted susceptibility 𝜒𝐺 𝑝,𝜆 defines a continuous function N𝑑 → [0, ∞]. That is, if 𝐺 𝑛 is a sequence in N𝑑 converging to some 𝐺 in N𝑑 𝐺 𝑛 then 𝜒𝐺 𝑝,𝜆 converges to 𝜒 𝑝,𝜆 as 𝑛 → ∞. We will prove Lemma 6.10.8 using a tilted version of the “𝜙 𝑝 (𝑆) argument” of Duminil-Copin and Tassion [DT16b]. Let 𝐺 and Δ be as above. For each 𝜆 ∈ R, 𝑝 ∈ [0, 1], and each finite set of vertices 𝑆 ∋ 𝑜 we consider the quantity 𝜙 𝑝,𝜆 (𝑆) defined by 𝜙 𝑝,𝜆 (𝑆) := 𝑝 ∑︁ 𝑆 P 𝑝 (𝑜 ← → 𝑥) ∑︁ 1 (𝑦 ∉ 𝑆)Δ𝜆 (𝑜, 𝑦) = 𝑝 𝑦∼𝑥 𝑥∈𝑆 ∑︁ 𝑥∈𝑆 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) ∑︁ 1 (𝑦 ∉ 𝑆)Δ𝜆 (𝑥, 𝑦), 𝑦∼𝑥 𝑆 where {𝑜 ← → 𝑥} denotes the event that 𝑜 and 𝑥 are connected by an open path only using vertices of 𝑆 and the equality between these two expressions follows from the cocycle identity Δ(𝑜, 𝑦) = Δ(𝑜, 𝑥)Δ(𝑥, 𝑦). Lemma 6.10.9. Let 𝐺 = (𝑉, 𝐸) be a connected, locally finite, nonunimodular (vertex-)transitive graph, let 𝑜 be a vertex of 𝐺 and let 𝑆 ∋ 𝑜 be a finite set of vertices. Then ∑︁ 𝑥∈𝑆 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) ≤ 𝜒 𝑝,𝜆 ≤ ∑︁ 1 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) 1 − 𝜙 𝑝,𝜆 (𝑆) 𝑥∈𝑆 for every 𝑝 ∈ [0, 1] and 𝜆 ∈ R such that 𝜙 𝑝,𝜆 (𝑆) < 1. In particular, if 𝜙 𝑝,𝜆 (𝑆) < 1 for some finite set of vertices 𝑆 ∋ 𝑜 then 𝜒 𝑝,𝜆 < ∞. 285 Proof of Lemma 6.10.9. The first inequality is trivial; we focus on the second. If 𝑜 is connected to some vertex 𝑧 by an open path in 𝐺 but not in 𝑆, this path must exit 𝑆 for the first time using 𝑆 some edge 𝑒 = 𝑥𝑦 with 𝑥 ∈ 𝑆 and 𝑦 ∉ 𝑆. On this event the three events {𝑜 ← → 𝑥}, {𝑒 open}, and {𝑦 ↔ Λ𝑧} occur disjointly, and we deduce from a union bound and the BK inequality that ∑︁ ∑︁ 𝑆 𝑆 P 𝑝 (𝑜 ↔ 𝑧) ≤ P 𝑝 (𝑜 ← → 𝑧) + 𝑝 P 𝑝 (𝑜 ← → 𝑥)P 𝑝 (𝑦 ← → 𝑧). 𝑥∈𝑆 𝑦∉𝑆 Let Λ ⊆ 𝑉 be a (finite or infinite) set of vertices containing 𝑆. Multiplying both sides by Δ𝜆 (𝑜, 𝑧), summing over 𝑧 ∈ Λ, and using the cocycle identity yields that ∑︁ P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) 𝑧∈Λ ≤ ∑︁ ≤ ∑︁ 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) + 𝑝 𝑥∈𝑆 ∑︁ ∑︁ 𝑆 P 𝑝 (𝑜 ← → 𝑥) 𝑥∈𝑆 𝑦∉𝑆 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) + 𝜙 𝑝,𝜆 (𝑆) sup ∑︁ P 𝑝 (𝑦 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) 𝑧∈Λ ∑︁ P 𝑝 (𝑦 ← → 𝑧)Δ𝜆 (𝑦, 𝑧). (6.10.2) 𝑦∈Λ 𝑧∈Λ 𝑥∈𝑆 If we knew that 𝜒 𝑝,𝜆 < ∞ we could conclude by rearranging ; a little care will be needed to conclude in a non-circular manner without this assumption. Suppose that there exists a finite set 𝑆 ∋ 𝑜 with 𝜙 𝑝,𝜆 (𝑆) < 1. For each finite set Λ ⊆ 𝑉 there exists a Í Í vertex 𝑤 with 𝑧∈Λ P 𝑝 (𝑤 ← → 𝑧)Δ𝜆 (𝑤, 𝑧) = sup𝑦∈Λ 𝑧∈Λ P 𝑝 (𝑦 ← → 𝑧)Δ𝜆 (𝑦, 𝑧) and an automorphism 𝛾 of 𝐺 sending 𝑜 to 𝑤. Using that ∑︁ ∑︁ ∑︁ P 𝑝 (𝑥 ← → 𝛾 −1 𝑧)Δ𝜆 (𝑥, 𝛾 −1 𝑧) = P 𝑝 (𝛾𝑥 ← → 𝑧)Δ𝜆 (𝛾𝑥, 𝑧) P 𝑝 (𝑥 ← → 𝑧)Δ𝜆 (𝑥, 𝑧) = 𝑧∈𝛾 −1 Λ 𝑧∈Λ 𝑧∈Λ for every 𝑥 ∈ 𝑉, we have that ∑︁ P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) = sup ∑︁ P 𝑝 (𝑦 ← → 𝑧)Δ𝜆 (𝑦, 𝑧). 𝑦∈𝛾 −1 Λ 𝑧∈𝛾 −1 Λ 𝑧∈𝛾 −1 Λ Thus, (6.10.2) implies the inequality ∑︁ ∑︁ ∑︁ 𝑆 P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) ≤ P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) + 𝜙 𝑝,𝜆 (𝑆) P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧), 𝑧∈𝛾 −1 Λ 𝑧∈𝛾 −1 Λ 𝑥∈𝑆 which, since Λ is finite, can be rearranged to yield that ∑︁ 𝑧∈𝛾 −1 Λ P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) ≤ ∑︁ 1 𝑆 P 𝑝 (𝑜 ← → 𝑥)Δ𝜆 (𝑜, 𝑥) 1 − 𝜙 𝑝,𝜆 (𝑆) 𝑥∈𝑆 286 when 𝜙 𝑝,𝜆 (𝑆) < 1. On the other hand, the choice of 𝛾 ensures that ∑︁ ∑︁ P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧), P 𝑝 (𝑜 ← → 𝑧)Δ𝜆 (𝑜, 𝑧) ≤ 𝑧∈𝛾 −1 Λ 𝑧∈Λ and the claim follows by taking Λ to exhaust the entire vertex set 𝑉. □ We now deduce Lemma 6.10.8 from Lemma 6.10.9 and Proposition 6.10.1. Proof of Lemma 6.10.8. Fix 𝑝 ∈ [0, 1] and 𝜆 ∈ R and let (𝐺 𝑛 )𝑛≥ be a sequence in N𝑑 converging 𝐺 𝑛 to some 𝐺 ∈ N𝑑 . We need to prove that 𝜒𝐺 𝑝,𝜆 converges to 𝜒 𝑝,𝜆 as 𝑛 → ∞. Let 𝐵𝐺 (𝑜, 𝑟) denote the 𝐺 𝑛 ball of radius 𝑟 around 𝑜 in 𝐺. The lower semicontinuity statement lim inf 𝜒𝐺 𝑝,𝜆 ≥ 𝜒 𝑝,𝜆 is immediate from the fact that ∑︁ 𝐵𝐺 (𝑜,𝑟) 𝜒𝐺 = sup P𝐺𝑝 (𝑜 ←−−−−→ 𝑥)Δ𝜆𝐺 (𝑜, 𝑥), 𝑝,𝜆 𝑟 𝑥∈𝐵𝐺 (𝑜,𝑟) since each function on the right hand side is continuous in 𝐺 by Proposition 6.10.1 and any supremum of continuous functions is lower semicontinuous. To conclude the proof it suffices to 𝐺 𝐺 𝐺 𝑛 prove that lim sup 𝜒𝐺 𝑝,𝜆 ≤ 𝜒 𝑝,𝜆 . The claim is trivial when 𝜒 𝑝,𝜆 = ∞ so we may assume that 𝜒 𝑝,𝜆 < ∞. Since the internal vertex boundaries of the balls 𝐵𝐺 (𝑜, 𝑟) are disjoint for different choices of 𝑟, we have that 𝜒𝐺 𝑝,𝜆 ≥ Í ∞ ∑︁ ∑︁ ∑︁ 𝐵𝐺 (𝑜,𝑟) 1 𝜆 P (𝑜 ← − − − − → 𝑥)Δ (𝑜, 𝑥) Δ𝜆 (𝑥, 𝑦) 1 (𝑦 ∉ 𝐵𝐺 (𝑜, 𝑟)) 𝑝 𝜆 (𝑜, 𝑦) Δ 𝑦∼𝑜 𝑦∼𝑥 𝑟=0 𝑥∈𝐵 (𝑜,𝑟) 𝐺 ∞ ∑︁ 1 = Í 𝜙 𝑝,𝜆 (𝐵𝐺 (𝑜, 𝑟)). 𝑝 𝑦∼𝑜 Δ𝜆 (𝑜, 𝑦) 𝑟=0 Since 𝜒𝐺 𝑝,𝜆 < ∞ it follows that 𝜙 𝑝,𝜆 (𝐵𝐺 (𝑜, 𝑟)) → 0 as 𝑟 → ∞. On the other hand, Proposition 6.10.1 and Lemma 6.10.9 imply that 𝑛 lim sup 𝜒𝐺 𝑝,𝜆 ≤ 𝑛→∞ 1 1 − 𝜙 𝑝,𝜆 (𝐵𝐺 (𝑜, 𝑟)) ∑︁ 𝐵𝐺 (𝑜,𝑟) P𝐺𝑝 (𝑜 ←−−−−→ 𝑥)Δ𝜆𝐺 (𝑜, 𝑥) 𝑥∈𝐵𝐺 (𝑜,𝑟) for every 𝑟 such that 𝜙 𝑝,𝜆 (𝐵𝐺 (𝑜, 𝑟)) < 1, and the claim follows since the sum on the right hand side is bounded above by 𝜒𝐺 □ 𝑝,𝜆 for every 𝑟 ≥ 1. Proof of Corollary 6.10.6. It suffices to prove the claim about 𝜒 𝑝 𝑐 ,1/2 , the claim about 𝑝 𝑡 − 𝑝 𝑐 following from this estimate. Suppose for contradiction that the claim does not hold, so that there 𝑛 exists a sequence (𝐺 𝑛 )𝑛≥1 in N𝑑 with 𝜒𝐺 → ∞ as 𝑛 → ∞. By taking a subsequence 𝑝 (𝐺 ),1/2 𝑐 𝑛 287 if necessary, we may assume that 𝐺 𝑛 converges to some 𝐺 ∈ N𝑑 as 𝑛 → ∞. It follows from Theorem 6.10.5 and Theorem 6.10.7 that lim𝑛→∞ 𝑝 𝑐 (𝐺 𝑛 ) = 𝑝 𝑐 (𝐺) < 𝑝 𝑡 (𝐺) and hence that there exists 𝑞 < 𝑝 𝑡 (𝐺) such that 𝑝 𝑐 (𝐺 𝑛 ) ≤ 𝑞 for all sufficiently large 𝑛. Lemma 6.10.8 then implies that 𝐺𝑛 𝐺𝑛 𝐺 < ∞, which contradicts the assumption that 𝜒 𝐺 𝑛 𝑛 𝜒𝑞,𝜆 → 𝜒𝑞,𝜆 → ∞ since 𝜒𝐺 ≤ 𝜒𝑞,𝜆 𝑝 𝑐 (𝐺 𝑛 ),1/2 𝑝 𝑐 (𝐺 𝑛 ),1/2 for all sufficiently large 𝑛. □ Sketch of proof of Theorem 6.10.3. Let 𝐺 = (𝑉, 𝐸) be a connected, locally finite, transitive graph, and for each 𝑝 ∈ [0, 1] define the two-point matrix 𝑇𝑝 ∈ R𝑉×𝑉 by 𝑇𝑝 (𝑢, 𝑣) = P 𝑝 (𝑢 ↔ 𝑣). We define ∥𝑇𝑝 ∥ 2→2 to be the operator norm of 𝑇𝑝 on 𝐿 2 (𝑉), which is infinite if 𝑇𝑝 does not define a bounded operator on 𝐿 2 (𝑉). The main result of [Hut20g] states that if 𝐺 satisfies the 𝐿 2 boundedness condition, meaning that ∥𝑇𝑝 𝑐 ∥ 2→2 is finite, then there exists 𝛿 > 0 h  i P 𝑝 (𝑛 ≤ |𝐾𝑜 | < ∞) ≍ 𝑛−1/2 exp −Θ | 𝑝 − 𝑝 𝑐 | 2 𝑛 (6.10.3) for every 𝑝 ∈ ( 𝑝 𝑐 − 𝛿, 𝑝 𝑐 + 𝛿) and 𝑛 ≥ 1. The following observations allow us to deduce Theorem 6.10.3 from this and Corollary 6.10.6: 1. The inequality ∥𝑇𝑝 ∥ 2→2 ≤ 𝜒 𝑝,𝜆 holds for every 𝑝 ∈ [0, 1] and 𝜆 ∈ R. This is proven in [Hut19b, Theorem 2.9]. Thus, Corollary 6.10.6 implies that the 𝐿 2 boundedness condition holds uniformly for nonunimodular transitive graphs, with constants depending only on the degree. 2. All the implicit constants in the results of [Hut20h] can be taken to depend only on the degree and on ∥𝑇𝑝 𝑐 ∥ 2→2 . (It may seem that they also depend on 𝑝 𝑐 , but this is redundant since 𝑝 𝑐 is bounded below by the reciprocal of the degree and bounded away from 1 by a constant depending only on ∥𝑇𝑝 ∥ 2→2 .) Although such a claim is not made explicit in that paper, it can be verified by going through the proof and using that all estimates derived from the triangle condition (e.g. on the percolation probability 𝜃 and the intrinsic one-arm) can be taken to depend only on the degree and the value of ∥𝑇𝑝 𝑐 ∥ 2→2 . This fact can in turn be seen from the derivation of mean-field critical behaviour from the triangle condition presented in [Hut]. Indeed, [Hut, Eq. 2.1 and Lemma 2.1] yield that the susceptibility satisfies the differential inequality "
2 # Í 𝜒 𝑝 |𝐾 | − 𝑢,𝑣∈𝐾 𝑇𝑝 (𝑢, 𝑣) 1 𝑑 Í E𝑝 𝜒𝑝 ≥ , 𝑑𝑝 𝜒 𝑝 𝑢,𝑣,𝑤∈𝐾 𝑇𝑝 (𝑢, 𝑣)𝑇𝑝 (𝑣, 𝑤) (1 − 𝑝)(log(1 − 𝑝)) 2 and we can bound the two sums appearing here by ∑︁ 𝑇𝑝 (𝑢, 𝑣) = ⟨𝑇𝑝 1, 1⟩𝐾 ≤ ∥𝑇𝑝 ∥ 2→2 ∥1∥ 2𝐾 = ∥𝑇𝑝 ∥ 2→2 |𝐾 | 𝑢,𝑣∈𝐾 288 and ∑︁ 𝑇𝑝 (𝑢, 𝑣)𝑇𝑝 (𝑢, 𝑤) = ⟨𝑇𝑝 1, 𝑇𝑝 1⟩𝐾 ≤ ∥𝑇𝑝 ∥ 22→2 ∥1∥ 2𝐾 = ∥𝑇𝑝 ∥ 22→2 |𝐾 | 𝑢,𝑣,𝑤∈𝐾 to obtain that ( 𝜒 𝑝 − ∥𝑇𝑝 ∥ 2→2 ) 2 𝑑 𝜒𝑝 ≥ . 𝑑𝑝 (1 − 𝑝)(log(1 − 𝑝)) 2 ∥𝑇𝑝 ∥ 22→2 This implies that 𝜀 𝜒 𝑝 𝑐 −𝜀 is bounded above and below by positive constants depending only on the degree and the value of ∥𝑇𝑝 𝑐 ∥ 2→2 ; the fact that all other implicit constants appearing in the consequences of the triangle condition can be taken to depend only on the degree and ∥𝑇𝑝 𝑐 ∥ 2→2 follows from this as can be seen from [Hut; Hut20h]. 3. The restriction that 𝑝 ∈ ( 𝑝 𝑐 − 𝛿, 𝑝 𝑐 + 𝛿) in (6.10.3) is only really needed for the lower bound, since the stated estimate is not sharp for 𝑝 very close to 0 or 1. Indeed, the claimed upper bound extends to all 0 ≤ 𝑝 ≤ 𝑝 𝑐 by monotonicity, and an upper bound of the same form follows for all 𝑝 𝑐 + 𝛿 ≤ 𝑝 ≤ 1 by [Hut20h, Proposition 3.1] and the methods of either [HH19] or [Hut23b]. We omit further details. □ Remark 6.10.1. Most the analysis of [Hut20h] can be skipped if one only wishes to establish our main results in the nonunimodular case, rather than the sharp uniform volume-tail estimate of Theorem 6.10.3. Indeed, the proof of [Hut20h, Proposition 3.1] yields in the transitive case that Ψ𝑝 (𝑆) := 𝜃 ( 𝑝)|𝑆| 1 ∑︁ ∑︁ 1 (𝑣 ∉ 𝑆)P 𝑝 (𝑣 ↔ ∞ off 𝑆) ≥ . 1 − 𝑝 𝑢∈𝑆 𝑣∼𝑢 𝑝(1 − 𝑝) 2 ∥𝑇𝑝 ∥ 22→2 (6.10.4) Since Ψ𝑝 (𝑆) is monotone in 𝑝, it follows from this together with Corollary 6.10.6, [Hut19b, Lemma 2.4] (which bounds the effect of changing 𝑝 on ∥𝑇𝑝 ∥ 2→2 ), and the mean-field lower bound that Ψ𝑝 (𝑆) ≥ 𝑐( 𝑝 − 𝑝 𝑐 )|𝑆|. for every connected, locally finite, nonunimodular transitive graph 𝐺 = (𝑉, 𝐸) and every finite set 𝑆 ⊆ 𝑉, where 𝑐 is a positive constant depending only on the degree of 𝐺. The fact that this implies a uniform upper bound on the volume tail of the form P 𝑝 (𝑛 ≤ |𝐾𝑜 | < ∞) ≤ exp[−𝑐(𝑑, 𝜀)𝑛] for every 𝑝 ≥ 𝑝 𝑐 + 𝜀 and 𝑛 ≥ 1 can easily be deduced from the methods of either [HH19] or [Hut23b]. Proof of Corollary 6.10.4. The analyticity of 𝜃 on ( 𝑝 𝑐 , 1] is already established in [HH19]. Since percolation with parameter 1 − (1 − 𝑝) 𝑛 can be thought of as the union of 𝑛 independent copies of 289 percolation with parameter 𝑝, we have that 𝜃 (1 − (1 − 𝑝) 𝑛 , 𝐺) ≥ (1 − 𝜃 ( 𝑝, 𝐺)) 𝑛 for every 𝑝 ∈ (0, 1) and 𝑛 ≥ 1, which implies the claimed bound on 𝜃 ( 𝑝, 𝐺) for 𝑝 close to 1 in conjunction with the mean-field lower bound and the fact that 𝑝 𝑐 is bounded away from zero for elements of N𝑑 . It remains to prove the uniform upper bound on the derivative. Given a connected, Í locally finite graph 𝐺 and 𝑝 ∈ [0, 1], the triangle diagram is defined by ∇ 𝑝 := 𝑥,𝑦 P 𝑝 (𝑜 ↔ 𝑥)P 𝑝 (𝑥 ↔ 𝑦)P 𝑝 (𝑦 ↔ 𝑜), and satisfies the inequality ∇ 𝑝 ≤ ∥𝑇𝑝 ∥ 32→2 ≤ 𝜒3𝑝,1/2 . The convergence of the triangle diagram at 𝑝 𝑐 is a well-known sufficient condition for mean-field percolation critical behaviour [AN84; BA91; Hut], and in particular, it follows from the results of [Hut] that if ∇ 𝑝 𝑐 < ∞ then there exist positive constants 𝑐 and 𝐶 depending only on ∇ 𝑝 𝑐 and the degree of 𝐺 such that 𝑐( 𝑝 − 𝑝 𝑐 ) ≤ 𝜃 ( 𝑝) ≤ 𝐶 ( 𝑝 − 𝑝 𝑐 ) (6.10.5) for every 𝑝 ≥ 𝑝 𝑐 . (Again, it may seem that these constants should also depend on 𝑝 𝑐 , but 𝑝 𝑐 is bounded below the reciprocal of the degree and bounded away from 1 by a quantity depending only on ∇ 𝑝 𝑐 .) Russo’s formula yields in our setting that   𝜃 ( 𝑝)𝑑 1 𝑑 𝜃 ( 𝑝) = E 𝑝 Ψ𝑝 (𝐾𝑜 ) 1 (|𝐾𝑜 | < ∞) ≤ E 𝑝 [|𝐾𝑜 | 1 (|𝐾𝑜 | < ∞)] 𝑑𝑝 1− 𝑝 1− 𝑝 for every 𝑝 > 𝑝 𝑐 , where Ψ𝑝 is defined in (6.10.4). (Note that Russo only applies directly for events depending on at most finitely many edges, but in our setting we may easily take a limit since, by Theorem 6.10.3, the contribution from distant edges is very small. The argument needed to do this is standard and is omitted.) The claim follows from this together with (6.10.5) and Theorem 6.10.3, Í −1 for some which implies that E 𝑝 [|𝐾𝑜 | 1 (|𝐾𝑜 | < ∞)] = ∞ 𝑛=1 P 𝑝 (𝑛 ≤ |𝐾 𝑜 | < ∞) ≤ 𝐶 | 𝑝 − 𝑝 𝑐 | constant 𝐶 depending only on the degree. □ 6.11 Proofs of the main theorems To streamline our proofs, let us introduce some language that will only be used in this subsection. Let H be a countably infinite set of (finite or infinite) transitive graphs. Say that an assignment 𝑝 : H → [0, 1] is supercritical if there is a constant 𝜀 > 0 such that for 𝜃 ((1 − 𝜀) 𝑝, 𝐺) ≥ 𝜀 for all but finitely many 𝐺 ∈ H , where 𝜃 ( 𝑝, 𝐺) is as defined in eq. (6.4.1). We say that H has a discrete percolation phase transition if limH 𝜃 ( 𝑝, 𝐺) = 1 for every supercritical assignment 𝑝. This generalises the definition of discrete percolation phase transition for families of finite transitive graphs given in Section 6.6. Recall the definition of irreducibly macromolecular from Section 6.9. Let us moreover say that H ⊂ F is irreducibly molecular if H is dense and irreducibly macromolecular. 290 Now say that H is of type... 1 if every graph in H is infinite and nonunimodular, and the vertex degrees in H are bounded; 2 if every graph in H is infinite and unimodular, and the vertex degrees in H are bounded; 3 if every graph in H is finite, and the vertex degrees in H are bounded; 4 if every graph in H is finite, limH deg 𝐺 = ∞, and H does not contain an infinite subset that is either macromolecular or has a discrete percolation phase transition; 5 if every graph in H is finite, limH deg 𝐺 = ∞, H is sparse and irreducibly macromolecular, and H does not contain an infinite subset that has a discrete percolation phase transition; 6 if every graph in H is finite, H does not have a discrete percolation phase transition, and H is irreducibly molecular; ...and say that H satisfies property... T if for every supercritical 𝑝, lim lim sup 𝜃 ( 𝑝, 𝐺) − P 𝑝 (|𝐾𝑜 | ≥ 𝑛) = 0; 𝑛→∞ L H if for every supercritical 𝑝 and for every infinite subset H ′ ⊆ H : – if H ′ converges locally to some infinite transitive graph 𝐺 ∞ , then lim′ |𝜃 ( 𝑝, 𝐺) − 𝜃 ( 𝑝, 𝐺 ∞ )| = 0; H – if limH ′ deg 𝐺 = ∞, then lim′ |𝜃 ( 𝑝, 𝐺) − mf( 𝑝 deg 𝐺)| = 0; H C if for every supercritical 𝑝 and for every infinite subset H ′ ⊆ H ∩ F , lim P 𝑝 (|∥giant∥ − 𝜃 ( 𝑝, 𝐺)| > 𝜀) = 0; H E if for every supercritical 𝑝, lim sup sup |𝜃 (𝑞, 𝐺) − 𝜃 ((1 − 𝛿)𝑞, 𝐺)| = 0. 𝛿↓0 H 𝑞∈[ 𝑝,1] 291 We will establish the validity of the following table. Formally, this means that if H has some type 𝑖, then for every property 𝑃, if the (𝑃, 𝑖)th entry of the following table is a tick (✓), then H has property 𝑃, whereas if the entry is a cross (✗), then H does not have property 𝑃. T L E C 1 2 3 4 5 6 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✗ ✗ ✓ ✗ Starting the table In this subsection, we continue to let H be a countably infinite set of (finite or infinite) transitive graphs. Lemma 6.11.1. The following portion of our table is correct. 1 2 3 ✓ T L E C 4 5 6 ✓ ✓ ✓ ✓ Proof. (T,1) This follows immediately from Theorem 6.10.3. (L,2) and (L,3) Suppose that H is of type 2 or 3, and consider an infinite subset H ′ ⊆ H that converges locally to some infinite transitive graph 𝐺 ∞ . By Proposition 6.4.1, lim 𝜃 ( 𝑝, 𝐺) = 𝜃 ( 𝑝, 𝐺 ∞ ) H for every constant 𝑝 > 𝑝 ∗ := lim supH 𝑝 𝑐 (𝐺). In particular, 𝜃 ( 𝑝, 𝐺 ∞ ) > 0 for all 𝑝 > 𝑝 ∗ . So ( 𝑝 ∗ , 1] ⊆ ( 𝑝 𝑐 (𝐺 ∞ ), 1], and thus by [Sch99], 𝜃 (·, 𝐺 ∞ ) is continuous on ( 𝑝 ∗ , 1]. Now consider an arbitrary (possibly non-constant) supercritical assignment 𝑝. Since 𝑝 is supercritical, there exists a constant 𝜀 > 0 such that 𝑝 ∈ [ 𝑝 ∗ + 𝜀, 1] for all but finitely many 𝐺 ∈ H . Recall that if a sequence of monotone functions defined on a closed and bounded interval converges pointwise to a continuous function, then the sequence actually converges uniformly. This readily implies that as 𝐺 → ∞ 292 with 𝐺 ∈ H , the density 𝜃 (·, 𝐺) converges uniformly to 𝜃 (·, 𝐺 ∞ ) on the interval [ 𝑝 ∗ + 𝜀, 1]. In particular, lim |𝜃 ( 𝑝, 𝐺) − 𝜃 ( 𝑝, 𝐺 ∞ )| = 0. H (T,4) This follows immediately from Proposition 6.8.1 and Theorems 6.1.1 and 6.2.1. (E,5) Suppose that H has type 5. Let H ′ ⊆ H be an infinite subset, and let 𝑞 1 , 𝑞 2 : H ′ → (0, 1) be a pair of assignments satisfying limH ′ 𝑞 2 /𝑞 1 = 1 and 𝑞 1 , 𝑞 2 ∈ [ 𝑝, 1] for all 𝐺 ∈ H ′. Since 𝜃 (·, 𝐺) is trivially continuous for each 𝐺 ∈ H , to establish E, it suffices to show that lim′ |𝜃 (𝑞 2 , 𝐺) − 𝜃 (𝑞 1 , 𝐺)| = 0. H | {z } ♣ By Lemma 6.2.2 and Proposition 6.6.1, this holds if limH ′ 𝑞 2 deg 𝐺 ∈ {0, ∞}. So it suffices for us to hanfdle the case when there is a constant 𝐶 < ∞ such that 𝑞 2 deg 𝐺 ∈ [1/𝐶, 𝐶] for all 𝐺 ∈ H ′. Fix an irreducibly macromolecular decomposition ( 𝐴, 𝐵) for H ′, and similarly to as in the statement of Proposition 6.9.1, for each 𝐺 ∈ H ′ and for both 𝑖 ∈ {1, 2}, let 𝜓𝑖 (𝐺) := mf(𝑞𝑖 (𝐺) deg 𝐺) and 𝑞𝑖∗ (𝐺) := 𝑞𝑖 (𝐺)𝜓𝑖 (𝐺) 2 . For all 𝐺 ∈ H ′ and 𝑛 ≥ 1, ♣ ≤ ♥1𝑛 + ♥2𝑛 + ♦𝑛 , where for both 𝑖 ∈ {1, 2}, ♥𝑖𝑛 := 𝜃 (𝑞𝑖 , 𝐺) − 𝜓𝑖 · P𝑞𝐵𝑖 (|𝐾𝑜 | ≥ 𝑛) , and ♦𝑛 := 𝜓2 · P𝑞𝐵2 (|𝐾𝑜 | ≥ 𝑛) − 𝜓1 · P𝑞𝐵1 (|𝐾𝑜 | ≥ 𝑛) . By Proposition 6.9.1, limH ′ ♥𝑖𝑛 = 0 for both 𝑖 ∈ {1, 2}. So it suffices to prove that limH ′ ♦𝑛 = 0. Using that 𝜓 and 𝑞 2 are trivially bounded above by 1, for all 𝐺 and 𝑛, ♦𝑛 ≤ P𝑞𝐵2 (|𝐾𝑜 | ≥ 𝑛) − P𝑞𝐵1 (|𝐾𝑜 | ≥ 𝑛) + |𝜓2 − 𝜓1 | . | {z } | {z } ♦2 ♦1𝑛 There can never be more than 𝑛 deg 𝐺 closed pivotal edges for the event {|𝐾𝑜 | ≥ 𝑛}. So by Russo’s formula, for all 𝐺 and 𝑛, the derivative of the function 𝑓𝐺,𝑛 ( 𝑝) := P𝐵𝑝 (|𝐾𝑜 | ≥ 𝑛) 293 satisfies ′ 𝑓𝐺,𝑛 ( 𝑝) ≤ 𝑛 deg 𝐺 1− 𝑝 for all 𝑝 ∈ (0, 1). In particular, by the mean-value theorem, for all 𝐺 and 𝑛, ♦1𝑛 ≤ (𝑞 2 − 𝑞 1 ) · 𝑛 deg 𝐺 . 1 − 𝐶/deg 𝐺 (6.11.1) 1 ′ Since limH deg 𝐺 = ∞, we have 1−𝐶/deg 𝐺 ≤ 2 for all but finitely many 𝐺 ∈ H . By plugging this and 𝑞 1 ≤ deg𝐶 𝐺 into eq. (6.11.1), we obtain  ♦1𝑛 ≤  𝑞2 − 1 · 2𝐶𝑛, 𝑞1 and hence limH ′ ♦1𝑛 = 0 for all 𝑛. Note that the function 𝑓 : [−∞, +∞] → [0, 1] given by 𝑓 (𝑥) := mf (𝑒 𝑥 ) for all 𝑥 ∈ R, 𝑓 (−∞) := 0, and 𝑓 (+∞) := 1, is uniformly continuous. Since   𝑞2 = 0, lim′ |log (𝑞 2 deg 𝐺) − log (𝑞 1 deg 𝐺)| = lim′ log H H 𝑞1 it follows that limH ′ ♦2 = 0. By combining this with our control of ♦1𝑛 , we have limH ′ ♦𝑛 = 0, as required. □ Lemma 6.11.2. The following portion of our table is correct. 1 T L E C ✓ 2 3 4 5 6 ✗ ✗ ✓ ✗ Proof. (C,1) and (C,2) These hold vacuously because if H is of type 1 or 2, then none of the graphs in H are finite. (T,5) Suppose that H is of type 5. Since H does not have a discrete percolation phase transition, by Corollary 6.6.2, there exists a supercritical assignment 𝑝 satisfying lim inf 𝑝 deg 𝐺 < ∞. H 294 It now follows easily from Claim 6.9.2 and the definition of macromolecular decompositions that there exists a constant 𝜀 > 0 such that   1 (6.11.2) lim sup P 𝑝 𝜀 deg 𝐺 ≤ |𝐾𝑜 | < deg 𝐺 ≥ 𝜀. 𝜀 H (With probability bounded away from zero, for infinitely many choices of 𝐺, every edge in the boundary of the equivalence class of 𝑜 is closed, but the cluster of 𝑜 contains a positive proportion of the vertices in the equivalence class of 𝑜.) For all 𝑛 ≥ 1 and 𝐺 ∈ H , by transitivity, ♥𝑛,𝐺 := 𝜃 ( 𝑝, 𝐺) − P 𝑝 (|𝐾𝑜 | ≥ 𝑛)
1 ∑︁ = P 𝑝 (𝑣 ∈ giant) − P 𝑝 (|𝐾𝑣 | ≥ 𝑛) |𝑉 | 𝑣∈𝑉      1 ∑︁ 1 1 ≥ P 𝑝 𝜀 deg 𝐺 ≤ |𝐾𝑣 | < deg 𝐺 − P 𝑝 |giant| < deg 𝐺 . |𝑉 | 𝑣 𝜀 𝜀 So by eq. (6.11.2), Theorem 6.2.1, and the fact that H is sparse, for all 𝑛 ≥ 1, lim sup ♥𝑛,𝐺 ≥ 𝜀. H Since 𝜀 is independent of 𝑛, this shows that H does not have property T. (T,6) Suppose that H is of type 6. By Theorem 6.1.1, H does not have the supercritical uniqueness property. So there exists a supercritical assignment 𝑝 and a constant 𝜀 > 0 such that 1 ∑︁ P 𝑝 (|𝐾𝑣 | ≥ 𝜀 |𝐺 | but 𝑣 ∉ giant) ≥ 𝜀. lim sup |𝑉 | 𝑣∈𝑉 H By arguing along the same lines as in case (T,5), it follows that for all 𝑛 ≥ 1, lim sup ♥𝑛,𝐺 ≥ 𝜀, H and hence that H does not have property T. (C,6) This follows immediately from [EH21a, Lemma 3.6]. 295 □ Finishing the table In this subsection, we continue to let H be a countably infinite set of (finite or infinite) transitive graphs. We will extend the portion of our table completed in the last subsection to complete the whole table. We will use “(𝑃, 𝑖)” as shorthand for the statement “if H is of type 𝑖, then H has property 𝑃”. Recall that F𝑑 is the subset of F of graphs with vertex degree exactly 𝑑, and recall that mf (𝜆) is the probability that a Poisson(𝜆)-branching process survives forever. The following result is easy to prove and known to experts. It is classical for percolation on complete graphs and hypercubes, and the standard arguments extend without change to the general case. There are many elementary ways to prove this result, so we will omit the details. (The arguments even work more generally for high-degree regular graphs that are not necessarily transitive.) Lemma 6.11.3. For every positive constant 𝐶, lim lim sup sup sup 𝑛→∞ 𝑑→∞ 𝐺∈F 𝑝∈[0,𝐶/𝑑] 𝑑 P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) − mf( 𝑝𝑑) = 0. Proof sketch. Let P𝜆 denote the law of a Poisson(𝜆)-branching process, and let 𝑁 be total number of offspring. Fix 𝐶 > 0. We have (e.g. by Dini’s theorem) that P𝜆 (𝑁 ≥ 𝑘) → mf (𝜆) as 𝑘 → ∞, uniformly over all 𝜆 ∈ [0, 𝐶]. Moreover, for every fixed 𝑘, we have (e.g. by comparing an exploration of 𝐾𝑜 to a Binomial branching process, by coupling to percolation on a regular tree, or by counting the number of trees of a given size) that P𝜆/deg 𝐺 (|𝐾𝑜 | ≤ 𝑘) − P𝜆 (|𝐾𝑜 | ≤ 𝑘) → 0 as 𝐺 → ∞ with 𝐺 ∈ F , uniformly over all 𝜆 ∈ [0, 𝐶]. So given any 𝜂 > 0, if we first pick 𝑛 sufficiently large, then pick 𝑑 sufficiently large (depending on 𝑛), we can ensure that 𝐺 (|𝐾𝑜 | ≥ 𝑛) − mf (𝜆) ≤ 𝜂 P𝜆/𝑑 for all 𝐺 ∈ F𝑑 and all 𝜆 ∈ [0, 𝐶]. □ Lemma 6.11.4. For every 𝑖 ∈ {1, . . . , 6}, (𝑇, 𝑖) ⇐⇒ Proof. 296 (𝐿, 𝑖). (𝑇, 𝑖) =⇒ (𝐿, 𝑖) Let H ′ ⊆ H be an arbitrary infinite subset that either (1) converges locally to some infinite transitive graph 𝐺 ∞ , or (2) satisfies limH ′ deg 𝐺 = ∞. 1. For all 𝐺 ∈ H and 𝑛 ≥ 1, ♥ := |𝜃 ( 𝑝, 𝐺) − 𝜃 ( 𝑝, 𝐺 ∞ )| ≤ 𝜃 ( 𝑝, 𝐺) − P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) + P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) − P𝐺𝑝 ∞ (|𝐾𝑜 | ≥ 𝑛) + P𝐺𝑝 ∞ (|𝐾𝑜 | ≥ 𝑛) − 𝜃 ( 𝑝, 𝐺 ∞ ) . | {z } | {z } | {z } ♥1𝑛 ♥2𝑛 ♥3𝑛 Since 𝑝 is supercritical, for all 𝑛, lim P 𝑝 (|giant| ≥ 𝑛) = 1, H and hence lim sup ♥1𝑛 = lim sup P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛 but 𝑜 ∉ giant) , H (6.11.3) H which tends to zero as 𝑛 tends to infinity if T holds. In particular, if T holds then   lim sup ♥ ≤ lim sup inf ♥1𝑛 + ♥2𝑛 + ♥3𝑛 𝑛≥1 H′ H′   1 2 3 ≤ lim sup lim sup ♥𝑛 + ♥𝑛 + ♥𝑛 𝑛→∞ H′ ≤ lim sup lim sup ♥1𝑛 + sup lim sup ♥2𝑛 + lim sup ♥3𝑛 = 0. 𝑛→∞ | H′ {z =0 by T H′ 𝑛≥1 } 𝑛→∞ | {z } | {z } =0 =0 2. Thanks to Corollary 6.6.2, the claim is trivial if lim′ 𝑝 deg 𝐺 = ∞. H So by passing to a further infinite subset if necessary, let us assume without loss of generality that sup 𝑝 deg 𝐺 < ∞, H′ which will allow us to invoke Lemma 6.11.3. For all 𝐺 ∈ H and 𝑛 ≥ 1, ♦ := |𝜃 ( 𝑝, 𝐺) − mf ( 𝑝 deg 𝐺)| ≤ ♥1𝑛 + P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) − mf ( 𝑝 deg 𝐺) . | {z } ♦2𝑛 297 In particular, if T holds then   lim sup ♦ ≤ lim sup inf ♦1𝑛 + ♦2𝑛 𝑛≥1 H′ H′   ≤ lim sup lim sup ♦1𝑛 + ♦2𝑛 H′ 𝑛→∞ ≤ lim sup lim sup ♦1𝑛 + lim sup lim sup ♦2𝑛 = 0. H′ 𝑛→∞ | (𝑇, 𝑖) ⇐= (𝐿, 𝑖) {z =0 by T H′ 𝑛→∞ } | {z } =0 by Lemma 6.11.3 Thanks to eq. (6.11.3), our goal is to show that lim lim ♥1𝑛 = 0, 𝑛→∞ H or equivalently, to show that this holds for every infinite subset H ′ ⊆ H that either (1) converges locally to some infinite transitive graph 𝐺 ∞ , or (2) satisfies limH deg 𝐺 = ∞. 1. By a similar argument as above, for all 𝐺 ∈ H and 𝑛 ≥ 1, ♥1𝑛 ≤ ♥ + ♥2𝑛 + ♥3𝑛 . So if L holds then lim sup lim sup ♥1𝑛 ≤ lim sup ♥ + sup lim sup ♥2𝑛 + lim sup ♥3𝑛 = 0. 𝑛→∞ H′ H′ H′ 𝑛≥1 | {z } =0 by L 𝑛→∞ | {z } | {z } =0 =0 2. By a similar argument as above, for all 𝐺 ∈ H and 𝑛 ≥ 1, ♥1𝑛 ≤ ♦ + ♦2𝑛 . So if L holds then lim sup lim sup ♥𝑛 ≤ lim sup ♦ + lim sup lim sup ♦2𝑛 = 0. 𝑛→∞ H′ H′ 𝑛→∞ | {z } | =0 by L H′ {z } =0 by Lemma 6.11.3 □ 298 Lemma 6.11.5. For all 𝑖 ∈ {1, . . . , 6}, (𝐿, 𝑖) =⇒ (𝐸, 𝑖). Proof. Let H ′ ⊆ H be an infinite subset, and let 𝑞 1 , 𝑞 2 : H ′ → (0, 1) be a pair of assignments satisfying limH ′ 𝑞 2 /𝑞 1 = 1 and 𝑞 1 , 𝑞 2 ∈ [ 𝑝, 1] for all 𝐺 ∈ H ′. To establish E, it suffices to show that lim′ |𝜃 (𝑞 2 , 𝐺) − 𝜃 ((𝑞 1 , 𝐺)| = 0. H | {z } ♥ Without loss of generality, we may further assume that either (1) H ′ converges locally to some infinite transitive graph 𝐺 ∞ , or (2) limH ′ deg 𝐺 = ∞. 1. For every 𝐺 ∈ H ′, ♥ ≤ |𝜃 (𝑞 2 , 𝐺) − 𝜃 (𝑞 2 , 𝐺 ∞ )| + |𝜃 (𝑞 2 , 𝐺 ∞ ) − 𝜃 (𝑞 1 , 𝐺 ∞ )| + |𝜃 (𝑞 1 , 𝐺 ∞ ) − 𝜃 (𝑞 1 , 𝐺)| . | {z } ♥1 So if L holds then lim sup ♥ ≤ lim sup ♥1 , H′ H′ and since 𝑞 1 is supercritical, 𝑏 := lim inf 𝑞 1 > 𝑝 𝑐 (𝐺 ∞ ). ′ H So by [Sch99], the function 𝜃 (·, 𝐺 ∞ ) is uniformly continuous on [𝑏, 1]. Since lim sup |𝑞 2 − 𝑞 1 | ≤ lim sup H′ H 𝑞2 − 1 = 0, 𝑞1 it follows that limH ′ ♥1 = 0, as required. 2. Arguing as in the case above, if L holds then lim sup ♥ ≤ lim sup |mf (𝑞 2 deg 𝐺) − mf(𝑞 1 deg 𝐺)| . | {z } H′ H′ ♥2 Note that the function 𝑓 : [−∞, +∞] → [0, 1] given by 𝑓 (𝑥) := mf (𝑒 𝑥 ) for all 𝑥 ∈ R, 𝑓 (−∞) := 0, and 𝑓 (+∞) := 1, is uniformly continuous. Since   𝑞2 lim′ |log (𝑞 2 deg 𝐺) − log (𝑞 1 deg 𝐺)| = lim′ log = 0, H H 𝑞1 it follows that lim supH ′ ♥2 = 0, as required. 299 □ Lemma 6.11.6. For all 𝑖 ∈ {3, 4, 5}, (𝐸, 𝑖) =⇒ (𝐶, 𝑖). Proof. Assume that H is of type 3,4, or 5. So every graph in H is finite, and H does not contain an infinite subset that is molecular. Fix a constant 𝜀 > 0. If lim sup P 𝑝 (∥giant∥ ≥ 𝜃 ( 𝑝, 𝐺) + 𝜀) > 0, (6.11.4) H then by Theorem 6.2.1, there is a sequence 𝛿 : H → [0, 1] with limH 𝛿 = 0 such that lim sup P (1+𝛿) (∥giant∥ ≥ 𝜃 ( 𝑝, 𝐺) + 𝜀) = 1, H and hence lim sup 𝜃 ((1 + 𝛿) 𝑝, 𝐺) ≥ 𝜃 ( 𝑝, 𝐺) + 𝜀, H which implies that E does not hold. The same reasoning shows that E does not hold if lim sup P 𝑝 (∥𝐾𝑜 ∥ ≤ 𝜃 ( 𝑝, 𝐺) − 𝜀) > 0. (6.11.5) H If H does not have property 𝐶, then eq. (6.11.4) or eq. (6.11.5) must hold for some choice of 𝜀 > 0. So, putting everything together, we have shown that if H does not have property 𝐶, then H does not have property 𝐸 either. □ Lemma 6.11.7. (𝐶, 5) ∧ (𝐿, 5) =⇒ (𝐸, 6). Proof. Suppose that (𝐶, 5) and (𝐿, 5) hold. Suppose that H is of type 6, and let ( 𝐴, 𝐵) be irreducibly macromolecular decomposition for H . For each 𝐺 ∈ H , let ∼ be an equivalence class inducing the macromolecular decomposition ( 𝐴, 𝐵). Now the fact that H has property 𝐸 follows easily from the fact that the subgraphs induced by equivalence classes of ∼ all have properties 𝐸 and 𝐶. Although this can be shown directly, for completeness, let us note that this follows for example from Claim 6.9.5. (Although this claim is proven in the setting that H is sparse and irreducibly macromolecular, the hypothesis that H is sparse is not used for this claim.) □ 300 Proofs of the main theorems By applying the results of Section 6.11, we can extend the entries we found in Section 6.11 to establish that all 4 × 6 entries of our table are correct. To deduce our main theorems from this table, the only non-trivial fact to check is that every infinite set of finite transitive graphs contains an infinite subset that is of some type 𝑖 ∈ {3, 4, 5, 6}. Even more precisely, it suffices to show that if H is macromolecular then H has an infinite subset that is irreducibly macromolecular. This is the content of the following lemma. Lemma 6.11.8. Every infinite set of finite transitive graphs that is macromolecular contains an infinite subset that is irreducibly macromolecular. Proof. Let H be an infinite set of finite transitive graphs that is macromolecular. Let 𝛿∗ ∈ [0, 1] be the largest constant such that for all 𝛿 ∈ [0, 𝛿∗ ] there exists a constant 𝜀 > 0 such that infinitely many graphs in H admit a macromolecular decomposition ( 𝐴, 𝐵) where 𝐴 is 𝛿-dense and |𝐸 (𝐵)| 1 ≤ . |𝐺 | 𝜀 Since H is itself macromolecular, we know that 𝛿∗ > 0. Now pick a constant 𝜀 > 0, an infinite 9 1 𝛿∗ -dense and |𝐸|𝐺(𝐵)| set I ⊆ 𝐻, and a collection (( 𝐴(𝐺), 𝐵(𝐺)) : 𝐺 ∈ I) such that 𝐴(𝐺) is 10 | ≤ 𝜀 for all 𝐺 ∈ I. It suffices to show that ( 𝐴(𝐺) : 𝐺 ∈ I) does not contain an infinite subset that is macromolecular. Suppose for contradiction that there exists an infinite subset J ⊆ I and a constant 𝜂 > 0 such that for each 𝐺 ∈ J , we can find an 𝜂-macromolecular decomposition ( 𝐴′ (𝐺), 𝐵′ (𝐺)) for 𝐴(𝐺). Note that every 𝐺 ∈ J now trivially admits a macromolecular decomposition ( 𝐴′, 𝐵′′) where |𝐸 (𝐵′′)| |𝐸 (𝐵)| |𝐸 (𝐵′)| 2 2 = + ≤ + . |𝐺 | |𝐺 | | 𝐴| 𝜀 𝜂 We will show that for infinitely many choices of 𝐺 ∈ I, the graph 𝐴′ is 34 𝛿∗ -dense, contradicting the maximality of 𝛿∗ . Note that for all 𝐺 ∈ J , deg 𝐴 = deg 𝐺 − 2 |𝐸 (𝐵)| 2 ≥ deg 𝐺 − , |𝐺 | 𝜀 and similarly, 2 |𝐸 (𝐵′)| 2 ≥ deg 𝐴 − . |𝑉 ( 𝐴)| 𝜂 ′ ′ ′ In particular, note that lim𝐺∈J | 𝐴 | = ∞ because | 𝐴 | ≥ deg 𝐴 for all 𝐺, and lim𝐺∈H deg 𝐺 = ∞. So without loss of generality, let us assume that every 𝐺 ∈ I satisfies deg 𝐴′ = deg 𝐴 − | 𝐴′ | ≥ 16 16 + . 𝜀𝛿∗ 𝜂𝛿∗ 301 For all 𝐺 ∈ I, since | 𝐴′ | must be a non-trivial divisor of | 𝐴|, we have | 𝐴′ | ≤ 12 | 𝐴|. Therefore, 2 2 deg 𝐴 𝛿∗ deg 𝐴′ deg 𝐺 − 𝜀 − 𝜂 ≥ − . ≥ 2 ′ 1 |𝐴 | | 𝐴| 4 | 𝐴| 2 9 9 In particular, since 𝐴 is 10 𝛿∗ -dense, 𝐴′ must be 2 · 10 𝛿∗ − 𝛿2∗ = 43 𝛿∗ -dense, as required. □ Now Theorem 6.1.3 follows immediately from Theorem 6.1.1 and row 𝐶 of our table. Theorem 6.1.5 follows from (𝐿, 1) and (𝐿, 2), since the set of non-unimodular infinite transitive graphs is a closed and open subset of the set of all infinite transitive graphs. Theorem 6.1.7 follows from (𝐿, 3) and (for the converse) (𝐿, 4) and (𝐿, 5). Theorem 6.1.11 follows from row 𝐸. 302 Chapter 7 303 EXISTENCE OF A PERCOLATION THRESHOLD ON FINITE TRANSITIVE GRAPHS Abstract Let (𝐺 𝑛 ) be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters ( 𝑝 𝑛 ) is a percolation threshold if for every 𝜀 > 0, the proportion ∥𝐾1 ∥ of vertices contained in the largest cluster under bond percolation P𝐺𝑝 satisfies both 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 1 lim P𝐺(1+𝜀) 𝑝𝑛 for some 𝛼 > 0, and 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 0 lim P𝐺(1−𝜀) 𝑝 for all 𝛼 > 0. 𝑛→∞ 𝑛→∞ 𝑛 We prove that (𝐺 𝑛 ) has a percolation threshold if and only if (𝐺 𝑛 ) does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville’s new proof of the sharpness of the phase transition for infinite graphs via couplings [Van22b] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [EH21b]. 7.1 Introduction Given a graph 𝐺, build a random spanning subgraph 𝜔 by independently including each edge with a fixed probability 𝑝. This model P𝐺𝑝 is called (Bernoulli bond) percolation. In their pioneering work on random graphs, Erdős and Rényi [ER60] proved that when 𝐺 is the complete graph on 𝑛 1+𝜀 vertices, percolation has a phase transition: as we increase 𝑝 from 1−𝜀 𝑛 to 𝑛 for any fixed 𝜀 > 0, a giant cluster suddenly emerges containing a positive proportion of the total vertices. Since then, there has been much interest in establishing this phenomenon for more general classes of finite graphs. However, as remarked in [Bol+10c], progress has been slow. In this paper, we solve this problem for arbitrary finite graphs that are (vertex-)transitive, meaning that for any two vertices 𝑢 and 𝑣, there is a graph automorphism mapping 𝑢 to 𝑣. This includes all Cayley graphs, and in particular, the complete graphs, hypercubes, and tori. Our setting of percolation on finite transitive graphs places us at the intersection of two wellestablished fields. Loosely speaking, one of these began in combinatorics with the work of Erdős and Rényi [ER59; ER60], whereas the other began in mathematical physics with the work of Broadbent and Hammersley [BH57a]. In the former, a subgraph of a finite graph is said to percolate if its largest cluster contains a positive proportion of the total vertices, whereas in the latter, a subgraph of an infinite transitive graph is said to percolate if its largest cluster is infinite. This leads to two different definitions of what it means to have a percolation phase transition. Let us start by making these precise. Here are the graph-theoretic conventions we will be using throughout: The volume of a graph 𝐺 = (𝑉, 𝐸) is simply the number of vertices |𝑉 |. We label the clusters (i.e. connected components) of a spanning subgraph of 𝐺 in decreasing order of volume by 𝐾1 , 𝐾2 , . . . . In a slight abuse of notation, we also write 𝐾𝑣 for the cluster containing a given vertex 𝑣. The density of a cluster 𝐾 is | ∥𝐾 ∥ := |𝐾 |𝑉 | , the proportion of vertices contained in 𝐾. Now let (𝐺 𝑛 ) be a sequence of finite graphs with volume tending to infinity. Following Bollobás, Borgs, Chayes, and Riordan [Bol+10c], we say that (𝐺 𝑛 ) has a percolation phase transition if there is a sequence of parameters ( 𝑝 𝑛 ) such that for every 𝜀 > 0, both1 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 1 lim P𝐺(1+𝜀) 𝑝 for some 𝛼 > 0, and 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 0 lim P𝐺(1−𝜀) 𝑝 for all 𝛼 > 0. 𝑛→∞ 𝑛→∞ 𝑛 𝑛 This is the subject of our paper: we characterise the existence of a percolation phase transition for finite transitive graphs. Let us mention that the question of a percolation phase transition on general finite graphs is attributed by the above authors of [Bol+10c] to Bollobás, Kohakayawa, and Łuksak [BKŁ92]. On the other hand, when 𝐺 is an infinite (locally finite) transitive graph, we define the critical parameter  𝑝 𝑐 := sup 𝑝 : P𝐺𝑝 (there exists an infinite cluster) = 0 . By Kolmogorov’s zero-one law, the probability of an infinite cluster under P𝐺𝑝 is zero when 𝑝 < 𝑝 𝑐 and one when 𝑝 > 𝑝 𝑐 . So in a trivial sense, 𝐺 always has a percolation phase transition. The real question is whether 𝑝 𝑐 < 1. (The fact that 𝑝 𝑐 > 0 is obvious by a branching argument.) So in this context, we often say that 𝐺 has a percolation phase transition to mean that 𝑝 𝑐 < 1. Hutchcroft and Tointon [HT21d] dealt with an analogue of this question for finite transitive graphs with bounded vertex degrees, i.e. the question of whether for a given sequence (𝐺 𝑛 ) of such graphs, there exists 𝑛 ∥ 𝛿 > 0 such that P𝐺 1−𝛿 (∥𝐾1 ≥ 𝛿) ≥ 𝛿 for all 𝑛. This is not the subject of our paper. To avoid any possible confusion, when a sequence of finite graphs (𝐺 𝑛 ) with volume tending to infinity has a percolation phase transition in the above sense of [Bol+10c], we will instead say that it has a percolation threshold, referring to the threshold sequence of parameters ( 𝑝 𝑛 ) in the definition. 1 We will use the convention that P𝐺 := P𝐺 if 𝑝 > 1 and P𝐺 := P𝐺 if 𝑝 < 0. 𝑝 𝑝 1 0 304 The main result of this paper is Theorem 8.1.1 below, which characterises the existence of a percolation threshold on a sequence of finite transitive graphs in terms of the presence of an infinite collection of molecular subsequences. We discovered molecular sequences with Hutchcroft in [EH21b] as the only obstacles to the supercritical giant cluster being unique. Interestingly, unlike the usual story for percolation on a new family of graphs (as told in the introduction of [Bol+10c], for example), uniqueness of the supercritical giant cluster came first, before the existence of a percolation threshold, and the former is key to our proof of the latter. See Section 7.1 for more background. Here we will just recall the definition of a molecular sequence before stating Theorem 8.1.1. Definition 7.1.1. Given an integer 𝑚 ≥ 2, we say that (𝐺 𝑛 ) is 𝑚-molecular if it is dense, meaning that lim inf 𝑛→∞ |𝐸 (𝐺 𝑛 )|2 > 0, and there is a constant 𝐶 < ∞ such that for every 𝑛, there is a set of |𝑉 (𝐺 𝑛 )| edges 𝐹𝑛 ⊆ 𝐸 (𝐺 𝑛 ) satisfying the following conditions: 1. 𝐺 𝑛 \𝐹𝑛 has 𝑚 connected components; 2. 𝐹𝑛 is invariant under the action of Aut 𝐺 𝑛 ; 3. |𝐹𝑛 | ≤ 𝐶 |𝑉 (𝐺 𝑛 )|. For example, the sequence of Cartesian products of complete graphs (𝐾𝑛 □𝐾𝑚 )𝑛≥1 is 𝑚-molecular. We say that (𝐺 𝑛 ) is molecular if it is 𝑚-molecular for some 𝑚 ≥ 2. Theorem 7.1.2. A sequence of finite connected transitive graphs with volume tending to infinity has a percolation threshold if and only if it contains an 𝑚-molecular subsequence for at most finitely many integers 𝑚. The condition that a sequence (𝐺 𝑛 ) contains 𝑚-molecular subsequences for infinitely many integers 𝑚 is extremely stringent. For example, we can rule it out if (𝐺 𝑛 ) is either sparse or dense, i.e. the edge density of 𝐺 𝑛 either tends to zero or remains bounded away from zero. Indeed, it is clear that a sparse sequence cannot contain any molecular subsequences, but also notice that since every 𝑚-molecular sequence (𝐺 𝑛 ) satisfies lim sup𝑛→∞ |𝐸 (𝐺 𝑛 )|2 ≤ 𝑚1 , a dense sequence can contain an |𝑉 (𝐺 𝑛 )| 𝑚-molecular subsequence for at most finitely many integers 𝑚. In particular, our result implies the existence of a percolation threshold for the complete graphs, hypercubes, and tori. Our result is new even under the additional hypothesis that the graphs have uniformly bounded vertex degrees2 , which is particularly relevant to percolation on infinite graphs. 2 One could imagine a family of sequences of such graphs that each has a percolation threshold but such that the (1+𝜀)-supercritical giant cluster density is not bounded away from zero over the entire family. Then by diagonalising, we could construct a sequence without a percolation threshold. 305 Most previous work on percolation on finite graphs treated specific sequences such as the complete graphs, hypercubes, and tori. Indeed, many authors have remarked how little work has been done on more general classes of finite graphs [ABS04c; Bor+05a; Bol+10c; Ben13b]. Alon, Benjamini, and Stacey [ABS04c]3 studied percolation on expanders with bounded vertex degrees. In particular, they proved that if each graph is 𝑑-regular for some fixed integer 𝑑 and has girth tending to infinity, then the sequence has a (constant) percolation threshold at 𝑝 𝑛 := 1/(𝑑 − 1). Borgs, Chayes, van der Hofstad, Slade, and Spencer [Bor+05a; Bor+05b] and Nachmias [Nac09] analysed the emergence of a cluster with volume of order |𝑉 (𝐺 𝑛 )| 2/3 for percolation on finite transitive graphs satisfying the triangle condition or a random walk return-probability condition, respectively, both of which enforce mean-field behaviour. Frieze, Krivelevich, and Martin [FKM04] proved that if a sequence of finite regular graphs is pseudorandom (an eigenvalue condition forcing the graph to be like the complete graph), then it has a percolation threshold at 𝑝 𝑛 := 1/deg(𝐺 𝑛 ) where deg(𝐺 𝑛 ) is the vertex degree of the 𝑛th graph. Bollobás, Borgs, Chayes, and Riordan [Bol+10c] studied percolation on arbitrary sequences of finite graphs (𝐺 𝑛 ) that are dense, meaning that lim inf 𝑛→∞ |𝐸 (𝐺 𝑛 )|2 > 0. The |𝑉 (𝐺 𝑛 )| following theorem from their paper will be used in our argument. Theorem 7.1.3 (Bollobás, Borgs, Chayes, Riordan 2010). Let (𝐺 𝑛 ) be a sequence of finite connected graphs with volume tending to infinity. Suppose that (𝐺 𝑛 ) is dense. For each 𝑛, let 𝜆 𝑛 be the largest eigenvalue of the adjacency matrix of 𝐺 𝑛 . Then (1/𝜆 𝑛 )𝑛≥1 is a percolation threshold for (𝐺 𝑛 ). Our proof of Theorem 8.1.1 does not build a percolation threshold by defining a natural candidate for the critical parameter of a finite graph in terms of, say, its vertex degrees or the largest eigenvalue of the adjacency matrix. In our setting, where we have no quantitative assumptions, we are forced to use a softer and more indirect approach. In particular, our proof of Theorem 8.1.1 says little about the rate at which the percolation probabilities tend to zero or to one. That said, as part of our proof we do obtain explicit lower bounds on the supercritical giant cluster density, which are analogous to the well-known mean-field lower bound for percolation on infinite graphs. Corollary 7.1.4. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity that does not contain an 𝑚-molecular subsequence for any 𝑚 > 𝑀, where 𝑀 is some positive integer. Let ( 𝑝 𝑛 ) be a percolation threshold, which exists by Theorem 8.1.1. Then for every 𝜀 > 0,   𝜀 𝐺𝑛 − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. P (1+𝜀) 𝑝 𝑛 ∥𝐾1 ∥ ≥ 𝑀 (1 + 𝜀) 3 Here Alon, Benjamini, and Stacey also make several conjectures about percolation on finite transitive graphs, which may interest the reader. 306 Moreover, by simply bounding a percolation threshold ( 𝑝 𝑛 ) below by the threshold for being dominated by a subcritical branching process and above by the threshold for connectivity, we obtain the following optimal bounds on its location. Proposition 7.1.5. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity. Let 𝑑𝑛 denote the vertex degree of 𝐺 𝑛 . If (𝐺 𝑛 ) has a percolation threshold ( 𝑝 𝑛 ), then it satisfies 1 log |𝑉 (𝐺 𝑛 )| (1 − 𝑜(1)) ≤ 𝑝 𝑛 ≤ (2 + 𝑜(1)) as 𝑛 → ∞. 𝑑𝑛 − 1 𝑑𝑛 When (𝐺 𝑛 ) has a percolation threshold ( 𝑝 𝑛 ) and converges locally to an infinite graph 𝐺, one might ask how the location of ( 𝑝 𝑛 ) relates to the critical parameter 𝑝 𝑐 and the uniqueness threshold 𝑝 𝑢 of 𝐺. See Remark 1.6 in [EH21b] for a discussion of this question. Let us simply note that ( 𝑝 𝑛 ) typically (but not always) converges to 𝑝 𝑐 . For example, this is the case when (𝐺 𝑛 ) is a sequence of transitive expanders and 𝐺 is nonamenable [BNP11b]. We conclude this discussion by explaining how our result relates to the general theory of sharp thresholds. Consider a large collection of independent random bits 𝑏 := (𝑏𝑖 )1≤𝑖≤𝑛 ∈ {0, 1}𝑛 each sampled according to the Bernoulli(𝑝) distribution for some 𝑝 ∈ [0, 1]. For many natural monotone events 𝐴 ⊆ {0, 1}𝑛 , the probability that 𝑏 belongs to 𝐴 has a sharp threshold: it increases from 𝑜(1) to 1− 𝑜(1) as 𝑝 increases across an interval of width 𝑜( 𝑝(1− 𝑝)). There is a general philosophy that a sharp threshold occurs if and only if 𝐴 is sufficiently symmetric/global. For example, the event that the Erdős-Rényi random graph is connected has a sharp threshold, but the event that it contains a triangle and the event that a particular edge is present do not. Theorem 8.1.1 can be understood as a kind of extension of this principle to the existence of a phase transition in a statistical mechanics model. Indeed, Theorem 8.1.1 says that "having a giant cluster" typically has a threshold around a critical parameter 𝑝 with a threshold width4 𝑜( 𝑝) whenever the underlying graph is transitive, which is a symmetry/homogeneity condition. However, we would like to emphasise that "having a giant cluster" is not a well-defined event, so it does not fall within the usual scope of sharp-threshold techniques. Molecular sequences and the supercritical phase Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity. If (𝐺 𝑛 ) has a percolation threshold ( 𝑝 𝑛 ), then a sequence of parameters (𝑞 𝑛 ) is called supercritical if 4 The threshold width fails to be 𝑜(1 − 𝑝) even for simple examples such as the sequence of cycles. 307 there exist 𝜀 > 0 and 𝑁 < ∞ such that for all 𝑛 ≥ 𝑁 satisfying 𝑞 𝑛 < 1, we have 𝑞 𝑛 ≥ (1 + 𝜀) 𝑝 𝑛 . As in [HT21d; EH21b], we generalise this definition to the situation that there may or may not be a percolation threshold by saying that a sequence of parameters (𝑞 𝑛 ) is supercritical if there exist 𝜀 > 0 and 𝑁 < ∞ such that for all 𝑛 ≥ 𝑁 satisfying 𝑞 𝑛 < 1, we have P (1−𝜀)𝑞 𝑛 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀. (The ‘𝑞 𝑛 < 1’ condition is a technicality that guarantees that (𝐺 𝑛 ) always admits at least one supercritical sequence of parameters, namely the sequence ( 𝑝 𝑛 ) with 𝑝 𝑛 := 1 for all 𝑛.) In our work with Hutchcroft [EH21b], we showed that not having a molecular subsequence is the geometric counterpart to the supercritical giant cluster being unique. Here is the precise result from that paper. Definition 7.1.6. We say that (𝐺 𝑛 ) has the supercritical uniqueness property if for every supercritical sequence of parameters (𝑞 𝑛 ), 𝑛 lim P𝐺 𝑞 𝑛 (∥𝐾2 ∥ ≥ 𝛼) = 0 𝑛→∞ for all 𝛼 > 0. Theorem 7.1.7 (Easo and Hutchcroft, 2021). A sequence of finite connected transitive graphs with volume tending to infinity has the supercritical uniqueness property if and only if it does not contain a molecular subsequence. A crucial step in the proof of the above theorem is that if (𝐺 𝑛 ) does not contain a molecular subsequence, then it has the sharp-density property. To prove Theorem 8.1.1, we only need to recall that the sharp-density property guarantees that for every density 𝛼 ∈ (0, 1], constant 𝜀 > 0, and supercritical sequence of parameters ( 𝑝 𝑛 ), lim inf P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛼) > 0 𝑛→∞ implies 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 1. lim P𝐺(1+𝜀) 𝑝 𝑛→∞ 𝑛 This implies that the event that “there exists a cluster with density at least 𝛼” undergoes a sharp threshold for each fixed 𝛼. Notice that this does not immediately imply the existence of a percolation threshold. One obstruction could be how thresholds are spaced: one could imagine that the   these 𝛼 , say, in which case there would be no percolation 𝛼-density threshold always occurs at √𝑛 𝑛≥1 threshold. The implication is not clear even if we additionally require that the 𝐺 𝑛 ’s have uniformly bounded vertex degrees: see our footnote on page 3, and see Conjecture 1.2 in [Ben+12] for an analogous situation in the context of expanders. 308 Proof strategy Most of our work goes into showing that if (𝐺 𝑛 ) does not contain a molecular subsequence, then it has a percolation threshold. In this section, we outline this step. To extend this result to the case that (𝐺 𝑛 ) contains an 𝑚-molecular subsequence for at most finitely many integers 𝑚, we apply the same argument to a molecular sequence’s constituent sequence of atoms. We will deduce that converse as an immediate consequence of a corollary from [EH21b]. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity. Suppose we want to prove that (𝐺 𝑛 ) has a percolation threshold. Although "having a giant cluster" is not an event, we could try taking the event {∥𝐾1 ∥ ≥ 𝛿𝑛 } as a proxy, where (𝛿𝑛 ) is a sequence tending to zero very slowly. Then as a candidate for the percolation threshold, we could take ( 𝑝 𝑛 ) where each 𝑝 𝑛 is defined to be the unique parameter satisfying 1 P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛿𝑛 ) = . 2 As a sanity check, notice that if (𝐺 𝑛 ) does have a percolation threshold, then by a diagonal argument, there is a percolation threshold ( 𝑝 𝑛 ) that arises in this way. To prove that our candidate ( 𝑝 𝑛 ) is in fact a percolation threshold, we need two ingredients. The first ingredient is that the emergence of a cluster of any constant density has a threshold about a critical parameter 𝑝 with a threshold width 𝑜( 𝑝). This immediately handles the subcritical half 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 0 for every of our task: since lim𝑛→∞ 𝛿𝑛 = 0, it guarantees that lim𝑛→∞ P𝐺(1−𝜀) 𝑝𝑛 𝛼, 𝜀 > 0. As mentioned in Section 7.1, the existence of these constant-density thresholds is implied by the sharp-density property, which holds whenever (𝐺 𝑛 ) has no molecular subsequences. The second ingredient is a universal lower bound on the supercritical giant cluster density. This says that for every 𝜀 > 0, there exists 𝛿 > 0 such that for every sequence of parameters ( 𝑝 𝑛 ), if 𝑛 ∥𝐾1 ∥ ≥ 𝛿. Together with the first ingredient, lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then lim inf 𝑛→∞ E𝐺(1+𝜀) 𝑝𝑛 this ensures that by taking (𝛿𝑛 ) to decay slowly enough, for every 𝜀 > 0, there exists 𝛿 > 0 with 𝑛 (∥𝐾1 ∥ ≥ 𝛿) = 1. We will prove that this too holds whenever (𝐺 𝑛 ) has no molecular lim𝑛→∞ P𝐺(1+𝜀) 𝑝𝑛 subsequences. This second ingredient is reminiscent of the mean-field lower bound from the study of percolation on infinite graphs. This says, for example, that every vertex 𝑣 in an infinite transitive graph 𝐺 satisfies 𝜀 . P𝐺(1+𝜀) 𝑝 𝑐 (𝐺) (𝐾𝑣 is infinite) ≥ 1+𝜀 Together with the exponential decay of |𝐾𝑣 | throughout the subcritical phase, this forms what is known as the sharpness of the phase transition for percolation on infinite graphs. This foundational 309 result was first proved in [Men86; AB87a; CC87] but has recently been reproved by more modern arguments in [DT16c; DRT19; Hut20d; Van22b]. The main obstacle to adapting these proofs to our finite setting is that they concern the event that the cluster at a vertex reaches a certain distance or exceeds a certain volume, whereas we care about the cluster’s volume as a proportion of the total vertices. Moreover, while the sharpness of the phase transition is completely general —- applying to all infinite transitive graphs — there is no universal lower bound on the supercritical giant cluster density that applies to every finite transitive graph5 . So to adapt one of these proofs to our setting, we need to include information about the underlying sequence of graphs in the argument itself, for example, that it has the supercritical uniqueness property, or equivalently, that it has no molecular subsequences. Very recently, Vanneuville [Van22b] gave a new proof of the sharpness of the phase transition via couplings. Unlike previous arguments, this one does not rely on a differential inequality. Vanneuville’s key insight was that by using an exploration process, we can upper bound the effect of conditioning on a certain decreasing event, namely the event that a vertex’s cluster does not reach a certain distance, by the effect of slightly decreasing the percolation parameter. Our strategy is to apply this argument but with the event that a vertex’s cluster has small density. Rather than building an exact monotone coupling of the conditioned percolation measure and the percolation measure with a smaller parameter, which is impossible, we will construct a coupling that is monotone outside of an error event. Then under the additional hypothesis that (𝐺 𝑛 ) has the supercritical uniqueness property, we will prove that this error event has probability tending to zero. Just as Vanneuville’s coupling immediately yields the mean-field lower bound, our approximately monotone coupling will tell us that for every sequence of parameters ( 𝑝 𝑛 ) and every constant 𝜀 > 0, if lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then 𝑛 ∥𝐾1 ∥ ≥ lim inf E𝐺(1+𝜀) 𝑝 𝑛→∞ 𝑛 𝜀 . 1+𝜀 Let us mention that for this step — establishing a universal lower bound on the supercritical giant cluster density when we have the supercritical uniqueness property — it is possible to instead adapt Hutchcroft’s proof of sharpness from [Hut20d], rather than Vanneuville’s new proof. The adaptation that we found of Hutchcroft’s proof is more involved than the argument presented here. For example, it invokes the universal tightness result from [Hut21d]. Invoking this auxiliary result 5 Consider the sequences (𝐾 □𝐶 ) 𝑛 𝑚 𝑛≥1 for each 𝑚 ≥ 3, where 𝐾 𝑛 is a complete graph and 𝐶𝑚 is a cycle. 310 also has the consequence that the universal lower bound we ultimately obtain is weaker than the mean-field bound established here. 7.2 The coupling lemma In this section, we control the effect on percolation of conditioning on the event that a vertex’s cluster has small density. In particular, we prove that the conditioned measure approximately stochastically dominates percolation of a slightly smaller parameter. As mentioned in Section 7.1, the argument in this section is inspired by [Van22b]. Lemma 7.2.1. Let 𝐺 = (𝑉, 𝐸) be a finite connected transitive graph with a distinguished vertex 𝑜. Let 𝑝 ∈ (0, 1) be a parameter and 𝛼 ∈ (0, 1) a density. Define  2ℎ1/2 𝛼 , , 𝛿 := 𝜃 := E 𝑝 ∥𝐾1 ∥ , ℎ := P 𝑝 ∥𝐾1 ∥ < 𝛼 or ∥𝐾2 ∥ ≥ 2 1−𝜃−ℎ and assume that 𝜃 + ℎ < 1 (so that 𝛿 is well-defined and positive). Then there is an event 𝐴 with P 𝑝 ( 𝐴 | ∥𝐾𝑜 ∥ < 𝛼) ≤ ℎ1/2 such that P (1−𝜃−𝛿) 𝑝 ≤st P 𝑝 (𝜔 ∪ 1 𝐴 = · | ∥𝐾𝑜 ∥ < 𝛼) , where ≤st denotes stochastic domination with respect to the usual partial order ⪯ on {0, 1} 𝐸 , and 1 𝐴 denotes the random configuration with every edge open on 𝐴 and every edge closed on 𝐴𝑐 . Proof. To lighten notation, set P̂ := P 𝑝 ( · | ∥𝐾𝑜 ∥ < 𝛼) and 𝑞 := (1 − 𝜃 − 𝛿) 𝑝. Our goal is to construct an approximately monotone coupling of P𝑞 and P̂. We will build this in the obvious way: by fixing an exploration process and building samples of P𝑞 and P̂ in terms of a common collection of 𝐸-indexed uniform random variables. The rest of the proof consists in controlling the failure of this coupling to be monotone. Fix an enumeration of the edge set 𝐸. Recall the following standard method for exploring a configuration 𝜔 from 𝑜: Start with all edges unrevealed. Iteratively reveal the unrevealed edge of smallest index that is connected to 𝑜 by an open path of revealed edges until there are none. Then iteratively reveal the unrevealed edge of smallest index from among all remaining unrevealed edges until there are none. Let 𝜌1 , . . . , 𝜌 |𝐸 | be the sequence of edges as they are revealed by this process. Let (F𝑡 ) 0≤𝑡≤|𝐸 | be the filtration associated to this exploration, i.e.
F𝑡 := 𝜎 𝜔 𝜌1 , . . . , 𝜔 𝜌𝑡 . We now use this exploration to construct a coupling of P𝑞 and P̂. Let (𝑈𝑒 ) 𝑒∈𝐸 be a collection of independent uniform-[0,1] random variables. Recursively define 𝜔 𝜌𝑡 := 1𝑈𝜌 ≤P̂(𝜌𝑡 open |F𝑡 −1 ) for 𝑡 311 
every 𝑡 to obtain a configuration 𝜔 with law P̂, and simply take 1𝑈𝑒 ≤𝑞 𝑒∈𝐸 for a configuration with law P𝑞 . This coupling is monotone (in the direction we want) on the edges 𝜌1 , 𝜌2 , . . . , 𝜌𝜏fail where 𝜏fail is the stopping time defined by 𝜏fail := inf{𝑡 : P̂ (𝜌𝑡+1 open | F𝑡 ) < 𝑞},
with the convention that inf ∅ := |𝐸 |. So 1𝑈𝑒 ≤𝑞 𝑒∈𝐸 ⪯ 𝜔 almost surely when 𝜏fail = |𝐸 |. Since
1𝑈𝑒 ≤𝑞 𝑒∈𝐸 ⪯ 1𝜏fail <|𝐸 | holds trivially when 𝜏fail < |𝐸 |, we know that
P𝑞 ≤st P̂ 𝜔 ∪ 1𝜏fail <|𝐸 | = · . So it suffices to verify that P̂ (𝜏fail < |𝐸 |) ≤ ℎ1/2 . By definition of 𝜏fail , we have the upper bound
P̂ 𝜌𝜏fail +1 open | F𝜏fail < 𝑞 a.s. when 𝜏fail < |𝐸 |. (7.2.1) Our first step is to prove a complementary lower bound. Say that an edge 𝑒 is pivotal if ∥𝐾𝑜 (𝜔\{𝑒})∥ < 𝛼 but ∥𝐾𝑜 (𝜔 ∪ {𝑒})∥ ≥ 𝛼. If 𝑒 is open and pivotal, then ∥𝐾𝑜 ∥ ≥ 𝛼, which is P̂-almost surely impossible. So

P̂ 𝜌𝜏fail +1 open | F𝜏fail = P̂ 𝜌𝜏fail +1 open and not pivotal | F𝜏fail a.s. when 𝜏fail < |𝐸 | . (7.2.2) Suppose we reveal the edges 𝜌1 , . . . , 𝜌𝜏fail and find that 𝜏fail < |𝐸 |. Note that 𝜌𝜏fail +1 is now almost surely determined. To finish building a sample of P̂, rather than continuing our exploration process, we could first sample every unrevealed edge except 𝜌𝜏fail +1 , then sample 𝜌𝜏fail +1 itself. The first stage will determine whether 𝜌𝜏fail +1 is pivotal. If it is not pivotal, then conditioning on the event {∥𝐾𝑜 ∥ < 𝛼} will have no effect in the second stage, i.e. the conditional probablity that 𝜌𝜏fail +1 is open will simply be 𝑝. So we can rewrite eq. (7.2.2) as

P̂ 𝜌𝜏fail +1 open | F𝜏fail = 𝑝 P̂ 𝜌𝜏fail +1 not pivotal | F𝜏fail  (7.2.3)
 = 𝑝 1 − P̂ 𝜌𝜏fail +1 pivotal | F𝜏fail a.s. when 𝜏fail < |𝐸 | . As in our argument for eq. (7.2.2), since it is P̂-almost surely impossible for an edge to be both open and pivotal, this further implies that 

 P̂ 𝜌𝜏fail +1 open | F𝜏fail = 𝑝 1 − P̂ 𝜌𝜏fail +1 closed and pivotal | F𝜏fail a.s. when 𝜏fail < |𝐸 | . (7.2.4) By combining inequality eq. (7.2.1) with eq. (7.2.3), we deduce that
P̂ 𝜌𝜏fail +1 pivotal | F𝜏fail > 0 312 a.s. when 𝜏fail < |𝐸 | . So when 𝜏fail < |𝐸 |, we can almost surely label the endpoints of 𝜌𝜏fail +1 by 𝑣 − and 𝑣 + such that 𝑣 − is connected to 𝑜 by a path of open edges among the revealed edges {𝜌1 , . . . , 𝜌𝜏fail }, whereas 𝑣 + is not. Consider a configuration 𝜔 with 𝜏fail < |𝐸 | in which 𝜌𝜏fail +1 is closed and pivotal. By the pigeonhole principle, since 𝐾𝑜 (𝜔 ∪ {𝜌𝜏fail +1 }) ≥ 𝛼, 𝐾𝑣 − ≥ 𝛼 2 or 𝐾𝑣 + ≥ 𝛼 . 2 (7.2.5) Since closing 𝜌𝜏fail +1 disconnects 𝐾𝑜 (𝜔 ∪ {𝜌𝜏fail +1 }), the endpoints 𝑣 − and 𝑣 + belong to distinct clusters of 𝜔. In particular, since 𝑣 − ∈ 𝐾𝑜 , we know that 𝑣 + ∉ 𝐾𝑜 . So eq. (7.2.5) implies that ∥𝐾𝑜 ∥ ≥ 𝛼 2 or 𝐾𝑣 + (𝜔\𝐾𝑜 ) ≥ 𝛼 , 2 where 𝐾𝑜 := 𝐾𝑜 ∪ 𝜕𝐾𝑜 and 𝜕𝐾𝑜 denotes the edge boundary of 𝐾𝑜 . Now 𝜔 was arbitrary, so by a union bound,    
𝛼 𝛼 P̂ 𝜌𝜏fail +1 closed and pivotal | F𝜏fail ≤ P̂ ∥𝐾𝑜 ∥ ≥ | F𝜏fail + P̂ 𝐾𝑣 + (𝜔\𝐾𝑜 ) ≥ | F𝜏fail 2 2 a.s. when 𝜏fail < |𝐸 | . (7.2.6) To bound the first term in inequality (7.2.6), notice that ∥𝐾𝑜 ∥ ≥ 𝛼2 and ∥𝐾𝑜 ∥ < 𝛼 together imply the bad event 𝐵 := {∥𝐾1 ∥ < 𝛼 or ∥𝐾2 ∥ ≥ 𝛼2 }. So because ∥𝐾𝑜 ∥ < 𝛼 occurs P̂-almost surely,  
𝛼 P̂ ∥𝐾𝑜 ∥ ≥ | F𝜏fail ≤ P̂ 𝐵 | F𝜏fail a.s. when 𝜏fail < |𝐸 | . (7.2.7) 2 To bound the second term in inequality (7.2.6), define a new stopping time 𝜏moat := max{𝑡 : 𝜌𝑡 ∈ 𝐾 𝑜 }. (This is defined with respect to the standard exploration described in the second paragraph of the current proof environment, not the modified exploration mentioned below eq. (7.2.2).) Just after we reveal 𝜌𝜏moat , since conditioning on the event {∥𝐾𝑜 ∥ < 𝛼} no longer has any effect, the distribution of the configuration on the unrevealed edges is simply   𝑜 P̂ 𝜔| 𝐸\𝐾𝑜 = · | F𝜏moat = P𝐺\𝐾 ≤st P 𝑝 a.s. (7.2.8) 𝑝 When 𝜏fail < |𝐸 |, we showed that 𝜌𝜏fail +1 almost surely has an endpoint belonging to 𝐾𝑜 , namely 𝑣 − . So 𝜏fail < |𝐸 | implies 𝜏fail < 𝜏moat almost surely. By applying this observation, inequality (7.2.8), 313 and transitivity, we obtain   h   i 𝛼 𝛼 P̂ 𝐾𝑣 + (𝜔\𝐾𝑜 ) ≥ | F𝜏fail = Ê P̂ 𝐾𝑣 + (𝜔\𝐾𝑜 ) ≥ | F𝜏moat | F𝜏fail 2 2 i h  𝛼 ≤ Ê P 𝑝 𝐾𝑣 + ≥ | F𝜏fail 2   𝛼 = P 𝑝 ∥𝐾𝑜 ∥ ≥ a.s. when 𝜏fail < |𝐸 | . 2 When ∥𝐾𝑜 ∥ ≥ 𝛼2 , we must have that 𝑜 ∈ 𝐾1 or ∥𝐾2 ∥ ≥ 𝛼2 . So by a union bound,    𝛼 𝛼 P̂ 𝐾𝑣 + (𝜔\𝐾𝑜 ) ≥ | F𝜏fail ≤ P 𝑝 (𝑜 ∈ 𝐾1 ) + P 𝑝 ∥𝐾2 ∥ ≥ 2 2 ≤ 𝜃 + ℎ a.s. when 𝜏fail < |𝐸 | , (7.2.9) where ℎ := P 𝑝 (𝐵) and 𝜃 := E 𝑝 ∥𝐾1 ∥, which satisfies 𝜃 = P 𝑝 (𝑜 ∈ 𝐾1 ) by transitivity. Plugging inequalities (7.2.6), (7.2.7) and (7.2.9) into eq. (7.2.4) gives a lower bound on P̂ 𝜌𝜏fail +1 open | F𝜏fail when 𝜏fail < |𝐸 |. By contrasting this with upper bound ?? and expanding the definition of 𝑞, we deduce that   
 2ℎ1/2 𝑝 1 − 𝜃 − ℎ − P̂ 𝐵 | F𝜏fail ≤ 𝑝 1 − 𝜃 − a.s. when 𝜏fail < |𝐸 | . 1−𝜃−ℎ In particular, ℎ1/2 a.s. when 𝜏fail < |𝐸 | . P̂ 𝐵 | F𝜏fail ≥ 1−𝜃−ℎ 
 By the law of total expectation, we know that Ê P̂ 𝐵 | F𝜏fail = P̂ (𝐵). So by Markov’s inequality,  
ℎ1/2 1−𝜃−ℎ P̂ (𝜏fail < |𝐸 |) ≤ P̂ P̂ 𝐵 | F𝜏fail ≥ P̂ (𝐵) . (7.2.10) ≤ 1−𝜃−ℎ ℎ1/2
By definition of P̂ and ℎ, we have the trivial bound P̂ (𝐵) ≤ P 𝑝 (𝐵) ℎ = . P 𝑝 (∥𝐾𝑜 ∥ < 𝛼) 1 − P 𝑝 (∥𝐾𝑜 ∥ ≥ 𝛼) If ∥𝐾𝑜 ∥ ≥ 𝛼, then 𝑜 ∈ 𝐾1 or ∥𝐾2 ∥ ≥ 𝛼. So similarly to the argument for inequality (7.2.9), a union ℎ bound gives P̂ (𝐵) ≤ 1−𝜃−ℎ . Plugging this into inequality (7.2.10) yields P̂ (𝜏fail < |𝐸 |) ≤ ℎ1/2 , as required. □ 7.3 Characterising the existence of a percolation threshold In this section, we prove Theorem 8.1.1 and Corollary 7.1.4. Our first step is to establish a mean-field lower bound on the supercritical giant cluster density for sequences of graphs without molecular subsequences. This is where we use the coupling lemma from Section 7.2. 314
Lemma 7.3.1. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity that does not contain a molecular subsequence. Fix 𝜀 > 0 and let ( 𝑝 𝑛 ) be any sequence of parameters. If lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then 𝑛 ∥𝐾1 ∥ ≥ lim inf E𝐺(1+𝜀) 𝑝 𝑛 𝑛→∞ 𝜀 . 1+𝜀 𝑛 ∥𝐾1 ∥ < Proof. Suppose for contradiction that lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0 but lim inf 𝑛→∞ E𝐺(1+𝜀) 𝑝𝑛 𝜀 1+𝜀 . By passing to a suitable subsequence, we may assume that 𝑛 ∥𝐾1 ∥ < lim sup E𝐺(1+𝜀) 𝑝 𝑛 𝑛→∞ 𝜀 . 1+𝜀 (7.3.1) Pick 𝛼 > 0 such that E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ ≥ 2𝛼 for all sufficiently large 𝑛. By Markov’s inequality applied to 1 − ∥𝐾1 ∥, this implies that P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝛼 for all sufficiently large 𝑛. Now for each 𝑛, define  𝛼 𝐺𝑛 𝑛 ∥𝐾 ∥ ∥𝐾 ∥ ∥𝐾 ∥ < 𝛼 or ≥ 𝜃 𝑛 := E𝐺(1+𝜀) and ℎ := P . 1 2 1 𝑛 𝑝𝑛 (1+𝜀) 𝑝 𝑛 2 Since (𝐺 𝑛 ) does not contain a molecular subsequence, it has the sharp-density and supercritical uniqueness properties from [EH21b]. (See Section 7.1 for more information.) So the fact that lim inf 𝑛→∞ P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛼) > 0 implies that  𝛼 𝐺𝑛 𝑛 ∥𝐾 ∥ (∥𝐾 ∥ ≥ < 𝛼) = 0 and lim P lim P𝐺(1+𝜀) = 0. 2 1 𝑝𝑛 𝑛→∞ (1+𝜀) 𝑝 𝑛 𝑛→∞ 2 So by a union bound, lim𝑛→∞ ℎ𝑛 = 0. Together with inequality (7.3.1), this guarantees that for all sufficiently large 𝑛, we have 𝜃 𝑛 + ℎ𝑛 < 1, allowing us to apply Lemma 7.2.1. By passing to a suitable tail of (𝐺 𝑛 ), we may assume that this holds for every 𝑛. So for every 𝑛, Lemma 7.2.1 says that there is an event 𝐴𝑛 ⊆ {0, 1} 𝐸 (𝐺 𝑛 ) with 𝑛 ( 𝐴𝑛 | ∥𝐾𝑜 ∥ < 𝛼) ≤ ℎ1/2 P𝐺(1+𝜀) 𝑛 𝑝 𝑛 (7.3.2) such that P𝐺 𝑛 1/2 2ℎ 1−𝜃 𝑛 − 1− 𝜃𝑛𝑛−ℎ𝑛  (1+𝜀) 𝑝 𝑛 𝑛 ≤st P𝐺(1+𝜀) 𝑝 𝑛
𝜔 ∪ 1 𝐴𝑛 = · | ∥𝐾𝑜 ∥ < 𝛼 . (7.3.3) When P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝛼, it follows by transitivity that P𝐺𝑝 𝑛𝑛 (∥𝐾𝑜 ∥ ≥ 𝛼) ≥ 𝛼2 . On the other hand, by inequalities (7.3.2) and (7.3.3), we know that P𝐺 𝑛 1/2 2ℎ 1−𝜃 𝑛 − 1− 𝜃𝑛𝑛−ℎ𝑛 (∥𝐾𝑜 ∥ ≥ 𝛼) ≤ ℎ1/2 𝑛 .  (1+𝜀) 𝑝 𝑛 315 2 So by taking 𝑛 sufficiently large that both P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛼) ≥ 𝛼 and ℎ1/2 𝑛 < 𝛼 , we can force ! 2ℎ1/2 𝑛 (1 + 𝜀) 𝑝 𝑛 ≤ 𝑝 𝑛 . (7.3.4) 1 − 𝜃𝑛 − 1 − 𝜃 𝑛 − ℎ𝑛 However, by inequality (7.3.1) and the fact that lim𝑛→∞ ℎ𝑛 = 0, 1 − 𝜃𝑛 − 2ℎ1/2 1 𝑛 > 1 − 𝜃 𝑛 − ℎ𝑛 1 + 𝜀 for all sufficiently large 𝑛, contradicting inequality (7.3.4). □ By the sharp-density property, we can convert this mean-field lower bound that holds in expectation into one that holds with high probability. Lemma 7.3.2. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity that does not contain a molecular subsequence. Fix 𝜀 > 0 and let ( 𝑝 𝑛 ) be any sequence of parameters. If lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then   𝜀 𝑛 ∥ ∥𝐾 ≥ P𝐺(1+𝜀) − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. 1 𝑝𝑛 1+𝜀 Proof. Let 𝛿 ∈ (0, 𝜀) be any constant. By Lemma 7.3.1, 𝑛 ∥𝐾1 ∥ ≥ lim inf E𝐺(1+𝜀−𝛿) 𝑝 𝑛→∞ 𝑛 𝜀−𝛿 . 1+𝜀−𝛿 So by Markov’s inequality applied to 1 − ∥𝐾1 ∥,   𝜀−𝛿 𝛿(𝜀 − 𝛿) 𝐺𝑛 lim inf P (1+𝜀−𝛿) 𝑝 ∥𝐾1 ∥ ≥ > 0. ≥ 𝑛 𝑛→∞ 1+𝜀 (1 + 𝛿)(1 + 𝜀 − 𝛿) (7.3.5) Since (𝐺 𝑛 ) does not contain a molecular subsequence, it has the sharp-density property from [EH21b]. (Recall our discussion in Section 7.1.) So inequality (7.3.5) implies that   𝜀−𝛿 𝐺𝑛 ∥𝐾1 ∥ ≥ lim P = 1. 𝑛→∞ (1+𝜀) 𝑝 𝑛 1+𝜀 Since 𝛿 was arbitrary, the result now follows by a diagonal argument. □ Our next step is to extend a version of this lower bound to a simple kind of molecular sequence. The idea is to break the graphs in the sequence into their constituent atoms then apply Lemma 7.3.2 to the sequence of atoms. 316 Lemma 7.3.3. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity. Assume that (𝐺 𝑛 ) is 𝑚-molecular for some 𝑚 ≥ 2 but does not contain a 𝑘molecular subsequence for any 𝑘 > 𝑚. Fix 𝜀 > 0 and let ( 𝑝 𝑛 ) be any sequence of parameters. If lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then   𝜀 𝐺𝑛 − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. P (1+𝜀) 𝑝 ∥𝐾1 ∥ ≥ 𝑛 𝑚(1 + 𝜀) Proof. By definition of an 𝑚-molecular sequence, for every 𝑛, we can pick a set of edges 𝐹𝑛 ⊆ 𝑛| 𝐸 (𝐺 𝑛 ) such that 𝐹𝑛 is Aut 𝐺 𝑛 -invariant, 𝐺 𝑛 \𝐹𝑛 has 𝑚 connected components, and |𝑉|𝐹 (𝐺 𝑛 )| is uniformly bounded. Let 𝐴𝑛 denote one of the connected components of 𝐺 𝑛 \𝐹𝑛 , which are all necessarily isomorphic to each other and transitive. Notice that ( 𝐴𝑛 ) does not contain a molecular subsequence. Indeed, if ( 𝐴𝑛 ) contained an 𝑟-molecular subsequence ( 𝐴𝑛 )𝑛∈𝐼 for some 𝑟 ≥ 2, then (𝐺 𝑛 )𝑛∈𝐼 would be an 𝑟𝑚-molecular subsequence of (𝐺 𝑛 ). (Every automorphism of 𝐺 𝑛 acts on the 𝑚 copies of 𝐴𝑛 by permuting the copies and applying an automorphism of 𝐴𝑛 to each.) For each finite graph 𝐺, let 𝜆(𝐺) denote the largest eigenvalue of the adjacency matrix for 𝐺. Recall that when 𝐺 is regular, 𝜆(𝐺) = deg(𝐺), the vertex degree of 𝐺. By Theorem 7.1.3, which is taken from [Bol+10c], since (𝐺 𝑛 ) and ( 𝐴𝑛 ) are sequences of dense graphs, they have 𝑛| percolation thresholds at (1/𝜆(𝐺 𝑛 ))𝑛≥1 and (1/𝜆( 𝐴𝑛 ))𝑛≥1 respectively. Since |𝑉|𝐹 (𝐺 𝑛 )| is uniformly deg 𝐴 bounded, lim𝑛→∞ deg 𝐺𝑛𝑛 = 1, and since every 𝐴𝑛 and 𝐺 𝑛 is regular (since transitive), this means 𝜆( 𝐴𝑛 ) that lim𝑛→∞ 𝜆(𝐺 = 1. So for any 𝛿 > 0, since the sequence ((1 + 𝛿) 𝑝 𝑛 ) is supercritical for (𝐺 𝑛 ), 𝑛) it is also supercritical for ( 𝐴𝑛 ). In particular, 𝐴𝑛 ∥𝐾1 ∥ > 0. lim inf E (1+𝛿) 𝑝 𝑛→∞ 𝑛 Now since ( 𝐴𝑛 ) has no molecular subsequences, it follows by Lemma 7.3.2 that   𝜀−𝛿 𝐴𝑛 − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. P (1+𝜀) 𝑝 ∥𝐾1 ∥ ≥ 𝑛 1+𝜀 Since 𝛿 > 0 was arbitrary, a diagonal argument gives   𝜀 𝐴𝑛 ∥𝐾 ∥ P (1+𝜀) − 𝑜(1) = 1 − 𝑜(1) ≥ 1 𝑝𝑛 1+𝜀 as 𝑛 → ∞. The result follows because 𝐴𝑛 is a subgraph of 𝐺 𝑛 and |𝑉 (𝐺 𝑛 )| = 𝑚 |𝑉 ( 𝐴𝑛 )|. □ We now extend this lower bound to sequences of graphs that have 𝑚-molecular subsequences for at most finitely many integers 𝑚. 317 Lemma 7.3.4. Let (𝐺 𝑛 ) be a sequence of finite connected transitive graphs with volume tending to infinity that does not contain an 𝑚-molecular subsequence for any 𝑚 > 𝑀, where 𝑀 is some positive integer. Fix 𝜀 > 0 and let ( 𝑝 𝑛 ) be any sequence of parameters. If lim inf 𝑛→∞ E𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ > 0, then   𝜀 𝐺𝑛 − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. P (1+𝜀) 𝑝 ∥𝐾1 ∥ ≥ 𝑛 𝑀 (1 + 𝜀) Proof. It is enough to show that for every subsequence (𝐺 𝑛 )𝑛∈𝐼 of (𝐺 𝑛 ), we can find a further subsequence (𝐺 𝑛 )𝑛∈𝐽 with 𝐽 ⊆ 𝐼 such that   𝜀 𝐺𝑛 P (1+𝜀) 𝑝 ∥𝐾1 ∥ ≥ − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞ with 𝑛 ∈ 𝐽. 𝑛 𝑀 (1 + 𝜀) Let (𝐺 𝑛 )𝑛∈𝐼 be a subsequence of (𝐺 𝑛 ). If (𝐺 𝑛 )𝑛∈𝐼 does not contain a molecular subsequence, then
𝜀 𝑛 ∥𝐾 ∥ by Lemma 7.3.2, we know that P𝐺(1+𝜀) ≥ − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞ with 𝑛 ∈ 𝐼. On 1 1+𝜀 𝑝𝑛 the other hand, if (𝐺 𝑛 )𝑛∈𝐼 does contain a molecular subsequence, then we can pick a subsequence (𝐺 𝑛 )𝑛∈𝐽 with 𝐽 ⊆ 𝐼 that is 𝑚-molecular with 𝑚 ∈ {2, . . . , 𝑀 } maximum. Then Lemma 7.3.3 tells 𝜀 𝑛 ∥𝐾1 ∥ ≥ 𝑚(1+𝜀) − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞ with 𝑛 ∈ 𝐽. □ us that P𝐺(1+𝜀) 𝑝 𝑛 We are now ready to prove that if a sequence of graphs has 𝑚-molecular subsequences for at most finitely many integers 𝑚, then it has a percolation threshold. As outlined in Section 7.1, the idea is to prove that the threshold for the emergence of a cluster of very slightly sublinear density is a percolation threshold. Lemma 7.3.5. If a sequence of finite connected transitive graphs with volume tending to infinity contains an 𝑚-molecular subsequence for at most finitely many integers 𝑚, then it has a percolation threshold. Proof. Let (𝐺 𝑛 ) be such a sequence of graphs. Let 𝑀 be a positive integer such that there are no 𝑚molecular subsequences with 𝑚 > 𝑀. For every density 𝛿 ∈ (0, 1) and each index 𝑛, define 𝑝 𝑛𝑐 (𝛿) to be the unique parameter satisfying P𝐺𝑝 𝑛𝑛(𝛿) (∥𝐾1 ∥ ≥ 𝛿) = 21 . Given any constants 𝜀, 𝛿 ∈ (0, 1), we 𝑐 know by Lemma 7.3.4 that  𝜀  𝑛 ∥𝐾 ∥ lim P𝐺(1+𝜀) = 1, ≥ 𝑛 1 𝑝 𝑐 (𝛿) 𝑛→∞ 2𝑀 𝜀 𝜀 < 𝑀 (1+𝜀) . So by a diagonal argument, for every 𝜀 ∈ (0, 1), there exists a sequence (𝛿𝑛𝜀 )𝑛≥1 since 2𝑀 in (0, 1) with lim𝑛→∞ 𝛿𝑛𝜀 = 0 such that  𝜀  𝑛 ∥𝐾 ∥ lim P𝐺(1+𝜀) ≥ = 1. 1 𝑝 𝑛𝑐 (𝛿 𝑛𝜀 ) 𝑛→∞ 2𝑀 318 Now by a further diagonal argument, there exists a fixed sequence (𝛿𝑛 ) in (0, 1) with lim𝑛→∞ 𝛿𝑛 = 0  such that for every 𝜀 ∈ 21 , 13 , 14 , . . . ,  𝜀  𝐺𝑛 ∥𝐾1 ∥ ≥ lim P = 1. 𝑛 𝑛→∞ (1+𝜀) 𝑝 𝑐 (𝛿 𝑛 ) 2𝑀 We claim that the sequence of parameters ( 𝑝 𝑛 ) given by 𝑝 𝑛 := 𝑝 𝑛𝑐 (𝛿𝑛 ) has the required properties. 𝑛 (∥𝐾1 ∥ ≥ 𝛼) ≠ 0 To verify the subcritical condition, suppose for contradiction that lim𝑛→∞ P𝐺(1−𝜀) 𝑝𝑛 for some 𝜀, 𝛼 ∈ (0, 1). By passing to a suitable subsequence, we may assume that 𝑛 (∥𝐾1 ∥ ≥ 𝛼) > 0. lim inf P𝐺(1−𝜀) 𝑝 𝑛→∞ 𝑛 Then by Lemma 7.3.4,  𝜀  = 1. lim P𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ ≥ 𝑛→∞ 2𝑀 𝜀 This contradicts the fact that for every 𝑛 that is sufficiently large to ensure 𝛿𝑛 ≤ 2𝑀 ,  𝜀  1 P𝐺𝑝 𝑛𝑛 ∥𝐾1 ∥ ≥ ≤ P𝐺𝑝 𝑛𝑛 (∥𝐾1 ∥ ≥ 𝛿𝑛 ) = . 2𝑀 2  To verify the supercritical condition, fix 𝜀 > 0. Pick any 𝜀′ ∈ 12 , 13 , 14 , . . . with 𝜀′ < 𝜀. Then by construction of (𝛿𝑛 ),     𝜀′ 𝜀′ 𝐺𝑛 𝐺𝑛 P (1+𝜀) 𝑝 𝑛 ∥𝐾1 ∥ ≥ ≥ P (1+𝜀′ ) 𝑝 ∥𝐾1 ∥ ≥ = 1 − 𝑜(1) as 𝑛 → ∞. □ 𝑛 2𝑀 2𝑀 We now verify that these sequences of graphs — those that have 𝑚-molecular subsequences for at most finitely many integers 𝑚 — are the only sequences to have a percolation threshold. This follows from the following corollary of Theorem 7.1.7 from [EH21b] that characterises the supercritical existence property. Definition 7.3.6. We say that (𝐺 𝑛 ) has the supercritical existence property if for every supercritical sequence of parameters (𝑞 𝑛 ), 𝑛 lim P𝐺 𝑞 𝑛 (∥𝐾1 ∥ ≥ 𝛼) = 1 𝑛→∞ for some 𝛼 > 0. Corollary 7.3.7 (Easo and Hutchcroft, 2021). A sequence of finite connected transitive graphs with volume tending to infinity has the supercritical existence property if and only if it contains an 𝑚-molecular subsequence for at most finitely many integers 𝑚. Notice that if a sequence of finite connected transitive graphs with volume tending to infinity has a percolation threshold, then it automatically has the supercritical existence property. So the ‘only if’ direction of Corollary 7.3.7 immediately implies the following lemma. 319 Lemma 7.3.8. If a sequence of finite connected transitive graphs with volume tending to infinity contains an 𝑚-molecular subsequence for infinitely many integers 𝑚, then it does not have a percolation threshold. Our main result, Theorem 8.1.1, now follows by combining Lemmas 7.3.5 and 7.3.8. We conclude this section by deducing Corollary 7.1.4 from Lemma 7.3.4. Proof of Corollary 7.1.4. For every 𝛿 ∈ (0, 𝜀), we know by definition of a percolation threshold 𝑛 ∥𝐾1 ∥ > 0. So by Lemma 7.3.4, that lim inf 𝑛→∞ E𝐺(1+𝛿) 𝑝𝑛   𝜀−𝛿 𝐺𝑛 − 𝑜(1) = 1 − 𝑜(1) as 𝑛 → ∞. P (1+𝜀) 𝑝 ∥𝐾1 ∥ ≥ 𝑛 𝑀 (1 + 𝜀) Since 𝛿 was arbitrary, the result follows by a diagonal argument. 7.4 □ Bounding the threshold location In this section, we give a proof of Proposition 7.1.5. This is simply the observation that existing arguments immediately imply bounds on the location of a percolation threshold (when it exists), which happen to be best-possible. The lower bound is a completely standard path-counting argument that we only include for completeness. The upper bound comes from bounding the percolation threshold by the connectivity threshold. As explained in [GLL21], we can estimate the connectivity threshold thanks to an upper bound by Karger and Stein on the number of approximate minimum cutsets in a graph [KS96]. The lower bound is sharp in the classical case of the complete graphs [ER60]. In fact, it is sharp for a very wide range of examples, including arbitrary dense regular graphs [Bol+10c], the hypercubes [AKS82a] (which are sparse yet have unbounded vertex degrees), and expanders with high girth and bounded vertex degrees [ABS04c]. However, it is less obvious that the upper bound is optimal in the sense that the constant ’2’ cannot be improved. For this, see the fat cycles construction in [GLL21]. Proof of Proposition 7.1.5. We start with the lower bound. This argument is just an optimisation  is not of the proof of Lemma 2.8 in [EH21b]. Fix 𝜀 ∈ (0, 1). It suffices to check that 𝑑1−𝜀 𝑛 −1 𝑛≥1 supercritical along any subsequence. There are at most 𝑑𝑛 (𝑑𝑛 − 1) 𝑟−1 simple paths of length 𝑟 starting at a particular vertex 𝑜 in each 𝐺 𝑛 . So for every 𝑛,  𝑟 ∑︁ 2 𝐺𝑛 𝑟−1 1 − 𝜀 E 1− 𝜀 |𝐾𝑜 | ≤ 𝑑𝑛 (𝑑𝑛 − 1) ≤ . 𝑑𝑛 − 1 𝜀 𝑑𝑛 −1 𝑟 ≥0 320 In particular, lim𝑛→∞ E𝐺1−𝑛 𝜀 ∥𝐾𝑜 ∥ = 0, and hence by transitivity, lim𝑛→∞ E 1− 𝜀 ∥𝐾1 ∥ = 0. 𝑑𝑛 −1 𝑑𝑛 −1 We now prove the upper bound. Every finite connected transitive graph has edge connectivity equal to its vertex degree [Mad71]. So by Theorem 4.1 from [GLL21], lim P𝐺 𝑛 𝑛→∞  In particular, 7.5 (2+𝑜(1)) log|𝑉 (𝐺𝑛 ) | 𝑑𝑛 (𝜔 is connected) = 1.  2 log|𝑉 (𝐺 𝑛 )| is not subcritical along any subsequence. 𝑑𝑛 𝑛≥1 □ Acknowledgements I thank Tom Hutchcroft for helpful comments on earlier drafts and for encouraging me to pursue this question of mine. I also thank an anonymous referee for their useful feedback and suggestions, which improved the clarity of the paper. 321 Chapter 8 322 SHARPNESS AND LOCALITY FOR PERCOLATION ON FINITE TRANSITIVE GRAPHS Abstract Let (𝐺 𝑛 ) = ((𝑉𝑛 , 𝐸 𝑛 )) be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that |𝑉𝑛 | → ∞ as 𝑛 → ∞. We say that percolation on 𝐺 𝑛 has a sharp phase transition (as 𝑛 → ∞) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on 𝐺 𝑛 has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on 𝐺 𝑛 (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in 𝐺 𝑛 coincides with the critical point for the emergence of an infinite cluster in the Benjamini-Schramm limit of (𝐺 𝑛 ), when this limit exists. 8.1 Introduction Given a graph 𝐺, build a random spanning subgraph 𝜔 by independently including each edge of 𝐺 with a fixed probability 𝑝 ∈ [0, 1]. The law of 𝜔 is called (Bernoulli bond) percolation and is denoted by P𝐺𝑝 . This simple model often undergoes a phase transition: for many natural choices of the underlying graph 𝐺, as 𝑝 increases past some critical value 𝑝 𝑐 (𝐺), the typical behaviour of the connnected components of 𝜔 changes abruptly. The study of this phenomenon has two origins, roughly coming from mathematical physics and combinatorics, respectively. The first origin is the 1957 work of Broadbent and Hammersley [BH57b] introducing percolation on the Euclidean lattice 𝐺 = Z𝑑 as a model for the spread of fluid through a porous medium. Note that Euclidean lattices are always (vertex-)transitive, meaning that for all vertices 𝑢 and 𝑣, there is a graph automorphism that maps 𝑢 to 𝑣. This is a way to formalise the notion that a graph is homogeneous or that its vertices are indistinguishable. For example, every Cayley graph of a finitely-generated group is transitive. In 1996, Benjamini and Schramm [BS96b] launched the systematic study of percolation on general infinite transitive graphs. A cornerstone of this theory is that percolation on an infinite transitive graph 𝐺 always undergoes a sharp phase transition. Let us recall what this means. We will write 𝑜 to denote an arbitrary vertex in 𝐺 and write |𝐾𝑜 | to denote the cardinality of its cluster, i.e. connected component in 𝜔.1 There is a trivial sense in which percolation on 𝐺 always undergoes a phase transition: by Kolmogorov’s 0-1 law, there exists some critical point 𝑝 𝑐 (𝐺) ∈ [0, 1] such that P𝐺𝑝 (there exists an infinite cluster) equals 0 for all 𝑝 < 𝑝 𝑐 (𝐺) and equals 1 for all 𝑝 > 𝑝 𝑐 (𝐺). Now the phase transition is said to be sharp if for all 𝑝 < 𝑝 𝑐 (𝐺), not only does P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) → 0 as 𝑛 → ∞, but in fact there exists a constant 𝑐(𝐺, 𝑝) > 0 such that P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝑒 −𝑐𝑛 for every 𝑛 ≥ 1.2 This was first proved in [AB87a; Men86] and now has multiple modern proofs [DT16a; DRT19; Hut20d; Van24]. The second origin is the 1960 work of Erdős and Rényi [ER60] investigating percolation on the complete graph 𝐺 𝑛 with 𝑛 vertices. This is the celebrated Erdős-Rényi (or simply random graph) model. The fundamental result is that percolation on 𝐺 𝑛 undergoes a sharp phase transition around 𝑝 = 1/𝑛 in the sense that for any fixed 𝜀 > 0, the cardinality of the largest cluster of 𝜔 under P𝐺𝑝 jumps from being3 Θ(log 𝑛) at 𝑝 = (1−𝜀)/𝑛 to being Θ(𝑛) at 𝑝 = (1+𝜀)/𝑛 with high probability as 𝑛 → ∞.4 Analogous results have since been established for certain other families of finite graphs with diverging degrees. For example, Ajtai, Komlós, and Szemerédi [AKS82b] and Bollobás, Kohakayawa, and Łuksak [BKŁ92] investigated percolation on the hypercube 𝐻𝑑 = {0, 1} 𝑑 , which has a sharp phase transition around 𝑝 = 1/𝑑. Note that every complete graph and hypercube is transitive. For a small sample of the vast literature on percolation on finite graphs, see, for example, [ABS04a; KLS20] on expanders, [FKM04] on pseudorandom graphs, [Bor+05a; Bor+05b; Bor+06; Nac09] on transitive graphs satisfying certain mean-field conditions, [Bol+10c] on dense graphs, and [Dis+24; DK24a; DK24b] on general graphs satisfying certain isoperimetric conditions. Between these two settings lies the less-developed theory of percolation on bounded-degree finite transitive graphs. This theory, which started in 2001, was initiated by Benjamini [Ben01b] and by Alon, Benjamini, and Stacey [ABS04a]. This concerns the asymptotic properties of percolation on a finite transitive graph 𝐺 = (𝑉, 𝐸) as |𝑉 | becomes large while the vertex degrees of 𝐺 remain bounded. As with the Erdős-Rényi model, here we are primarily interested in the phase transition for the emergence of a giant cluster, i.e. a cluster containing Θ(|𝑉 |) vertices, and we will call the phase transition sharp if the size of the largest cluster jumps from Θ(log |𝑉 |) to Θ(|𝑉 |). (See Section 8.1 for precise definitions.) At the same time, this theory is closely related to percolation on infinite transitive graphs via the local (Benjamini-Schramm) topology on the set of all transitive 1 More generally, 𝐾 𝑢 denotes the cluster containing a vertex called 𝑢. 2 Some people use sharpness to mean slightly different things e.g. the exponential decay of point-to-point connection probabilities for 𝑝 < 𝑝 𝑐 together with the mean-field lower bound for 𝑝 > 𝑝 𝑐 . 3 Given functions 𝑓 , 𝑔 : N → (0, ∞), we write 𝑓 (𝑛) = Θ(𝑔(𝑛)) to mean that there are constants 𝑐 > 0 and 𝐶 < ∞ such that 𝑐𝑔(𝑛) ≤ 𝑓 (𝑛) ≤ 𝐶𝑔(𝑛) for all 𝑛, i.e. 𝑓 (𝑛) = 𝑂 (𝑔(𝑛)) and 𝑔(𝑛) = 𝑂 ( 𝑓 (𝑛)). 4 When 𝑝 > 1 or 𝑝 < 1, we define P to be P or P respectively. 𝑝 1 0 323 graphs. Indeed, with respect to this topology, every infinite set G of finite transitive graphs with bounded degrees is relatively compact, and every graph in the boundary of G is infinite. Despite this close relation between infinite transitive graphs and bounded-degree finite transitive graphs, our understanding of percolation on infinite transitive graphs is quite far ahead. Roughly speaking, we can think of the theory of percolation on infinite transitive graphs as the theory of percolation on microscopic (i.e. 𝑂 (1)) scales in bounded-degree finite transitive graphs. In this sense, the finite graph theory generalises the infinite graph theory. (A limitation of this maxim is that not every infinite transitive graph can be locally approximated by finite transitive graphs.) In particular, certain basic questions in the finite graph theory have no natural analogues in the infinite graph theory. For example, the uniqueness/non-uniqueness of giant clusters is not directly related to the uniqueness/non-uniqueness of infinite clusters, which is instead related to the microscopic metric distortion of giant clusters [EH21a, Remark 1.6]. In this paper we investigate the following pair of closely related questions. An affirmative answer to the second question provides a direct way to move results and conjectures about infinite transitive graphs to finite transitive graphs. 1. Does percolation on a large bounded-degree finite transitive graph 𝐺 have a sharp phase transition? 2. If a finite transitive graph 𝐺 and an infinite transitive graph 𝐻 are close in the local sense, does the critical point for the emergence of a giant cluster in 𝐺 approximately coincide with the critical point for the emergence of an infinite cluster in 𝐻? Unfortunately, the answer to both of these questions in general is no. For example, take the
∞ sequence Z𝑛 × Z 𝑓 (𝑛) 𝑛=1 for any 𝑓 : N → N growing fast. This sequence always converges locally to Z2 , where the critical point for the emergence of an infinite cluster is 𝑝 𝑐 = 12 . On the other hand, provided that 𝑓 grows sufficiently fast, the threshold for the emergence of a giant cluster in Z𝑛 × Z 𝑓 (𝑛) will be as in the sequence of cycles, around 𝑝 𝑐 = 1. Moreover, for percolation of any fixed parameter 𝑝 ∈ ( 12 , 1) on Z𝑛 × Z 𝑓 (𝑛) , the order of the largest cluster will then typically be much larger than logarithmic but much smaller than linear in the total number of vertices. (See [EH23b, Example 5.1] for some more discussion of these sequences.) The problem is that these graphs are long and thin, coarsely resembling long cycles. In particular, after suitably rescaling, their graph metrics (rapidly) converge in the Gromov-Hausdorff metric to the unit circle. In this paper we prove that this is the only possible obstacle. 324 Locality Question (2) above is the finite analogue of Schramm’s locality conjecture. This conjecture was (equivalently) that for all 𝜀 > 0 there exists 𝑅 < ∞ such that for every pair of infinite transitive graphs 𝐺 and 𝐻 that are not one-dimensional5 , if the ball of radius 𝑅 in 𝐺 is isomorphic to the ball of radius 𝑅 in 𝐻, then | 𝑝 𝑐 (𝐺) − 𝑝 𝑐 (𝐻)| ≤ 𝜀. This conjecture formalised the idea that the critical point of an infinite transitive graph should generally be entirely determined by the graph’s small-scale, local geometry. By building on earlier progress, especially the work of Contreras, Martineau, and Tassion [CMT22], we verified this conjecture in our joint work with Hutchcroft [EH23b]. Schramm’s locality conjecture for infinite transitive graphs also spurred research on locality in other settings, including much research on the analogue of our question (2) about locality for finite graphs but where the hypothesis that the finite graphs are transitive is replaced by the hypothesis that they are expanders [BNP11a; Sar21b; RS22b; ABS23]. It may be surprising, from the perspective of percolation on infinite transitive graphs, that in fact sharpness and locality for finite transitive graphs are equivalent. That is to say, if we restrict ourselves to any particular infinite set G of bounded-degree finite transitive graphs, then the answers to questions (1) and (2) in the introduction will always coincide. (See Proposition 8.2.9 for a precise statement.) Indeed, if G satisfies locality, then one can easily extract sharpness for G from the sharpness of the phase transition on every infinite transitive graph that is a local limit of graphs in G, and the converse, that sharpness implies locality, can also be established with a little more work. One reason that this equivalence may be surprising is because for infinite transitive graphs, sharpness always holds, even for Z, whereas locality requires that the graphs are not onedimensional. To make sense of this, consider that for infinite transitive graphs, locality corresponds to a version of sharpness that is uniform in the choice of the graph, whereas in the context of finite transitive graphs, the only meaningful notion of sharpness is necessarily uniform. Given the similarity between locality for finite and infinite graphs, one may wonder why the present paper is necessary: why does the proof of locality for infinite transitive graphs not also imply (perhaps after some additional bookkeeping) locality and hence sharpness for finite transitive graphs? The most fundamental reason is that the approach to proving locality in [EH23b] relied inherently on the sharpness of the phase transition, which in our setting is what we are trying to prove! Let us be a little more precise. In the proof of [EH23b], we have an infinite transitive graph 𝐺 and a parameter 𝑝 that we want to show satisfies 𝑝 ≥ 𝑝 𝑐 (𝐺). The bulk of the argument in [EH23b] involves delicately propagating point-to-point connection lower bounds across larger 5 An infinite transitive graph is one-dimensional if and only if the graph is quasi-isometric to Z. 325 and larger scales to ultimately establish that for some function 𝑓 : N → (0, 1) tending to zero slower than exponentially, P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑓 (dist(𝑢, 𝑣)) for all vertices 𝑢 and 𝑣. Since point-to-point connection probabilities are decaying slower than exponentially, the conclusion 𝑝 ≥ 𝑝 𝑐 (𝐺) then follows from the sharpness of the phase transition on infinite transitive graphs. In a finite graph adaptation of this argument, at this final stage we would need to invoke the sharpness of the phase transition for finite transitive graphs, making the argument circular. One might hope to circumvent this problem by improving the locality argument so that the function 𝑓 does not tend to zero at all. Unfortunately, 𝑓 tends to zero because the propagation of point-to-point lower bounds in the locality argument is lossy, i.e. a lower bound of 𝜀𝑖 at scale 𝑛𝑖 is propagated to a lower bound of 𝜀𝑖+1 at scale 𝑛𝑖+1 where 𝜀𝑖+1 ≪ 𝜀𝑖 , which seems completely unavoidable to us with current technology. We will exploit the fact that the locality argument produces an explicit choice for 𝑓 that decays much slower than exponentially (even slower than algebraically). So for this final step, one only needs a weaker kind of quasi-sharpness of the phase transition to conclude. The new idea in the present paper is to directly establish this quasi-sharpness by applying quantitative versions of the proofs of two results that are a priori quite unrelated to locality: the uniqueness of the supercritical giant cluster [EH21a] and the existence of a percolation threshold [Eas23] on finite transitive graphs. In short, we can think of the existence of a percolation threshold as the weakest possible kind of quasisharpness. In general, if we allow graphs to have unbounded degrees (as we did in [Eas23]), then the implicit rates of convergence can be arbitrarily slow. Luckily, now assuming bounded degrees as we may in the present paper, we can plug into our argument in [Eas23] a quantitatively strong version of the uniqueness of the supercritical giant cluster from [EH21a] to get a quantitatively strong quasi-sharpness that suffices to conclude the proof of locality. There are also quite serious obstacles to adapting to finite graphs the part of the proof of locality leading up to this application of sharpness. To illustrate, say we tried to run the locality argument on an infinite transitive graph that is one-dimensional. What would go wrong? We would encounter a scale where we are unable to efficiently propagate connection lower bounds because two otherwise complementary arguments simultaneously break down. The breakdown of the first argument implies that 𝐺 cannot be one-ended (𝐺 is6 the Cayley graph of a finitely-presented group but its minimal cutsets are not coarsely connected), while the breakdown of the second implies that 𝐺 must have finitely many ends (𝐺 has polynomial growth because 𝐺 contains a large ball with small tripling). From this we deduce that 𝐺 is two-ended, thereby successfully identifying that 𝐺 was one-dimensional. On a finite transitive graph, these end-counting arguments are not applicable. This will require us to make the locality argument more finitary, even in the setting of infinite 6 Technically this applies to a certain graph 𝐺 ′ that approximates 𝐺. 326 transitive graphs, which is of independent interest. Unfortunately, this end-counting argument is so deeply embedded in the proof of [EH23b] that it will take some work to reorganise the high-level multi-scale induction in [EH23b] in order to isolate and make explicit the relevant part. Another obstacle is that the definition of exposed spheres, whose special connectivity properties played a pivotal role in [CMT22; EH23b], degenerates on finite transitive graphs. As part of our argument, we introduce the exposed sphere in a finite transitive graph, justify our definition (Lemma 8.3.19), and thereby establish that from the perspective of part of our argument, arbitrary finite transitive graphs can be treated like infinite transitive graphs that are one-ended. We hope that these basic geometric objects can be of use in future work on finite transitive graphs, analogously to their infinite counterparts. Statement of the main result Graphs will always be assumed to be connected, simple, countable, and locally finite. In a slight abuse of language, we identify together all graphs that are isomorphic to each other.7 Let G be an infinite set of finite transitive graphs. Note that G is countable. We will write lim𝐺∈G to denote limits taken with respect to some (and hence every) enumeration of G. We may omit references to 𝐺 and G when this does not cause confusion. Given a graph 𝐺, we will also assume by default that 𝑉 and 𝐸 refer to the sets of vertices and edges in 𝐺. Given a percolation configuration 𝜔, we write |𝐾1 | to denote the cardinality of the largest cluster. A sequence 𝑝 : G → (0, 1) is said to be a percolation threshold if for every constant 𝜀 > 0, we have8 lim P (1+𝜀) 𝑝 (|𝐾1 | ≥ 𝛼 |𝑉 |) = 1 for some constant 𝛼 > 0, whereas lim P (1−𝜀) 𝑝 (|𝐾1 | ≥ 𝛽 |𝑉 |) = 0 for every constant 𝛽 > 0. Note that when a percolation threshold exists, it is unique up to multiplication by 1 + 𝑜(1). So in this sense, we may refer to the percolation threshold for G, when one exists. Now assume that G has bounded degrees, i.e. there exists 𝑑 ∈ N such that for every 𝐺 ∈ G, every vertex in 𝐺 has degree at most 𝑑. By [Eas23], G always has a percolation threshold, say 𝑝. We say that percolation on G has a sharp phase transition if for every constant 𝜀 > 0, there exists a constant 𝐴 < ∞ such that lim P (1−𝜀) 𝑝 (|𝐾1 | ≥ 𝐴 log |𝑉 |) = 0. 1 > 0 (see [Eas23, Proposition 5]) Conversely, it is not hard to show in general that lim inf 𝑝 ≥ 𝑑−1 and that the complementary bound on |𝐾1 | always holds in the sense that if lim inf(1 − 𝜀) 𝑝 > 0 7 So a “graph” 𝐺 is really a graph-isomorphism equivalence class of graphs. 8 This equation means that for some (and hence every) enumeration G = {𝐺 , 𝐺 , . . .} where 𝐺 = (𝑉 , 𝐸 ), we 1 2 𝑛 𝑛 𝑛 have 𝑛 lim P𝐺 (|𝐾1 | ≥ 𝛼 |𝑉𝑛 |) = 1. (1+𝜀) 𝑝 (𝐺𝑛 ) 𝑛→∞ 327 then there exists 𝐴 < ∞ such that lim P (1−𝜀) 𝑝 (|𝐾1 | ≥ 𝐴1 log |𝑉 |) = 1 (see Proposition 8.2.6). Given a transitive graph 𝐺, we write 𝑜 to denote an arbitrary vertex, and we write 𝐵𝐺 𝑛 to denote the graph-metric ball of radius 𝑛 centred at 𝑜, viewed as a rooted subgraph of 𝐺. We also write 𝐺 9 Gr(𝑛) for the number of vertices in 𝐵𝐺 𝑛 , and define 𝑆 𝑛 to be the sphere of radius 𝑛. The local (aka Benjamini-Schramm) topology on the set of all transitive graphs is the metrisable10 topology with respect to which a sequence (𝐺 𝑛 ) converges to 𝐺 if and only if for 𝑟 ∈ N, the balls 𝐵𝑟𝐺 𝑛 and 𝐵𝑟𝐺 are ∞ converges locally isomorphic for all sufficiently large 𝑛. For example, the sequence of tori (Z2𝑛 )𝑛=1 to Z2 . Given metric spaces 𝑋 and 𝑌 , the Gromov-Hausdorff distance between 𝑋 and 𝑌 , denoted distGH (𝑋, 𝑌 ), is the infimum over all 𝜀 > 0 such that there exists a metric space 𝑍 and isometric embeddings 𝜙 : 𝑋 → 𝑍 and 𝜓 : 𝑌 → 𝑍 such that the Hausdorff distance between the images of 𝜙 and 𝜓 in 𝑍 is at most 𝜀. Given a graph 𝐺 and 𝑟 > 0, we write 𝑟𝐺 for the rescaled graph metric 2 ∞ of 𝐺 where all distances are multiplied by 𝑟. For example, the sequence of rescaled tori ( 2𝜋 𝑛 Z𝑛 ) 𝑛=1 Gromov-Hausdorff converges to the continuum torus 𝑆 1 × 𝑆 1 with the 𝐿 1 metric, where 𝑆 1 is the unit circle. The scaling limits that arise like this, as a Gromov-Hausdorff limit of a sequence of diameter-rescaled finite transitive graphs, are explored in [BFT17]. The main result of our paper resolves the problems of sharpness and locality for all bounded-degree finite transitive graphs that are not one-dimensional in a certain coarse-geometric sense. Theorem 8.1.1. Let G be an infinite set of finite transitive graphs with bounded degrees. Suppose
𝜋 that there does not exist an infinite subset H ⊆ G such that diam 𝐺 𝐺 𝐺∈H Gromov-Hausdorff converges to the unit circle. Then both of the following statements hold: 1. Percolation on G has a sharp phase transition. 2. If G converges locally to an infinite transitive graph 𝐻, then the constant sequence 𝑝 : 𝐺 ↦→ 𝑝 𝑐 (𝐻) is the percolation threshold for G. In fact, if either of these two statements is false, then there exists an infinite subset H ⊆ G such that for every 𝐺 ∈ H ,  𝜋  𝑒 (log diam 𝐺) 1/9 1 distGH 𝐺, 𝑆 ≤ . diam 𝐺 diam 𝐺 We interpret the upper bound on Gromov-Hausdorff distance as a bound on the rate of convergence of the large scale geometry of graphs in H towards the unit circle as their diameters tend to infinity. 9 We generalise these to non-integer 𝑛 by setting 𝐵 𝐺 := 𝐵 𝐺 and by defining 𝑆 𝐺 and Gr(𝑛) analogously. 𝑛 𝑛 ⌊𝑛⌋ 10 This topology is induced by the metric dist(𝐺, 𝐻) := exp(− max{𝑛 : 𝐵 𝐺  𝐵 𝐻 }), for example. 𝑛 𝑛 328 In this sense, graphs in H converge to the unit circle faster than do the tori {Z𝑛 × Z𝑒 (log 𝑛) 8 }𝑛≥1 , and in particular faster than do the polynomially-stretched tori {Z𝑛 × Z𝑛𝐶 }𝑛≥1 for any constant 𝐶. On the other hand, by using arguments specific to Euclidean tori [EH21a, Example 5.1], items 1 and 2 only fail once we reach exponentially-stretched tori {Z𝑛 × Z𝐶 𝑛 }𝑛≥1 for a constant 𝐶. So our rate is not sharp in this special case, even if we could improve the exponent 1/9, which we did not try to optimise. Perhaps these exponentially-stretched tori are worst possible, in which case the optimal (log diam 𝐺) 1/9 log diam 𝐺 bound on the rate should be on the order of diam 𝐺 instead of 𝑒 diam 𝐺 . There are also stronger ways that one could hope to describe the “one-dimensionality” of H . (See the discussion at the end of Section 8.1.) Previous work and strategy of the proof Recall from our earlier discussion that sharpness and locality for finite transitive graphs are equivalent. (See Proposition 8.2.9.) In this paper we will prove sharpness directly. At a high level, our idea is to apply arguments derived from the proofs of four existing results in succession: (1) The sharpness of the phase transition for infinite transitive graphs; (2) The locality of the critical point for infinite transitive graphs; (3) The uniqueness of the supercritical giant cluster on finite transitive graphs; (4) The existence of a percolation threshold on finite transitive graphs. Below we discuss each of these works and how they feature in our argument. To prove sharpness, we will start with a sequence 𝑝 : G → (0, 1) where P 𝑝 has a cluster larger than a large multiple of log |𝑉 | with good probability. Then given any 𝜀 > 0, we will show that P (1+𝜀) 𝑝 has a giant cluster with high probability. Since inf 𝑝 > 0, we can replace P (1+𝜀) 𝑝 by P 𝑝+𝜀 , or equivalently, P 𝑝+4𝜀 . We will split the jump 𝑝 → 𝑝 + 4𝜀 into four little hops 𝑝 → 𝑝 + 𝜀 → . . . → 𝑝 + 4𝜀. After each hop, we will prove something stronger about the connectivity properties of percolation at the current parameter. Each hop is the subject of one section, discussed below, and involves one of the four works listed above. Large clusters → local connections The sharpness of the phase transition for infinite transitive graphs is the statement that for every infinite transitive graph 𝐺 and every 𝑝 < 𝑝 𝑐 (𝐺), there is constant 𝑐(𝐺, 𝑝) > 0 such that P𝐺𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝑒 −𝑐𝑛 for every 𝑛 ≥ 1. Some people use slightly different definitions. For example, some replace this exponential tail on |𝐾1 | by the exponential decay of connection probabilities, which is 𝜀 as part of a priori weaker, and some include the mean-field lower bound 𝜃 ((1 + 𝜀) 𝑝 𝑐 (𝐺)) ≥ 1+𝜀 the definition. The analogue of the mean-field lower bound for finite transitive graphs was already established in full generality in our earlier work [Eas23, Corollary 4]. This statement of sharpness 329 for infinite transitive graphs looks similar to our definition of sharpness for an infinite set G of finite transitive graphs with bounded degrees. Indeed, let 𝑝 be the percolation threshold for G. By a simple union bound, if for all 𝜀 > 0 there exists 𝐶 (G, 𝜀) < ∞ such that P𝐺(1−𝜀) 𝑝(𝐺) (|𝐾𝑜 | ≥ 𝑛) ≤ 𝐶𝑒 −𝑛/𝐶 for all 𝑛 ≥ 1 and 𝐺 ∈ G, then percolation on G has a sharp phase transition. With a little more work (see Proposition 8.2.9), one can show that the converse holds too. So a natural approach towards proving sharpness for finite transitive graphs is to try to adapt an existing proof of sharpness for infinite transitive graphs. Before explaining what is wrong with this approach, notice that something must go wrong because these arguments are completely general, applying to every infinite transitive graph - including Z, whereas as illustrated by the sequence of stretched tori, some hypothesis on the geometry of finite transitive graphs is required for sharpness to hold. The problem is not that the arguments cannot be run, but rather that they do not address the right question. Roughly speaking, the issue is that a cluster that grows faster than every particular microscopic scale is not automatically macroscopic. Slightly more precisely, given an infinite graph 𝐺 and parameter 𝑝, if inf 𝑛≥1 P 𝑝 (|𝐾𝑜 | ≥ 𝑛) > 0, then under P 𝑝 there is an infinite cluster almost surely. However, for an infinite set G of finite graphs and a sequence of parameters 𝑝, if inf 𝑛≥1 lim inf G P 𝑝 (|𝐾𝑜 | ≥ 𝑛) > 0, then it does not necessarily follow that under P 𝑝 there is a giant cluster with high probability. While proofs of sharpness for infinite transitive graphs do not directly yield Theorem 8.1.1, our first step is still to adapt and run one of these proofs on finite transitive graphs. We will also apply Hutchcroft’s idea [Hut20a] of using his two-ghost inequality to convert point-to-sphere bounds into point-to-point lower bounds. (This was also the first step of [EH23b].) Together, this will establish that after the first hop, P 𝑝+𝜀 satisfies a point-to-point lower bound on a large constant scale. In this section we will also use an elementary spanning tree argument to prove a kind of “reverse” implication that if P 𝑝 was instead assumed to satisfy such a point-to-point lower bound, then it would follow that P 𝑝 has a cluster much larger than log |𝑉 | with high probability. This reverse direction is not relevant to proving Theorem 8.1.1, but we will apply it to establish the equivalence of different characterisations of sharpness on finite transitive graphs and in particular to prove the equivalence of items 1 and 2 in Theorem 8.1.1. Local connections → global connections Earlier we discussed the proof of the locality of the critical point for infinite transitive graphs [EH23b] and the obstructions to using the same argument to prove locality for finite transitive graphs. The primary obstruction was the application of sharpness, which we explained could be 330 replaced by a good enough quantitative quasi-sharpness. If we had organised the argument in the present paper as a direct proof of locality, rather than of sharpness, then this quasi-sharpness would be supplied by the following two hops (𝑝 + 2𝜀 → 𝑝 + 3𝜀 → 𝑝 + 4𝜀). For the current hop (𝑝 + 𝜀 → 𝑝 + 2𝜀), we will run the part of the proof of locality for infinite graphs leading up to this application of sharpness (after dealing with the challenges that we discussed this entails) to propagate the microscopic point-to-point lower bound at 𝑝 + 𝜀 to a global point-to-point lower bound at 𝑝 + 2𝜀. More precisely, we prove that if G does not contain a sequence converging rapidly to the unit circle, then for some explicit and slowly-decaying function 𝑓 : N → (0, 1), all but finitely many graphs 𝐺 ∈ G satisfy min P 𝑝+2𝜀 (𝑢 ↔ 𝑣) ≥ 𝑓 (|𝑉 |). 𝑢,𝑣∈𝑉 Global connections → unique large cluster In the supercritical phase of percolation on a bounded-degree finite transitive graph, there is exactly one giant cluster with high probability. This had been conjectured by Benjamini and was verified in our joint work with Hutchcroft [EH21a]. It is important to note that this result actually does not rely on the existence of a percolation threshold. To make sense of this, we need a definition of the supercritical phase that is agnostic to the existence of a percolation threshold. Let G be an infinite set of bounded-degree finite transitive graphs, and let 𝑞 : G → (0, 1) be a sequence of parameters. If 𝐺 admits a percolation threshold 𝑝, then the natural definition for 𝑞 being supercritical is that lim inf 𝑞/𝑝 > 1. To make this independent of the existence of 𝑝, we say that 𝑞 is supercritical if there exists a sequence 𝑞′ : G → (0, 1) and a constant 𝜀 > 0 such that lim inf 𝑞/𝑞′ > 1 and lim inf P𝑞 ′ (|𝐾1 | ≥ 𝜀 |𝑉 |) ≥ 𝜀. In this language, the main result of [EH21a] is that for every supercritical sequence 𝑞, the number of vertices |𝐾2 | contained in the second largest cluster satisfies lim P𝑞 (|𝐾2 | ≥ 𝛿 |𝑉 |) = 0 for every constant 𝛿 > 0. The argument in [EH21a] is fully quantitative. In particular, if we slightly weaken the hypothesis that 𝑞 is supercritical by replacing the constant 𝜀 > 0 in the definition of “𝑞 is supercritical” by a slowly decaying sequence 𝜀 : G → (0, 1), then we can still deduce that under P𝑞 the largest cluster is much larger than all other clusters with high probability. What we need is the same conclusion but with the alternative hypothesis that 𝛿 := min𝑢,𝑣∈𝑉 P𝑞 ′ (𝑢 ↔ 𝑣) tends to zero slowly. This is certainly possible in principle because by Markov’s inequality, the lower bound min𝑢,𝑣∈𝑉 P𝑞 ′ (𝑢 ↔ 𝑣) ≥ 2𝜀 always implies the lower bound P𝑞 ′ (|𝐾1 | ≥ 𝜀 |𝑉 |) ≥ 𝜀. Unfortunately, this approach ultimately requires that 𝛿 tends to zero extremely slowly, too slowly for our purposes. Fortunately, the argument in [EH21a] turns out to run much more efficiently if we directly supply the hypothesis 331 that min𝑢,𝑣∈𝑉 P𝑞 ′ (𝑢 ↔ 𝑣) ≥ 𝜀 rather than the hypothesis that P𝑞 ′ (|𝐾1 | ≥ 𝜀 |𝑉 |) ≥ 𝜀. Indeed, a significant loss in the proof in [EH21a] is due to the conversion of the latter into the former. We will apply this to deduce from the global point-to-point lower bound at P 𝑝+2𝜀 that under P 𝑝+3𝜀 , the largest cluster is much larger than all other clusters with high probability. However, note that a priori this largest cluster might not be giant, i.e. we may still have |𝐾1 | = 𝑜(|𝑉 |) with high probability. In particular, our proof is not complete at this stage, which is why we need the fourth hop. Unique large cluster → giant cluster Every infinite set of finite transitive graphs with bounded degrees G admits a percolation threshold. We verified this in [Eas23] by combining [EH21a; Van23]. The reader may find it surprising that the uniqueness of the supercritical giant cluster comes first, before the existence of a percolation threshold. Indeed, this is opposite to the order in the classical story for the Erdős-Rényi model, for example. On the other hand, the reader may suspect that the result is obvious because standard sharp threshold techniques imply that for every sequence 𝛼, the event {|𝐾1 | ≥ 𝛼 |𝑉 |} always has a sharp threshold.11 The challenge is to prove that every sequence 𝛼 that decays sufficiently slowly has a common sharp threshold. To prove this we embedded the fact that the supercritical giant cluster is unique into Vanneuville’s proof of the sharpness of the phase transition for infinite transitive graphs. In [Eas23], we did not give any explicit bounds because we were working without the hypothesis that G has bounded degrees. At this level of generality, there actually exist (very particular) sequences that do not admit a percolation threshold, and even for those that do, the implicit rates of convergence can be arbitrarily bad. However, our argument is itself fully quantitative. In particular, we will explain how it can still be run under an explicit weaker version of the uniqueness of the giant cluster. This will allow us to deduce from the global two-point lower bound under P 𝑝+2𝜀 and the uniqueness of the largest cluster under P 𝑝+3𝜀 that there is a giant cluster under P 𝑝+4𝜀 with high probability, completing our proof of Theorem 8.1.1. Further discussion Let us further explore the connection between percolation on finite and infinite transitive graphs. First, let us remark on how to canonically define 𝑝 𝑐 for finite graphs. By [Eas23], there exists a universal function 𝑝 𝑐 : F → (0, 1), where F is the set of all finite transitive graphs, such that for every infinite set G of finite transitive graphs with bounded degrees, the restriction 𝑝 𝑐 | G is the 11 We say that a sequence of events ( 𝐴(𝐺)) 𝐺 ∈ G has a sharp threshold if there exists a sequence 𝑝 such that lim sup 𝑞/𝑝 < 1 implies P𝑞 ( 𝐴) = 0 and lim inf 𝑞/𝑝 > 1 implies P𝑞 ( 𝐴) = 1 for every sequence 𝑞. 332 percolation threshold for G. Let us fix such a function 𝑝 𝑐 for the rest of this section. Now, thanks to Theorem 8.1.1, we can roughly12 interpret this as the unique continuous extension with respect to the local topology of the usual percolation threshold 𝑝 𝑐 for infinite transitive graphs to the set of finite transitive graphs. Let 𝐻 be an infinite transitive graph, and let G be an infinite set of finite transitive graphs with bounded degrees that does not contain a sequence approximating the unit circle in the sense that H does in Theorem 8.1.1. Let (𝑉𝑛 ) be an exhaustion of 𝐻 by finite sets, and let 𝐾∞ denote the set of vertices contained in infinite clusters. By a second-moment calculation, under P𝐻𝑝 for any 𝑝, |𝐾∞ ∩ 𝑉𝑛 | → 𝜃 𝐻 ( 𝑝) := P𝐻𝑝 (𝑜 ↔ ∞) |𝑉𝑛 | in probability as 𝑛 tends to infinity. In this sense, 𝜃 𝐻 ( 𝑝) captures the density of the union of the infinite clusters. In a finite graph 𝐺, we define the giant density to be ∥𝐾1 ∥ := |𝑉1 | |𝐾1 |. In conjunction with the main result of [EH23+a], Theorem 8.1.1 implies that if G converges locally to 𝐻, then for every constant 𝑝 ∈ (0, 1)\{𝑝 𝑐 (𝐻)}, the density ∥𝐾1 ∥ under P𝐺𝑝 converges in probability to the density 𝜃 𝐻 ( 𝑝) under P𝐻𝑝 as we run through 𝐺 ∈ G. In this sense, the infinite cluster phenomenon on infinite transitive graphs is a good model for the giant cluster phenomenon on finite transitive graphs. Similar ideas are discussed in Benjamini’s original work [Ben01b]. In light of this, our results let us easily move statements about infinite transitive graphs to the setting of finite transitive graphs. Here are three examples. For all three, remember that G is assumed to be a family of graphs satisfying the hypotheses of Theorem 8.1.1. First, it is wellknown that 𝑝 𝑐 (𝐻) < 1 if (and only if) 𝐻 is not one-dimensional [DGRSY20]. By the conclusion of Theorem 8.1.1, it immediately follows13 that sup𝐺∈G 𝑝 𝑐 (𝐺) < 1, i.e. there exists 𝜀 > 0 such that P1−𝜀 (|𝐾1 | ≥ 𝜀 |𝑉 |) ≥ 𝜀 every 𝐺 ∈ G. This conclusion is not new; we simply wish to illustrate how easily it follows from Theorem 8.1.1. Indeed, Hutchcroft and Tointon established this fundamental result under a weaker (essentially optimal!) version of the hypothesis that G is not one-dimensional, (almost) fully resolving a conjecture of Alon, Benjamini, and Stacey [ABS04c]. Second, it is a major open conjecture that 𝜃 𝐻 (·) is continuous if (and only if) 𝐻 is not one-dimensional. Following the discussion in our previous paragraph, this conjecture would immediately imply the following statement, which says that the giant cluster emerges gradually: Let P denote the law of the standard monotone coupling (𝜔 𝑝 : 𝑝 ∈ [0, 1]) of the percolation measures (P 𝑝 : 𝑝 ∈ [0, 1]), and define 12 It is unique (up to 𝑜(1)) and continuous whenever we restrict to an infinite set G that is compact in the local
𝜋 1 > 0. topology and satisfies inf 𝐺 ∈ G distGH diam 𝐺 𝐺, 𝑆 13 One just needs to verify that G cannot converge locally to a one-dimensional infinite transitive graph, e.g. by Lemma 8.3.20. 333 𝛼( 𝑝) := 𝐾1 (𝜔 𝑝 ) . Then for all 𝜀 > 0 there exists 𝛿 > 0 such that   lim P sup [𝛼( 𝑝 + 𝛿) − 𝛼( 𝑝)] ≤ 𝜀 = 1. 𝐺∈G (8.1.1) 𝑝 For example, thanks to [Hut16], we can already deduce from this relation between infinite and finite transitive graphs that eq. (8.1.1) holds whenever our family G has exponential growth on microscopic scales, e.g. for the sequence (Z3𝑛 × 𝐺 ℎ(𝑛) ) where (𝐺 𝑛 ) is a sequence of transitive expanders and ℎ : N → N tends to infinity arbitrarily slowly. (See the discussion in [EH23b, Example 5.1] of stretched tori, in which the giant cluster does not emerge gradually.) By the uniqueness of the supercritical giant cluster [EH21a], finite transitive graphs satisfying eq. (8.1.1) also automatically satisfy the conclusion of [ABS04a, Conjecture 1.1]. This links the well-known continuity conjecture for infinite transitive graphs to this conjecture about the uniqueness of the largest cluster in finite transitive graphs. Third, it is conjectured that the uniqueness threshold 𝑝 𝑢 (𝐻) satisfies 𝑝 𝑐 (𝐻) < 𝑝 𝑢 (𝐻) if and only if 𝐻 is nonamenable. Again by our discussion in previous paragraph, this conjecture would imply that if the Cheeger constant on graphs in G is uniformly bounded below on microscopic scales, then percolation on G has a phase in which there is a giant cluster whose metric distortion tends to infinity. (See [EH21a, Remark 1.6].) What can be said when the Cheeger constant is uniformly bounded below on larger scales? In the limit, this connects the 𝑝 𝑐 vs 𝑝 𝑢 question to the existing theory of percolation on expanders. This opens the door to many directions for future work, adapting questions and techniques from percolation on infinite transitive graphs to finite transitive graphs. For example, what can be said about supercritical sharpness? Since the continuity conjecture for infinite transitive graphs would imply the unique giant cluster conjecture of [ABS04a, Conjecture 1.1] (possibly with a weaker onedimensionality condition), might [ABS04a, Conjecture 1.1] be a stepping stone towards continuity that is easier to establish? Another direction for future work is to improve the rate of convergence in Theorem 8.1.1. One could also explore stronger notions of one-dimensionality. The GromovHausdorff metric only considers the coarse geometry of graphs, ignoring how densely vertices are packed (i.e. the volume growth on small scales). It is natural to expect that graphs in which vertices are packed more densely can afford to have a more one-dimensional coarse geometry before percolation arguments break down. For example, consider the product of a torus with a long cycle versus the product of an expander with a long cycle. In the work of Hutchcroft and Tointon [HT21a], one-dimensionality was characterised more stringently14 in terms of the relationship of |𝑉 | volume to diameter, for example, by requiring that |𝑉 | ≤ (diam 𝐺) 1+𝜀 or log|𝑉 | = 𝑜(diam 𝐺). One 14 This is indeed stronger than asking for Gromov-Hausdorff convergence to the unit circle, by the results of [BFT17]. 334 could also investigate questions such as sharpness without bounded degrees. For example, [EH21a; Eas23; EH23+a] did not require this hypothesis, thus linking the story of percolation on infinite transitive graphs to the classical Erdős-Rényi model. Acknowledgement We thank Tom Hutchcroft for helpful comments on an earlier draft. 8.2 Large clusters → local connections In this section we will adapt a proof of the sharpness of the phase transition for infinite transitive graphs to finite transitive graphs. By combining this with an idea of Hutchcroft [Hut20a] to convert voume-tail bounds into point-to-point bounds, we will prove the following proposition. This roughly says that if for some percolation parameter 𝑝, the largest cluster contains much more than log |𝑉 | vertices with good probability, then for percolation of any higher parameter 𝑝 + 𝜂, we have a uniform point-to-point lower bound on a divergently large scale. Later, in Section 8.2 we will prove a kind of converse to this statement, and in Section 8.2 we will use this converse to give equivalent characterisations of sharpness for finite transitive graphs. Proposition 8.2.1. Let 𝐺 be a finite transitive graph with degree 𝑑. Let 𝜂 > 0. There exists 𝑐(𝑑, 𝜂) > 0 such that for all 𝑝 ∈ (0, 1) and 𝜆 ≥ 1, P 𝑝 (|𝐾1 | ≥ 𝜆 log |𝑉 |) ≥ 1 𝑐 |𝑉 | 𝑐 =⇒ min 𝑢∈𝐵 𝑐 log(𝜆) − 1 P 𝑝+𝜂 (𝑜 ↔ 𝑢) ≥ 𝑐 𝜂2 . 20 We have chosen to adapt Vanneuville’s recent proof of sharpness for infinite graphs [Van24]. This involves ghost fields. Given a graph 𝐺, a ghost field of intensity 𝑞 ∈ (0, 1) is a random set of vertices 𝑔 ⊆ 𝑉 distributed according to (Bernoulli) site percolation of parameter 𝑞.15 We denote its law by Q𝑞 and write P 𝑝 ⊗ Q𝑞 for the joint law of independent samples 𝜔 ∼ P 𝑝 and 𝑔 ∼ Q𝑞 . One reason to introduce ghost fields is that it can be easier to work with the event {𝑜 ↔ 𝑔} when 𝑞 = 1/𝑛 than to work with the closely related event {|𝐾𝑜 | ≥ 𝑛}. The following is [Van24, Theorem 2]. This can also be deduced from [Hut20d] with different constants. This says that starting from any percolation parameter 𝑝, if we decrease 𝑝 by a suitable amount, then the volume of the cluster at the origin will have an exponential tail under the new parameter. This is proved by a variant of Vanneuville’s stochastic comparison technique from [Van23], which we will describe in more detail in Section 8.5. 15 Some authors use a slightly different parameterisation. When we write “a ghost field of intensity 𝑞 ∈ (0, 1)”, they write “a ghost field of intensity ℎ > 0” for the same object, where 𝑞 = 1 − 𝑒 −ℎ . 335 Lemma 8.2.2. Let 𝐺 be a transitive graph. Given 𝑝 ∈ (0, 1) and ℎ > 0, define 𝜔 𝜇 𝑝,ℎ := P 𝑝 ⊗ Q1−𝑒 −ℎ (𝑜 ← → 𝑔). Then for all 𝑚 ≥ 1, P (1−𝜇 𝑝,ℎ ) 𝑝 (|𝐾𝑜 | ≥ 𝑚) ≤ P 𝑝 (|𝐾𝑜 | ≥ 𝑚) −ℎ𝑚 𝑒 . 1 − 𝜇 𝑝,ℎ Vanneuville proved this lemma when 𝐺 is infinite, but his proof also works verbatim when 𝐺 is finite. The following easy corollary of this lemma says (contrapositively) that if the cluster at the origin is much larger than log |𝑉 | with reasonable probability, then after sprinkling, the cluster at the origin is at least mesoscopic with good probability. Corollary 8.2.3. Let 𝐺 be a finite transitive graph. For all 𝜀 > 0 there exists 𝑐(𝜀) > 0 such that for all 𝑝 ∈ (0, 1) and 𝑛, 𝜆 ≥ 1, P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝜀 =⇒ P (1−2𝜀) 𝑝 (|𝐾𝑜 | ≥ 𝜆 log |𝑉 |) ≤ 1 . 𝑐𝜆 𝑐 |𝑉 | 𝑛 Proof. Suppose that P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝜀. We may assume that 𝜀 < 1/2, otherwise the result is 1 trivial. Define ℎ := 𝑛1 log 1−𝜀 and 𝑞 := 1 − 𝑒 −ℎ . By a union bound, 𝜔 𝜔 𝜇 𝑝,ℎ := P 𝑝 ⊗ Q𝑞 (𝑜 ← → 𝑔) ≤ P 𝑝 (|𝐾𝑜 | ≥ 𝑛) + P 𝑝 ⊗ Q𝑞 (𝑜 ← → 𝑔 | |𝐾𝑜 | < 𝑛). We now bound these two terms individually. By hypothesis, P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝜀. By our choice of ℎ and 𝑞, 𝜔 P 𝑝 ⊗ Q𝑞 (𝑜 ← → 𝑔 | |𝐾𝑜 | < 𝑛) ≤ 1 − 𝑒 −ℎ𝑛 = 𝜀. Therefore 𝜇 𝑝,ℎ ≤ 2𝜀. So by Lemma 8.2.2, P (1−2𝜀) 𝑝 (|𝐾𝑜 | ≥ 𝜆 log |𝑉 |) ≤ P 𝑝 (|𝐾𝑜 | ≥ 𝜆 log |𝑉 |) −ℎ𝜆 log|𝑉 | 𝜆 1 1 |𝑉 | − 𝑛 log 1− 𝜀 . 𝑒 ≤ 1 − 2𝜀 1 − 2𝜀  1 . So the claim holds with 𝑐 := min 1 − 2𝜀, log 1−𝜀 □ To convert the fact that the cluster at the origin is at least mesoscopic with good probability into a uniform point-to-point lower bound on a divergently large scale, we will apply Hutchcroft’s volumetric two-arm bound [Hut20a, Corollary 1.7], stated below as Theorem 8.2.4. This applies in our setting because every finite transitive graph is unimodular, and in this case we can trivially drop the hypothesis that at least one of the clusters is finite in the definition of T𝑒,𝑛 . This tells us that it is always unlikely that the endpoints of a given edge belong to distinct large clusters. 336 Theorem 8.2.4. Let 𝐺 be a unimodular transitive graph with degree 𝑑. There exists 𝐶 (𝑑) < ∞ such that for all 𝑒 ∈ 𝐸, 𝑛 ≥ 1, and 𝑝 ∈ (0, 1)  1− 𝑝 P 𝑝 (T𝑒,𝑛 ) ≤ 𝐶 𝑝𝑛  1/2 , where T𝑒,𝑛 is the event that the endpoints of 𝑒 belong to distinct clusters, each of which contains at least 𝑛 vertices, and at least one of which is finite. Hutchcroft showed in [Hut20a] that this can be used to convert volume-tail bounds into point-topoint bounds. This was also used in [EH23b]. Here is the quantitative output of his argument, stated in the case of finite graphs. Corollary 8.2.5. Let 𝐺 be a finite transitive graph with degree 𝑑. There exists 𝐶 (𝑑) < ∞ such that for all 𝑛, 𝑟 ≥ 1 and 𝑝 ∈ (0, 1), min P 𝑝 (𝑜 ↔ 𝑢) ≥ P 𝑝 (|𝐾𝑜 | ≥ 𝑛) 2 − 𝑢∈𝐵𝑟 𝐶𝑟 . 𝑝𝑟+1 𝑛1/2 Proof. Let 𝑢 ∈ 𝐵𝑟 . By Harris’ inequality and a union bound, P 𝑝 (𝑜 ↔ 𝑢) ≥ P 𝑝 (|𝐾𝑜 | ≥ 𝑛) 2 − P 𝑝 (|𝐾𝑜 | ≥ 𝑛 and |𝐾𝑢 | ≥ 𝑛 but 𝑜 ↮ 𝑢). The second term on the right can now be bounded by [EH23b, Lemma 2.6]. (In that lemma the hypothesis that 𝐺 is infinite and 𝑝 < 𝑝 𝑐 can be replaced by the hypothesis that 𝐺 is finite.) □ We now combine Corollary 8.2.3 and Corollary 8.2.5 to establish Proposition 8.2.1.
Proof of Proposition 8.2.1. Fix 𝑝 ∈ (0, 1), 𝜆 ≥ 1, and 𝜂 > 0. Let 𝜀 := 𝜂4 and let 𝑐 1 𝜂4 > 0 be the corresponding constant from Corollary 8.2.3. We may assume that 𝜂 ≤ 12 , 𝑐 1 < 1, and |𝑉 | > 1. Suppose that P 𝑝 (|𝐾1 | ≥ 𝜆 log |𝑉 |) ≥ 𝑐|𝑉1 | 𝑐 . By a union bound, P 𝑝 (|𝐾𝑜 | ≥ 𝜆 log |𝑉 |) ≥ 1𝑐+1 . Let 𝑐|𝑉 | 𝑐1𝜆 𝑐1𝜆 𝑛 := 2 . Since 𝑛 − 1 > 𝑐 1 and (1 − 2𝜀)( 𝑝 + 𝜂) ≥ 𝑝, it follows by Corollary 8.2.3 that  P 𝑝+𝜂  𝜂 𝑐1𝜆 |𝐾𝑜 | ≥ ≥ . 2 4 Let 𝐶1 (𝑑) < ∞ be the constant from Corollary 8.2.5. Let 𝑟 ≥ 1 be arbitrary. Then by Corollary 8.2.5,  𝜂 2 𝐶1𝑟 min P 𝑝+𝜂 (𝑜 ↔ 𝑢) ≥ −   1/2 . 𝑢∈𝐵𝑟 4 𝑐1𝜆 𝑟+1 ( 𝑝 + 𝜂) 2 337 Note that 𝑟 ≤ 𝜂−𝑟 because 𝜂 ≤ 21 . So there is a constant 𝐶2 (𝑑, 𝜂) < ∞ such that 𝐶1𝑟  ( 𝑝 + 𝜂) 𝑟+1 𝑐1𝜆 2  1/2 ≤ 𝐶2 . 𝜂2𝑟 𝜆1/2 2 𝜂 Now there exists 𝑐 2 (𝑑, 𝜂) > 0 such that 𝑟 := 𝑐 2 log(𝜆) − 𝑐12 satisfies 𝜂2𝑟𝐶𝜆21/2 ≤ 80 . Then by our above work (when 𝑟 ≥ 1, otherwise the inequality anyway holds trivally),  𝜂  2 𝜂2 𝜂2 min P 𝑝+𝜂 (𝑜 ↔ 𝑢) ≥ − = . 𝑢∈𝐵𝑟 4 80 20 Therefore the claim holds with 𝑐 := min{𝑐 1 , 𝑐 2 }. □ Local connections → large clusters In this subsection we prove the following proposition. This imples that if there is a uniform pointto-point lower bound on a divergently large scale, then the largest cluster contains much more than log |𝑉 | vertices with high probability. This is a kind of converse to Proposition 8.2.1. This will be used in the next subsection to prove the equivalence of different notions of sharpness. Proposition 8.2.6. Let 𝐺 be a finite transitive graph. For all 𝛿 > 0 there exists 𝑐(𝛿) > 0 such that for all 𝑝 ∈ (0, 1) and 𝑟 ≥ 1 with |𝐵𝑟 | ≤ |𝑉 | 1/10 ,
1 . min P 𝑝 𝑜 ↔ 𝑢) ≥ 𝛿 =⇒ P 𝑝 (|𝐾1 | ≥ 𝑐 |𝐵𝑟 | log |𝑉 | ≥ 1 − 𝑢∈𝐵𝑟 |𝑉 | 3/4 The next lemma converts point-to-point connection lower bounds on one scale into volume-tail lower bounds on all scales. The idea is to approximately cover the graph by a large number of balls on which the point-to-point lower bound holds then glue together large clusters from multiple balls. Lemma 8.2.7. Let 𝐺 be a finite transitive graph. For all 𝛿 > 0 there exists 𝑐(𝛿) > 0 such that for | all 𝑝 ∈ (0, 1) and 𝑛, 𝑟 ≥ 1 satisfying 𝑛 ≤ 𝑐|𝑉 |𝐵𝑟 | , 𝑛 min P 𝑝 (𝑜 ↔ 𝑢) ≥ 𝛿 =⇒ 𝑢∈𝐵𝑟 P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝑐𝑒 − 𝑐 | 𝐵𝑟 | . Proof. Fix 𝛿 > 0, 𝑝 ∈ (0, 1), and 𝑛, 𝑟 ≥ 1. Suppose that min𝑢∈𝐵𝑟 P 𝑝 (𝑜 ↔ 𝑢) ≥ 𝛿. Let 𝑊 be a maximal (with respect to inclusion) set of vertices such that 𝑜 ∈ 𝑊 and dist𝐺 (𝑢, 𝑣) ≥ 2𝑟 for all distinct 𝑢, 𝑣 ∈ 𝑊. Build a graph 𝐻 with vertex set 𝑊 by including the edge {𝑢, 𝑣} if and only if dist𝐺 (𝑢, 𝑣) ≤ 5𝑟 and 𝑢 ≠ 𝑣. By maximality of 𝑊, the graph 𝐻 is connected. Let 𝑇 be a spanning tree for 𝐻. Let 𝑓 : 𝑊\{𝑜} → 𝑊 be a function encoding 𝑇 where ‘ 𝑓 (𝑢) = 𝑣’ means that the edge {𝑢, 𝑣} is present in 𝑇 and dist𝑇 (𝑜, 𝑣) < dist𝑇 (𝑜, 𝑢). Extend this to a function 𝑓 : 𝑊 → 𝑊 by setting
𝑓 (𝑜) := 𝑜. By Markov’s inequality, every 𝑢 ∈ 𝑉 satisfies P 𝑝 |𝐾𝑢 ∩ 𝐵𝑟 (𝑢)| ≥ 2𝛿 |𝐵𝑟 | ≥ 2𝛿 .16 By 16 In this proof, P 𝑝 and |𝐵𝑟 | refer to 𝐺, not to 𝑇 or 𝐻. 338 Harris’ inequality, every edge {𝑢, 𝑣} in 𝐻 satisfies P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝛿5 . So by Harris’ inequality again, for every 𝑢 ∈ 𝑊, the event 𝐴𝑢 that 𝑢 ↔ 𝑓 (𝑢) and |𝐾𝑢 ∩ 𝐵𝑟 (𝑢)| ≥ 2𝛿 |𝐵𝑟 | satisfies P 𝑝 ( 𝐴𝑢 ) ≥ 𝛿5 · 2𝛿 = 𝛿6 2. | Let 𝑐(𝛿) > 0 be a small constant to be determined. Suppose that 𝑛 ≤ 𝑐|𝑉 |𝐵𝑟 | . By maximality of 𝑊, the balls {𝐵2𝑟 (𝑢) : 𝑢 ∈ 𝑊 } cover 𝑉, and hence |𝑉 | ≤ |𝑊 | · |𝐵2𝑟 |. So provided that 𝑐 is sufficiently small, |𝑉 | |𝑉 | 2𝑛 𝑛 |𝑊 | ≥ ≥ ≥ . ≥ |𝐵2𝑟 | 𝑐 |𝐵𝑟 | 𝛿 |𝐵𝑟 | |𝐵𝑟 | 2 l m 2𝑛 In particular, we can find a 𝑇-connected set of vertices 𝑈 ⊆ 𝑊 such that 𝑜 ∈ 𝑈 and |𝑈| = 𝛿|𝐵 . 𝑟| If 𝐴𝑢 holds for every 𝑢 ∈ 𝑈 then |𝐾𝑜 | ≥ 2𝛿 · |𝐵𝑟 | · |𝑈| ≥ 𝑛. So by Harris’ inequality, provided 𝑐 is sufficiently small, ! P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ P 𝑝 Ù 𝐴𝑢 𝑢∈𝑈 l m  6  𝛿 |2𝑛𝐵𝑟 | 𝑛 𝛿 ≥ ≥ 𝑐𝑒 − 𝑐 | 𝐵𝑟 | . 2 □ The following is a second-moment calculation for the number of vertices contained in large clusters.17 In the proof, it will be convenient to introduce partial functions to encode partially-revealed percolation configurations. Recall that a partial function 𝑓 : 𝐴 ⇀ 𝐵 is a function 𝐴′ → 𝐵 for some 𝐴′ ⊆ 𝐴, i.e. for every 𝑎 ∈ 𝐴, either 𝑓 (𝑎) ∈ 𝐵 or 𝑓 (𝑎) = ‘undefined’. We denote this set 𝐴′ on which 𝑓 is defined by dom( 𝑓 ). Given partial functions 𝑓 and 𝑔, the override 𝑓 ⊔ 𝑔 is the partial function with dom( 𝑓 ⊔ 𝑔) = dom( 𝑓 ) ∪ dom(𝑔) that is equal to 𝑓 on dom( 𝑓 ) and is equal to 𝑔 on dom(𝑔)\ dom( 𝑓 ). We write Var 𝑝 to denote the variance of a random variable under P 𝑝 . Lemma 8.2.8. Let 𝐺 be a finite transitive graph. For all 𝑛 ≥ 0 and 𝑝 ∈ (0, 1), the random set 𝑋 := {𝑢 ∈ 𝑉 : |𝐾𝑢 | ≥ 𝑛} satisfies Var 𝑝 |𝑋 | ≤ 𝑛2 · E 𝑝 |𝑋 | . Proof. Let P be the joint law of a uniformly random automorphism of 𝐺, denoted 𝜙, and three configurations 𝜔1 , 𝜔2 , 𝜔3 sampled according to P 𝑝 , where all four of these random variables are independent. Given a configuration 𝜔 : 𝐸 → {0, 1}, let 𝜔ˆ : 𝐸 ⇀ {0, 1} be the partial function encoding the edges revealed in an exploration of the cluster at 𝑜 from inside (with respect to an arbitrary fixed ordering of 𝐸) that is halted as soon as the event {|𝐾𝑜 (𝜔)| ≥ 𝑛} is determined by the states of the revealed edges. Define 𝜔 := ( 𝜔ˆ 1 ⊔ 𝜙( 𝜔ˆ 2 )) ⊔ 𝜔3 . By transitivity, the law of 𝜙(𝑜) 17 We were inspired by a weaker (degree-dependent) version of this argument that arose during joint work with Hutchcroft towards [EH23+a], which was made redundant and thus did not appear in the final version of that work. 339 is uniform on 𝑉, and by a standard cluster-exploration argument, P(𝜔 = · | 𝜙) = P 𝑝 almost surely. This lets us rewrite Var 𝑝 |𝑋 | as ∑︁   |𝑋 | Var 𝑝 = P 𝑝 (𝑢, 𝑣 ∈ 𝑋) − P 𝑝 (𝑢 ∈ 𝑋) · P 𝑝 (𝑣 ∈ 𝑋) 𝑢,𝑣 " # 1 ∑︁ = |𝑉 | E 𝑝 |𝑋 | · P 𝑝 (𝑢 ∈ 𝑋 | 𝑜 ∈ 𝑋) − P 𝑝 (𝑜 ∈ 𝑋) |𝑉 | 𝑢   = |𝑉 | E 𝑝 |𝑋 | · P (𝜙(𝑜) ∈ 𝑋 (𝜔) | 𝑜 ∈ 𝑋 (𝜔)) − P 𝑝 (𝑜 ∈ 𝑋) . (8.2.1) Consider the sets of vertices 𝐴1 := 𝐾𝑜 ( 𝜔ˆ 1 ) and 𝐴2 := 𝐾𝑜 ( 𝜔ˆ 2 ), which are defined purely in terms of the open edges in 𝜔ˆ 1 and 𝜔ˆ 2 respectively, i.e. all edges with ‘undefined’ state are treated as closed. Note that 𝑜 ∈ 𝑋 (𝜔) if and only if 𝑜 ∈ 𝑋 (𝜔1 ). Moreover, given that 𝑜 ∈ 𝑋 (𝜔1 ), if 𝜙(𝑜) ∈ 𝑋 (𝜔) then either 𝑜 ∈ 𝑋 (𝜔2 ) or 𝐴1 ∩ 𝜙( 𝐴2 ) ≠ ∅. So by a union bound and independence, P (𝜙(𝑜) ∈ 𝑋 (𝜔) | 𝑜 ∈ 𝑋 (𝜔)) ≤ P 𝑝 (𝑜 ∈ 𝑋) + P ( 𝐴1 ∩ 𝜙( 𝐴2 ) ≠ ∅ | 𝑜 ∈ 𝑋 (𝜔1 )) . (8.2.2) In particular, by eq. (8.2.1), it suffices to verify that P ( 𝐴1 ∩ 𝜙( 𝐴2 ) ≠ ∅ | 𝜔1 , 𝜔2 , 𝜔3 ) ≤ 𝑛2 |𝑉 | a.s. (8.2.3) Consider arbitrary deterministic sets of vertices 𝐵1 and 𝐵2 . By transitivity, the law of 𝜙(𝑢) for any fixed vertex 𝑢 is uniform over 𝑉. So by a union bound, ∑︁ ∑︁ |𝐵1 | |𝐵1 | |𝐵2 | P(𝐵1 ∩ 𝜙(𝐵2 ) ≠ ∅) ≤ = . P(𝜙(𝑢) ∈ 𝐵1 ) = |𝑉 | |𝑉 | 𝑢∈𝐵 𝑢∈𝐵 2 2 Equation (8.2.3) now follows by applying this to the sets 𝐴1 and 𝐴2 , which almost surely satisfy | 𝐴1 | , | 𝐴2 | ≤ 𝑛. □ We now combine Lemmas 8.2.7 and 8.2.8 to prove Proposition 8.2.6. Proof of Proposition 8.2.6. Let 𝛿 > 0, 𝑝 ∈ (0, 1), and 𝑟 ≥ 1. Suppose that |𝐵𝑟 | ≤ |𝑉 | 1/10 and min𝑢∈𝐵𝑟 P 𝑝 (𝑜 ↔ 𝑢) ≥ 𝛿. Let 𝑐 1 (𝛿) > 0 be the constant from Lemma 8.2.7. Let 𝑛 := 𝑐 |𝐵𝑟 | log |𝑉 | for a small constant 𝑐(𝛿) > 0 to be determined. Since |𝐵𝑟 | 2 ≤ |𝑉 | 2/10 , provided 𝑐 is small, 𝑛 := 𝑐 |𝐵𝑟 | log |𝑉 | ≤ 𝑐 1 |𝑉 | . |𝐵𝑟 | So Lemma 8.2.7 yields − 𝑛 −𝑐 P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≥ 𝑐 1 𝑒 𝑐1 | 𝐵𝑟 | = 𝑐 1 |𝑉 | 𝑐1 ≥ 𝑐 1 |𝑉 | −1/100 , 340 provided 𝑐 is small. By transitivity, it follows that the random set 𝑋 := {𝑢 ∈ 𝑉 : |𝐾𝑜 | ≥ 𝑛} satisfies E 𝑝 |𝑋 | ≥ 𝑐 1 |𝑉 | 99/100 . So by Chebychev’s inequality and Lemma 8.2.8, P 𝑝 (|𝐾1 | ≥ 𝑛) = 1 − P 𝑝 (|𝑋 | = 0) ≥ 1 − Var 𝑝 |𝑋 | 𝑛2 .
2 ≥ 1 − 𝑐 1 |𝑉 | 99/100 E 𝑝 |𝑋 | The conclusion follows because, provided 𝑐 is small,  𝑛2 𝑐 1 |𝑉 | = 99/100 (𝑐 |𝐵𝑟 | log |𝑉 |) 2 𝑐 1 |𝑉 | 99/100 1/10 log |𝑉 | 𝑐 1 |𝑉 | 99/100 𝑐 |𝑉 | ≤ 2 ≤ 1 |𝑉 | 3/4 . □ Equivalent notions of sharpness In this subsection we apply results from earlier in Section 8.2 to prove the following proposition. In the statement and the proof, we take for granted that G always admits a percolation threshold [Eas23]. Item 2 is analogous to the standard definition of sharpness for percolation on an infinite transitive graph. The fact that items 1 and 2 are equivalent is why we decided to label our version of “sharpness” for finite transitive graphs as such. Item 3 is analogous to the locality of the critical parameter for infinite transitive graphs. It is perhaps surprising that sharpness and locality are equivalent for finite graphs but not for infinite graphs. One way to make sense of this is that locality for infinite graphs is equivalent to a uniform (in the choice of graph) version of sharpness for infinite graphs, and for finite graphs, the only meaningful notion of sharpness is necessarily uniform. Proposition 8.2.9. For every infinite set G of finite transitive graphs with bounded degrees, the following are equivalent: 1. Percolation on G has a sharp phase transition. 2. For every subcritical sequence of parameters 𝑝, there exists a constant 𝐶 (G, 𝑝) < ∞ such that for all 𝐺 ∈ G and all 𝑛 ≥ 1, P 𝑝 (|𝐾𝑜 | ≥ 𝑛) ≤ 𝐶𝑒 −𝑛/𝐶 . 3. If an infinite subset H ⊆ 𝐺 converges locally to an infinite transitive graph 𝐻, then the constant sequence 𝐺 ↦→ 𝑝 𝑐 (𝐻) is the percolation threshold for H . We will prove that 3 =⇒ 2 =⇒ 1 =⇒ 3. For the first step, we apply Corollary 8.2.3 and compactness. 341 Proof that item 3 implies item 2. Assume that item 3 holds. Our goal is to prove that item 2 holds. Since G has bounded degrees, G is relatively compact in the local topology. In particular, we may assume without loss of generality that G converges locally to some infinite transitive graph 𝐺. (If item 2 is false, then we can find an infinite subset H ⊆ G such that item 2 is false for every sequence in H .) Now fix a subcritical sequence 𝑝 for G. By item 3, after passing to a tail of G if necessary, there exists a constant 𝜀 > 0 such that 𝑝(𝐺) ≤ (1 − 𝜀) 𝑝 𝑐 (𝐻) for every 𝐺 ∈ G. Pick 𝑟 ≥ 1 such that (|𝐾𝑜 | ≥ 𝑟) ≤ 𝜀/4. By passing to a further tail of G if necessary, we may assume that P𝐻 (1−𝜀/2) 𝑝 𝑐 (𝐻) 𝐵𝑟𝐺  𝐵𝑟𝐻 for every 𝐺 ∈ G. Then P𝐺(1−𝜀/2) 𝑝 (𝐻) (|𝐾𝑜 | ≥ 𝑟) ≤ 𝜀/4 for every 𝐺 ∈ G. Let 𝑐(𝜀) > 0 𝑐 𝑛 be the constant from Corollary 8.2.3. For every 𝑛 ≥ 1 and 𝐺 ∈ G, Corollary 8.2.3 with 𝜆 := log|𝑉 | tells us that P𝐺(1−𝜀) 𝑝 𝑐 (𝐻) (|𝐾𝑜 | ≥ 𝑛) ≤ P𝐺(1−𝜀/2) (1−2·𝜀/4) 𝑝 𝑐 (𝐻) (|𝐾𝑜 | ≥ 𝑛) ≤ 1 . 𝑐𝑒 𝑐𝑛/𝑟 Take 𝐶 := 𝑟/𝑐. The conclusion now follows by monotonicity because 𝑝(𝐺) ≤ (1 − 𝜀) 𝑝 𝑐 (𝐻) for every 𝐺 ∈ G. □ The second step is a simple union bound. Proof that item 2 implies item 1. Given a subcritical sequence 𝑝, let 𝐶 (G, 𝑝) < ∞ be the constant guaranteed to exist by item 2. Then for every 𝐺 ∈ G, P 𝑝 (|𝐾𝑜 | ≥ 2𝐶 log |𝑉 |) ≤ 𝐶𝑒 − 2𝐶 log|𝑉 | 𝐶 = 𝐶 , |𝑉 | 2 and hence by a union bound, P 𝑝 (|𝐾1 | ≤ 2𝐶 log |𝑉 |) ≥ 1 − |𝑉 | So lim P 𝑝 (|𝐾1 | ≤ 2𝐶 log |𝑉 |) = 1, as required. 𝐶 |𝑉 | 2 =1− 𝐶 . |𝑉 | □ We now turn to the third step, 1 =⇒ 3. Fix a choice of percolation threshold 𝑝 𝑐 : G → (0, 1), and think of this as an extension of the usual critical points 𝑝 𝑐 for percolation on the infinite transitive graphs that make up the boundary of G. Then our goal is to show that, assuming item 1, the function 𝑝 𝑐 is continuous as we approach the boundary of G from the interior. We split this into two parts: upper- and lower-semicontinuity. For lower-semicontinuity, we will apply a finite graph version of an argument of Pete [Pet, Section 14.2], which was based on the mean-field lower bound for infinite transitive graphs. For upper-semicontinuity, we will combine Corollary 8.2.5 and Proposition 8.2.6. 342 Proof that item 1 implies item 3. Suppose for contradiction that H ⊆ 𝐺 is an infinite subset that converges locally to some infinite transitive graph 𝐻, but the constant sequence 𝐺 ↦→ 𝑝 𝑐 (𝐻) is not a percolation threshold for H . By passing to a subsequence, we may assume without loss of generality that there is a constant 𝜀 > 0 such that either 𝑝 𝑐 (𝐺) ≤ (1 − 𝜀) 𝑝 𝑐 (𝐻) for every 𝐺 ∈ H , or 𝑝 𝑐 (𝐺) ≥ (1 + 𝜀) 𝑝 𝑐 (𝐻) for every 𝐺 ∈ H . Call these Case 1 and Case 2, corresponding to (a violation of) lower- and upper-semicontinuity in our discussion above. (Case 1) Since 𝑝 𝑐 : G → (0, 1) is a percolation threshold, there exists a constant 𝛿 > 0 such that P𝐺(1+𝜀) 𝑝 (𝐺) (|𝐾𝑜 | ≥ 𝛿 |𝑉 |) ≥ 𝛿 for every 𝐺 ∈ H . So by monotonicity, P𝐺(1−𝜀2 ) 𝑝 (𝐻) (|𝐾𝑜 | ≥ 𝛿 |𝑉 |) ≥ 𝛿 𝑐 𝑐 for every 𝐺 ∈ H . For every 𝑟 ≥ 1, there exists 𝐺 ∈ H such that 𝛿 |𝑉 | > 𝐵𝑟𝐻 and 𝐵𝑟𝐺  𝐵𝑟𝐻 , and hence P𝐻 (|𝐾𝑜 | ≥ 𝑟) ≥ P𝐺(1−𝜀2 ) 𝑝 (𝐻) (𝑜 ↔ 𝑆𝑟 ) ≥ P𝐺(1−𝜀2 ) 𝑝 (𝐻) (|𝐾𝑜 | ≥ 𝛿 |𝑉 |) ≥ 𝛿. (1−𝜀 2 ) 𝑝 (𝐻) 𝑐 𝑐 𝑐 In particular, P𝐻 (|𝐾𝑜 | = ∞) > 0, a contradiction. (1−𝜀 2 ) 𝑝 (𝐻) 𝑐 (Case 2) Let 𝐴 ≥ 1 be a given arbitrary constant. It suffices to prove that the parameter 𝑝 := (1 + 𝜀/2) 𝑝 𝑐 (𝐻) satisfies limG P𝐺𝑝 (|𝐾1 | ≥ 𝐴 log |𝑉 |) = 1. Set 𝛿 := P𝐻𝑝 (𝑜 ↔ ∞) > 0. Let 𝑑 be the vertex degree of 𝐻, and note that 𝑝 𝑐 (𝐻) > 1/𝑑, as this is well-known to hold for every infinite transitive graph. So by Corollary 8.2.5, there is a constant 𝐶 (𝑑) < ∞ such that for all 𝑛, 𝑟 ≥ 1 and 𝐻 all 𝐺 ∈ H with 𝐵𝐺 𝑛  𝐵𝑛 , 𝐶𝑟𝑑 𝑟+1 min P𝐺𝑝 (𝑜 ↔ 𝑢) ≥ 𝛿2 − 1/2 , 𝑛 𝑢∈𝐵𝑟𝐺  2 1/2  𝑛 𝑟 satisfies and in particular, (using that 𝑟 ≤ 𝑑 for all 𝑟 ≥ 1) the radius 𝑟 (𝑛) := log𝑑 𝛿2𝐶𝑑 2 min P𝐺𝑝 (𝑜 ↔ 𝑢) ≥ 𝛿2 − 𝑢∈𝐵𝑟𝐺(𝑛) 𝛿2 𝛿2 = . 2 2 Let 𝑐(𝛿2 /2) > 0 be the constant from Proposition 8.2.6. Fix 𝑛 sufficiently large that 𝑐 · 𝑟 (𝑛) ≥ 𝐴. 1/10 𝐺 𝐻 By passing to a tail of H if necessary, let us assume that 𝐵𝐺 for every 𝑛  𝐵𝑛 and |𝐵𝑟 (𝑛) | ≤ |𝑉 | 𝐺 ∈ H . Then by Proposition 8.2.6, for all 𝐺 ∈ H ,   P𝐺𝑝 |𝐾1 | ≥ 𝑐|𝐵𝑟𝐺(𝑛) | log |𝑉 | ≥ 1 − 1 . |𝑉 | 3/4 In particular, since 𝑐|𝐵𝑟𝐻(𝑛) | ≥ 𝑐𝑟 (𝑛) ≥ 𝐴, we deduce that limG P𝐺𝑝 (|𝐾1 | ≥ 𝐴 log |𝑉 |) = 1 as required. □ 343 8.3 Local connections → global connections In this section, we will apply the proof from [EH23b] that the critical point for percolation on (non-one-dimensional) infinite transitive graphs is local. As explained in the introduction, we need to both make this argument more finitary and adapt it to finite transitive graphs. We can roughly think of the proof of [EH23b] in two parts: First, if 𝐺 does not satisfy certain geometric properties around scale 𝑛, which include that 𝐺 is finitely-ended, then 𝐺 must satisfy a certain statement I𝑛 about the propagation of connection bounds around scale 𝑛. Second, if 𝐺 does satisfy these geometric properties and 𝐺 is one-ended, then 𝐺 again satisfies I𝑛 . Together, these two parts imply that if 𝐺 does not satisfy I𝑛 , then 𝐺 must actually be two-ended and hence one-dimensional. By looking at the proof of the second part, we can pinpoint where one-endedness is used, namely as a hypothesis in [EH23b, Lemma 5.8]. [EH23b, Lemma 5.8] concerns certain (𝑜, ∞)-cutsets called exposed spheres. The lemma says that if 𝐺 satisfies nice geometric properties around scale 𝑛 and is one-ended, then the exposed spheres around scale 𝑛 are in some sense well-connected. We took this from [CMT22, Lemma 2.1 and 2.7], where the authors deduced it from a theorem of Babson and Benjamini [BB99b]. By reading Timar’s proof [Tim07] of this theorem of Benjamini and Babson, we see that if an exposed sphere is not well-connected, then not only is 𝐺 multiply-ended, but this is actually witnessed by the exposed sphere itself in the sense that its removal from 𝐺 would create multiple infinite components. From this we can conclude that 𝐺 must in fact start to look one-dimensional from around scale 𝑛. This is how we will make this step from [EH23b] finitary. To adapt the argument to finite transitive graphs, we will additionally need to introduce the notion of the exposed sphere in a finite transitive graph and prove that finite transitive graphs can, for the purpose of part of our argument, be treated like infinite transitive graphs that are one-ended. Unfortunately, this application of Babson-Benjamini is deeply embedded in the proof of [EH23b] as it is currently written. So it will take some work to restructure the multi-scale induction in [EH23b] to isolate the relevant part. To avoid repetition, we have deferred the details of arguments that are implicit in [EH23b] to the appendix, thereby keeping many of the arguments in this section high-level. Ultimately we will prove the following proposition, which contains this finite-graph finitary refinement of locality. While we have written this for finite graphs, the same argument yields the analogous finitary refinement for infinite graphs. Proposition 8.3.1. Let 𝐺 be a finite transitive graph with degree 𝑑. Define   𝜋 9 1 and 𝛾 + := 𝑒 (log 𝛾) . 𝐺, 𝑆 · diam 𝐺 𝛾 := distGH diam 𝐺 344 For all 𝜀, 𝜂 > 0 there exists 𝜆(𝑑, 𝜀, 𝜂) < ∞ such that for all 𝑝 ∈ (0, 1), min P 𝑝 (𝑜 ↔ 𝑢) ≥ 𝜂 𝑢∈𝐵𝜆 =⇒ + 1/2 min P 𝑝+𝜀 (𝑜 ↔ 𝑢) ≥ 𝑒 −(log log 𝛾 ) . 𝑢∈𝐵 𝛾 + This proof of Proposition 8.3.1 is by induction. In Section 8.3, we describe the high-level structure of this induction, which is essentially the same as in [EH23b, Section 3.2], except for two differences. The first difference is that we have reworded the induction to say “we can keep propagating until and unless we reach a scale where the geometry is bad”, with a separate lemma that says “if the geometry is bad at scale 𝑛, then 𝐺 starts to look one-dimensional from around scale 𝑛”. In contrast, the induction in the earlier work simply says “if 𝐺 is not one-dimensional, then we can keep propagating forever”. The second difference is that the induction in the earlier work is slightly coarser in the sense that it groups multiple inductive steps of the argument we present here into a single inductive step. The additional detail in the present version is necessary to close the gap between the last scale from which we can propagate connection bounds and the first scale at which we can prove that the geometry “is bad”. The individual inductive steps are all implicit in [EH23b, Sections 4 and 6]. We will justify these in the appendix. In Section 8.3, we will prove something like the “base case” of the induction. This follows by a compactness argument from some intermediary results in [EH23b] and [CMT22]. In Section 8.3, we will prove that “if the geometry is bad at scale 𝑛, then 𝐺 starts to look onedimensional from around scale 𝑛”. This subsection is a refinement of [EH23b, Section 5], but for the reasons discussed, it will require some new ideas. The logic of the induction For the entirety of this subsection, fix a finite transitive graph 𝐺 with degree 𝑑, and define 𝛾 as in the statement of Proposition 8.3.1. We will describe the repeated-sprinkling multi-scale induction argument used to prove Proposition 8.3.1 (which is adapted from [EH23b]) as a deterministic colouring process evolving over time. At every time 𝑡 ∈ R, every scale18 𝑛 ∈ [3, ∞) can be coloured orange or green (or both, or neither — i.e. uncoloured), encoding a statement19 about the connec𝑡 tivity properties of percolation of parameter20 𝜙(𝑡) := 1 − 2−𝑒 over distances of approximately 𝑛. 18 It would have been more natural to consider scales 𝑛 ∈ N rather than 𝑛 ∈ [3, ∞). We chose the latter to avoid rounding issues and so that log log 𝑛 is always positive. 19 Formally, this colouring can be encoded as a function colour : [3, ∞) × R → P ({orange, green}), where P (𝑋) means the powerset of 𝑋. We say “𝑛 is green at time 𝑡” to mean that colour(𝑛, 𝑡) ∋ orange. Similar statements are formalised analogously. 20 This choice of parameterisation appears implicitly in [EH23b] as the natural choice for arguments that involve repeated sprinkling. Indeed, our function 𝜙 is the function Spr( 𝑝; 𝜆) from [EH23b, Section 3.1] evaluated at (1/2; 𝑡). 345 To lighten notation, let 𝛿(𝑛) := 𝑒 −(log log 𝑛) denote the standard small-quantity associated to each scale 𝑛. Now we colour a scale 𝑛 orange at time 𝑡 to mean that 1/2 min P𝜙(𝑡) (𝑜 ↔ 𝑢) ≥ 𝛿(𝑛). 𝑢∈𝐵𝑛 We also define the move-right (aka increase-scale) function 𝑅 : 𝑛 ↦→ 𝑒 (log 𝑛) , and write 𝑅 𝑘 := 𝑅 ◦ . . . ◦ 𝑅 for the 𝑘-fold composition of 𝑅 with itself. Now to prove Proposition 8.3.1, it suffices to prove the following lemma. 9 Lemma 8.3.2. For all 𝜀 > 0 and 𝑛1 ≥ 3 there exists 𝑛2 (𝑑, 𝜀, 𝑛1 ) < ∞ such that for all 𝑡 ∈ R with 𝑡 ≤ 𝜀1 , if [𝑛1 , 𝑛2 ] is orange at time 𝑡, then 𝑅(𝛾) is orange at time 𝑡 + 𝜀. Proof of Proposition 8.3.1 given Lemma 8.3.2. Fix 𝜀, 𝜂 > 0. Let 𝑛1 (𝜂) be the smallest integer satisfying 𝑛1 ≥ 3 and 𝛿(𝑛1 ) ≤ 𝜂. Let 𝛼(𝜀) > 0 be the unique real satisfying 𝜙(1/𝛼) = 1 − 𝜀. Let 𝑛2 (𝑑, 𝛼 ∧ 𝜀, 𝑛1 ) < ∞ be the constant that is guaranteed to exist by Lemma 8.3.2. We claim that we can take 𝜆 := 𝑛2 . Indeed, let 𝑝 ∈ (0, 1) and suppose that min𝑢∈𝐵𝑛2 P 𝑝 (𝑜 ↔ 𝑢) ≥ 𝜂. Define 𝑡 := 𝜙−1 ( 𝑝). The claim is trivial if 𝑝 ≥ 1 − 𝜀, so we may assume that 𝑡 ≤ 1/𝛼. By monotonicity of the function 𝛿(·), the interval [𝑛1 , 𝑛2 ] is orange at time 𝑡. So by applying Lemma 8.3.2, 𝑅(𝛾) is orange at time 𝑡 + 𝜀. Since (by calculus) 𝜙 is 1-Lipschitz, 𝑅(𝛾) is also orange at time 𝜙−1 ( 𝑝 + 𝜀), which is the required conclusion. □ We say that a set 𝑀 ⊆ N is a certain colour if every 𝑚 ∈ 𝑀 is that colour. Given a statement 𝐴 about a colouring at an implicit time 𝑡, we define 𝑠( 𝐴) := inf{𝑡 : 𝐴 is true at time 𝑡} where inf ∅ := +∞. For example, 𝑠 ({10, 12} is orange) := inf{𝑡 : 10 and 12 are both orange at time 𝑡} ∈ [−∞, +∞]. As a first approximation to our induction, imagine we knew that for every scale 𝑛 with 𝑛 ≤ 𝛾, 𝑠(𝑅(𝑛) is orange) ≤ 𝑠(𝑛 is orange) + 𝛿(𝑛). (8.3.1) Suppose for simplicity that some positive integer 𝑟 satisfies 𝑅𝑟 (𝑛2 ) = 𝛾 and that 𝑛2 exceeds some large universal constant. Then by repeatedly applying eq. (8.3.1), we could deduce that 𝑟 h    i ∑︁ 𝑠 (𝑅(𝛾) is orange) − 𝑠 ( [𝑛1 , 𝑛2 ] is orange) ≤ 𝑠 𝑅 𝑘+1 (𝑛2 ) is orange − 𝑠 𝑅 𝑘 (𝑛2 ) is orange ≤ 𝑘=0 𝑟 ∑︁  𝑘  𝛿 𝑅 (𝑛2 ) ≤ 𝑘=0 346 ∞ ∑︁ 𝑘=0 𝑒 − ( 9 log log 𝑛2 ) 𝑘 1/2 ≤ 2𝛿(𝑛2 ). Since 𝛿(𝑛2 ) → 0 as 𝑛2 → ∞, this would certainly imply Lemma 8.3.2. Rather than prove something as direct as eq. (8.3.1), we will have to bring into play a new colour, green. 𝑘 , we write start(𝜈) := 𝜈 and end(𝜈) := 𝜈 for its start and end Given a finite path 𝜈 = (𝜈𝑖 )𝑖=0 0 𝑘 Ð𝑘 vertices, |𝜈| := 𝑘 for its length, and given 𝑟 ≥ 0, we write 𝐵𝑟 (𝜈) := 𝑖=0 𝐵𝑟 (𝜈𝑖 ) for the associated tube. Given 𝑚, 𝑛 ≥ 1 and 𝑝 ∈ (0, 1), define the corridor function (which we take from [CMT22]),   𝜔∩𝐵𝑛 (𝜈) 𝜅 𝑝 (𝑚, 𝑛) := inf P 𝑝 start(𝜈) ←−−−−−→ end(𝜈) . 𝜈:|𝜈|≤𝑚 Notice that we always have 𝜅 𝑝 (𝑚, 𝑛) ≤ min𝑢∈𝐵𝑚 P 𝑝 (𝑜 ↔ 𝑢). Let us also define the set of low100 growth scales L := {𝑛 ≥ 3 : Gr(𝑛) ≤ 𝑒 (log 𝑛) }. Now we colour a scale 𝑛 green at time 𝑡 to mean that 𝑛 is orange and either 𝑛 ∉ L or 𝜅 𝜙(𝑡) (𝑅 2 (𝑛), 𝑛) ≥ 𝛿(𝑅(𝑛)). We will use this new colour to help us propagate orange by controlling the time taken for orange scales to turn green and for green scales to turn nearby scales orange. The next lemma says that green scales quickly turn nearby scales orange. If we see that 𝑛 is green at some time 𝑡 but 𝑛 ∈ L, then 𝜅 𝜙(𝑡) (𝑅 2 (𝑛), 𝑛) ≥ 𝛿(𝑅(𝑛)), which trivially implies that [𝑅(𝑛), 𝑅 2 (𝑛)] is already orange. So the content of this lemma is that if instead 𝑛 ∉ L, then we can efficiently propagate a point-to-point connection lower bound from scale 𝑛 to scales in [𝑅(𝑛), 𝑅 2 (𝑛)]. The proof of this is implicit in [EH23b, Section 4]. The argument uses some ghost-field technology that works more efficiently around scales 𝑛 where Gr(𝑛) is large. See the appendix for details. Lemma 8.3.3. There exists 𝑛0 (𝑑) < ∞ such that for all 𝑛 ≥ 𝑛0 , 𝑠([𝑅(𝑛), 𝑅 2 (𝑛)] is orange) ≤ 𝑠(𝑛 is green) + 𝛿(𝑛). The next lemma sometimes lets us control how long it takes for a scale 𝑛 to become green after turning orange. If 𝑛 ∉ L, then this time is trivially zero. So it suffices to consider 𝑛 ∈ L. We might hope for a statement like the following: there exists 𝑛0 (𝑑) < ∞ such that for all 𝑛 ≥ 𝑛0 with 𝑛 ∈ L, 𝑠(𝑛 is green) ≤ 𝑠(𝑛 is orange) + 𝛿(𝑛). (8.3.2) Our next lemma is less satisfying in two ways.21 First, we can only prove an upper bound like eq. (8.3.2) when 𝑛 belongs to a particular distinguished subset T(𝑐, 𝜆) of L. Second, our upper bound is in terms of a mysterious quantity Δ rather than something explicit like 𝛿(𝑛). So to use this lemma, we will need to: (1) Deal with scales 𝑛 ∈ L\T(𝑐, 𝜆), and (2) Find a way to upper bound Δ explicitly. For completeness, we will now define T(𝑐, 𝜆) and Δ, but the reader should feel free 21 We need to use the non-one-dimensionality hypothesis somewhere. 347 to skip these definitions for now because they are not necessary to follow the high-level induction argument being developed in this subsection. Given constants 𝑐, 𝜆 > 0, let T(𝑐, 𝜆) be the set of scales 𝑛 ∈ L such that 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes at every scale in an interval of the form [𝑚, 𝑚 1+𝑐 ] that is contained in [𝑛1/3 , 𝑛1/(1+𝑐) ]. We will recall the definition of plentiful polylog tubes, taken from [EH23b, Section 5], in Section 8.3. Given 𝑚, 𝑛 ≥ 1, we define Piv[𝑚, 𝑛] to be the event that in the restricted configuration 𝜔 ∩ 𝐵𝑛 , there are at least two clusters that each contain an open path from 𝐵𝑚 to 𝑆𝑛 . For each scale 𝑛 and time 𝑡, let 𝑈𝑡 (𝑛) be the uniqueness zone defined to be the maximum integer 𝑏 ≤ 81 𝑛1/3 satisfying P𝜙(𝑡) (Piv[4𝑏, 𝑛1/3 ]) ≤ (log 𝑛) −1 . The associated cost is  1/4  log log 𝑛 Δ𝑡 (𝑛) := . (log 𝑛) ∧ log Gr(𝑈𝑡 (𝑛)) Note that the cost is small if Gr(𝑈𝑡 (𝑛)) ≥ (log 𝑛) 𝐶 for a big constant 𝐶. The proof of the next lemma is implicit in [EH23b, Section 6], where, together with Hutchcroft, we used plentiful tubes to run an orange-peeling argument inspired by the one in [CMT22]. See the appendix for details. Lemma 8.3.4. For all 𝑐 > 0 there exist 𝜆(𝑑, 𝑐), 𝑛0 (𝑑, 𝑐), 𝐾 (𝑑, 𝑐) < ∞ such that the following holds for all 𝑛 ≥ 𝑛0 with 𝑛 ∈ T(𝑐, 𝜆). For all 𝑡 ∈ R, if 𝑛 is orange at time 𝑡 and 𝐾Δ𝑡 (𝑛) ≤ 1 then 𝑠(𝑛 is green) ≤ 𝑡 + 𝐾Δ𝑡 (𝑛). We now turn to the problem of finding an explicit upper bound on the cost Δ = Δ𝑡 (𝑛) of a scale 𝑛 at a time 𝑡. Define the move-left (aka decrease-scale) function 𝐿 : 𝑛 ↦→ (log 𝑛) 1/2 . The next lemma provides such an upper bound if the much smaller scale 𝐿(𝑛) happens to already be green at time 𝑡. The proof of this is implicit in [EH23b, Section 6.3]. Notice that to upper bound Δ𝑡 (𝑛) is to lower bound Gr(𝑈𝑡 (𝑛)). It is easy to check that in the setting of this lemma, 𝑈𝑡 (𝑛) ≥ 𝐿 (𝑛). The proof of the lemma establishes that when 𝐿(𝑛) is green and 𝑛 ∈ L, either Gr(𝐿 (𝑛)) is big (as a function of 𝐿(𝑛)) or we can find a better lower bound on 𝑈𝑡 (𝑛) than the trivial bound that is 𝐿(𝑛). See the appendix for details. Lemma 8.3.5. There exists 𝑛0 (𝑑) < ∞ such that the following holds for all 𝑛 ∈ L with 𝑛 ≥ 𝑛0 . For all 𝑡 ∈ R, if 𝐿(𝑛) is green at time 𝑡 then Δ𝑡 (𝑛) ≤ 1 . log log 𝑛 Lemmas 8.3.4 and 8.3.5 together provide an explicit upper bound on the time it takes for a scale 𝑛 that is orange to become green if the much smaller scale 𝐿 (𝑛) happens to already be green, at least 348 until we encounter a scale 𝑛 ∈ L\T(𝑐, 𝜆). Of course, this says nothing about how long we have to wait for for at least one orange scale to become green in the first place. We will return to this shortly, but for now, consider the following method for rapidly propagating orange once we have a big interval of green. Suppose that at some time 𝑡, some interval of the form [𝐿 (𝑛), 𝑛] is green. By Lemma 8.3.3, since 𝑅 −1 (𝑛) ∈ [𝐿 (𝑛), 𝑛], we will not have to wait long for [𝑛, 𝑅(𝑛)] to turn orange. By Lemmas 8.3.4 and 8.3.5, since 𝐿 (𝑚) ∈ [𝐿(𝑛), 𝑛] for every 𝑚 ∈ [𝑛, 𝑅(𝑛)], we will not then have to wait long for [𝑛, 𝑅(𝑛)] to turn green. By Lemma 8.3.3 again, we will not then have to wait long for [𝑅(𝑛), 𝑅 2 (𝑛)] to turn orange, and so forth. We can repeat this indefinitely until and unless we encounter a scale 𝑛 ∈ L\T(𝑐, 𝜆). In conjunction with Lemma 8.3.4, the next lemma lets us control long it takes to get this big interval of green in the first place, starting from an even bigger interval of orange. We prove this in Section 8.3. We will use a compactness argument to reduce this to an analogous statement about an arbitrary infinite unimodular transitive graph 𝐺, which is then addressed by results in either [CMT22] or [EH23b] according to whether 𝐺 has polynomial or superpolynomial growth. Lemma 8.3.6. For all 𝜀 > 0 and 𝑛1 ≥ 3, there exists 𝑛2 (𝑑, 𝜀, 𝑛1 ) < ∞ such that the following holds for all 𝑡 ∈ R with 𝑡 ≤ 𝜀1 . If [𝑛1 , 𝑛2 ] is orange at time 𝑡, then for some 𝑚 satisfying 𝐼 := [𝐿(𝑚), 𝑚] ⊆ [𝑛1 , 𝑛2 ], we have sup Δ𝑡+𝜀 (𝑛) ≤ 𝜀. 𝑛∈𝐼∩L At this point, the lemmas we have accumulated allow us to rapidly propagate orange, starting from a big interval of orange, until and unless we encounter a scale 𝑛 ∈ L\T(𝑐, 𝜆). To prove Lemma 8.3.2, we need this propagation to keep going until we encounter the scale 𝛾. The next lemma lets us ensure that we will encounter 𝛾 before we encounter L\T(𝑐, 𝜆). We will prove this in Section 8.3. This is the analogue in our setting of [EH23b, Section 5]. While the random walk arguments that make up [EH23b, Subection 5.2] work equally well in our setting, the geometric arguments in [EH23b, Subsection 5.1] will require some new ideas. Lemma 8.3.7. There exist 𝑐(𝑑) > 0 such that for all 𝜆 ≥ 1, there exists 𝑛0 (𝑑, 𝜆) < ∞ such that inf{𝑛 ∈ L\T(𝑐, 𝜆) : 𝑛 ≥ 𝑛0 } ≥ 𝛾. Let us conclude by formalising the above sketch of the fact that Lemma 8.3.2, which we know implies Proposition 8.3.1, can be reduced to the rest of the lemmas introduced in this subsection. 349 Proof of Lemma 8.3.2 given Lemmas 8.3.3 to 8.3.7. Fix 𝜀 > 0 and 𝑛1 ≥ 3. We may assume that 𝜀 < 1. Let 𝑐(𝑑) > 0 be the constant from Lemma 8.3.7. Let 𝜆(𝑑), 𝑢 0 (𝑑), 𝐾 (𝑑) be the constants “𝜆(𝑑, 𝑐), 𝑛0 (𝑑, 𝑐), 𝐾 (𝑑, 𝑐)” from Lemma 8.3.4 for this choice of 𝑐. We may assume that 𝐾 ≥ 1. Let 𝑢 1 (𝑑) be the constant “𝑛0 (𝑑)” from Lemma 8.3.3. Let 𝑢 2 (𝑑) be the constant “𝑛0 (𝑑)” from Lemma 8.3.5. Let 𝑢 3 (𝑑) be the constant “𝑛0 (𝑑, 𝜆)” from Lemma 8.3.7, with the above choice of
Í Í 𝜆. Note that 𝑖 𝛿 𝑅𝑖 (3) < ∞ and 𝑖 log log1 𝑅𝑖 (3) < ∞. Let 𝑖0 (𝑑, 𝜀) be the smallest non-negative integer such that ∞ ∞ ∑︁ ∑︁
𝜀 1 𝜀 𝛿 𝑅𝑖 (3) ≤ and ≤ , 𝑖 5 5𝐾 log log 𝑅 (3) 𝑖=𝑖 𝑖=𝑖 0 0 and set 𝑢 4 (𝑑, 𝜀) := 𝑅𝑖0 (3). Set 𝑢 5 (𝑑, 𝜀, 𝑛1 ) := max{𝑢 0 , 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 , 𝑛1 }. Let 𝑢 6 (𝑑, 𝜀, 𝑛1 ) be the
𝜀 , 𝑢 5 ” from Lemma 8.3.6. We claim that the conclusion holds with 𝑛2 (𝑑, 𝜀, 𝑛1 ) := constant “𝑛2 𝑑, 5𝐾 𝑅(𝑢 6 ). Let 𝑡 ∈ R with 𝑡 ≤ 𝜀1 , and suppose that [𝑛1 , 𝑛2 ] is orange at time 𝑡. By Lemma 8.3.6, there exists 𝑚 with 𝐼 := [𝐿(𝑚), 𝑚] ⊆ [𝑢 5 , 𝑢 6 ] such that sup𝑛∈𝐼∩L 𝐾Δ𝑡+ 5𝜀 (𝑛) ≤ 𝜀5 ≤ 1. Consider the possibility that 𝐼 ∩ (L\T(𝑐, 𝜆)) ≠ ∅. Then by Lemma 8.3.7, 𝛾 ≤ 𝑢 6 . In particular, 𝑅(𝛾) ≤ 𝑛2 , and hence 𝑅(𝛾) is already orange at time 𝑡. Since we are trivially done in that case, let us assume to the contrary that 𝐼 ∩ (L\T(𝑐, 𝜆)) = ∅. Then by Lemma 8.3.4, 𝑠 (𝐼 is green) ≤ 𝑡 + 2𝜀 𝜀 + sup 𝐾Δ𝑡+ 5𝜀 (𝑛) ≤ 𝑡 + . 5 𝑛∈𝐼∩L 5 Let 𝑘 be the largest non-negative integer such that 𝑅 𝑘 (𝑚) < 𝛾. (We may assume that such an integer exists, otherwise 𝛾 ≤ 𝑢 6 and hence we are trivially done as above.) We claim that for all 𝑖 ∈ {0, . . . , 𝑘 − 1},    
𝐾 𝑖+1 𝑖 𝑖−1 𝑠 [𝐿 (𝑚), 𝑅 (𝑚)] is green ≤ 𝑠 [𝐿 (𝑚), 𝑅 (𝑚)] is green + 𝛿 𝑅 (𝑚) + . log log 𝑅𝑖 (𝑚) (8.3.3) Indeed, fix an arbitrary index 𝑖 ∈ {0, . . . , 𝑘 − 1} and an arbitrary time 𝑠 ∈ R at which [𝐿 (𝑚), 𝑅𝑖 (𝑚)]  
is green. By Lemma 8.3.3, the interval 𝑅𝑖 (𝑚), 𝑅𝑖+1 (𝑚) is orange at time 𝑠 + 𝛿 𝑅𝑖−1 (𝑚) . By  
  Lemma 8.3.5, since 𝐿 𝑅𝑖 (𝑚), 𝑅𝑖+1 (𝑚) ⊆ 𝐿(𝑚), 𝑅𝑖 (𝑚) , sup 𝑛∈ [ 𝑅 𝑖 (𝑚),𝑅 𝑖+1 (𝑚) ] ∩L Δ𝑠+𝛿 ( 𝑅𝑖−1 (𝑚) ) (𝑛) ≤ 1 . log log 𝑅𝑖 (𝑚)   By Lemma 8.3.7, we know that 𝑅𝑖 (𝑚), 𝑅𝑖+1 (𝑚) ∩ (L\T(𝑐, 𝜆)) = ∅, and since 𝑅𝑖 (𝑚) ≥ 𝑢 4 , we know that log log𝐾𝑅𝑖 (𝑚) ≤ 𝜀5 ≤ 1. So by Lemma 8.3.4,      𝐾 𝑖 𝑖+1 𝑖−1 , 𝑠 𝑅 (𝑚), 𝑅 (𝑚) is green ≤ 𝑠 + 𝛿 𝑅 (𝑚) + log log 𝑅𝑖 (𝑚) 350 establishing eq. (8.3.3). By repeated applying eq. (8.3.3), it follows by induction that 𝑠   ∞   𝐾 2𝜀 ∑︁  𝑖−1 4𝜀 𝛿 𝑅 (𝑚) + + , 𝐿(𝑚), 𝑅 (𝑚) is green ≤ 𝑡 + ≤ 𝑡 + 𝑖 (𝑚) 5 5 log log 𝑅 𝑖=0 𝑘   where in the second inequality we used the fact that 𝑅 −1 (𝑚) ≥ 𝑢 4 . By maximality of 𝑘, we know   that 𝑅 −1 (𝛾) ∈ 𝐿 (𝑚), 𝑅 𝑘 (𝑚) . So by Lemma 8.3.3,      𝑠 (𝑅(𝛾) is orange) − 𝑠 𝐿 (𝑚), 𝑅 𝑘 (𝑚) is green ≤ 𝑠 ([𝛾, 𝑅(𝛾)] is orange) − 𝑠 𝑅 −1 (𝛾) is green   𝜀 −1 ≤ 𝛿 𝑅 (𝛾) ≤ 𝛿 (𝑢 4 ) ≤ . 5 Therefore, as required, 𝑠 (𝑅(𝛾) is orange) ≤ 𝑡 + 4𝜀 𝜀 + = 𝑡 + 𝜀. 5 5 □ Base case of the induction In this subsection we prove Lemma 8.3.6. By a compactness argument, we will reduce this to the following simpler statement about individual infinite transitive graphs. Although we defined Δ𝑡 (𝑛) and L in the context of finite transitive graphs, let us use the exact same definitions for infinite transitive graphs. Lemma 8.3.8. Let 𝐺 be a unimodular infinite transitive graph. For every 𝑡 ∈ R with 𝜙(𝑡) > 𝑝 𝑐 (𝐺), lim sup Δ𝑠 (𝑛) 1L (𝑛) = 0. 𝑛→∞ 𝑠≥𝑡 Our first goal is to prove this lemma. As mentioned earlier, to show that Δ𝑡 (𝑛) is small, we need to show that Gr(𝑈𝑡 (𝑛)) ≥ (log 𝑛) 𝐶 for a large constant 𝐶. The following lemma22 from [EH23b, Corollary 2.4] tells us in particular that if Gr(𝑛) is not too big with respect to 𝑛, then the uniqueness zone for 𝑛 is always at least of order log 𝑛. The proof of this result was essentially already contained in [CMT22], which in turn was inspired by [Cer15]. Lemma 8.3.9. Let 𝐺 be a unimodular transitive graph of degree 𝑑. Fix 𝜂 ∈ (0, 1) and 𝜀 ∈ (0, 1/2). There exists 𝑐(𝑑, 𝜂, 𝜀) > 0 such that for every 𝑛 ≥ 1 and 𝑝 ∈ [𝜂, 1],  log Gr(𝑛) P 𝑝 (Piv[𝑐 log 𝑛, 𝑛]) ≤ 𝑐𝑛  12 −𝜀 . 22 In the version in [EH23b], we also required that 𝑛 is larger than some constant depending on 𝑑, 𝜂, 𝜀, but that is redundant. 351 In an infinite transitive graph, if 𝑈𝑡 (𝑛) ≳ log 𝑛 then trivially Gr(𝑈𝑡 (𝑛)) ≳ log 𝑛, and hence Δ𝑡 (𝑛) is bounded above by a (possibly large) constant. Our goal is to improve this argument so that this constant can be made arbitrarily small. We will do this by improving either the bound on 𝑈𝑡 (𝑛) or the bound on Gr(𝑈𝑡 (𝑛)) given 𝑈𝑡 (𝑛). When 𝐺 has superpolynomial growth, this is easy: we can use the trivial bound on 𝑈𝑡 (𝑛), but then use the fact that Gr(𝑈𝑡 (𝑛)) ≳ (𝑈𝑡 (𝑛)) 𝐶 for any particular constant 𝐶 23 . When 𝐺 has polynomial growth, we will apply the following more delicate result from [CMT22, Proposition 6.1] to improve our bound on 𝑈𝑡 (𝑛). (For background on transitive graphs of polynomial and superpolynomial growth, see [TT21a].) Lemma 8.3.10. Let 𝐺 be an infinite transitive graph of polynomial growth. For every 𝑝 > 𝑝 𝑐 (𝐺) there exist 𝜒(𝐺, 𝑝) ∈ (0, 1) and 𝐶 (𝐺, 𝑝) < ∞ such that for every 𝑛 ≥ 1 and 𝑞 ∈ [ 𝑝, 1],    𝜒 P𝑞 Piv 𝑒 (log 𝑛) , 𝑛 ≤ 𝐶𝑛−1/4 . Proof of Lemma 8.3.8. Suppose that 𝑡 ∈ R satisfies 𝜙(𝑡) > 𝑝 𝑐 (𝐺). Note that 𝜙(𝑡) > 1/𝑑 because 𝑝 𝑐 (𝐺) ≥ 1/(𝑑 − 1) (as this holds for every infinite transitive graph). Let 𝑐 1 (𝑑, 1/𝑑, 1/6) > 0 be the constant from Lemma 8.3.9. Then for every sufficiently large 𝑛 ∈ L,  h   i   log Gr(𝑛1/3 )  12 − 61  (log 𝑛) 100  13 1 , ≤ ≤ sup P𝜙(𝑠) Piv 𝑐 1 log 𝑛1/3 , 𝑛1/3 ≤ 1/3 1/3 log 𝑛 𝑐1𝑛 𝑐1𝑛 𝑠≥𝑡 𝑐1 and hence inf 𝑠≥𝑡 𝑈𝑠 (𝑛) ≥ ⌊ 14 𝑐 1 log(𝑛1/3 )⌋ = 13 log 𝑛. In particular, " # 1/4 log log 𝑛
lim sup sup Δ𝑠 (𝑛) 1L (𝑛) ≤ lim sup . 𝑐1 (log 𝑛) ∧ log Gr 13 log 𝑛 𝑛→∞ 𝑠≥𝑡 𝑛→∞ 𝑐 log Gr ( 131 log 𝑛) If 𝐺 has superpolynomial growth, then → ∞ and hence sup𝑠≥𝑡 Δ𝑠 (𝑛) 1L (𝑛) → 0 as log log 𝑛 𝑛 → ∞. So we may assume to the contrary that 𝐺 has polynomial growth. Let 𝜒(𝐺, 𝜙(𝑡)) and 𝐶 (𝐺, 𝜙(𝑡)) be the constants from Lemma 8.3.10. Then for every sufficiently large 𝑛 ≥ 1,  h i 𝜒 1/3 1 sup P𝜙(𝑠) Piv 𝑒 ( log ( 𝑛 )) , 𝑛1/3 ≤ 𝐶 (𝑛1/3 ) −1/4 ≤ , log 𝑛 𝑠≥𝑡 and hence inf 𝑠≥𝑡 𝑈𝑠 (𝑛) ≥ Gr(𝑈𝑠 (𝑛)) ≥ 𝑈𝑠 (𝑛), 1 ( log ( 𝑛1/3 )) 4𝑒 𝜒 ≥ 𝑒 (log 𝑛) 𝜒/2 . In particular, using the trivial bound   1/4   log log 𝑛     lim sup sup Δ𝑠 (𝑛) 1L (𝑛) ≤ lim sup   = 0. 𝜒/2   𝑛→∞ 𝑛→∞ 𝑠≥𝑡  (log 𝑛) ∧ log 𝑒 (log 𝑛)    23 This was the idea in [EH23b, Section 3], where it sufficed to consider sequences converging to graphs of superpolynomial growth. 352 □ Next we will use a compactness argument to deduce Lemma 8.3.6 from Lemma 8.3.8. Given 𝑑 ∈ N, let U𝑑 be the space of all unimodular transitive graphs with degree 𝑑 endowed with the local topology. Recall that every finite transitive graph is unimodular. By [Hut20a, Corollary 5.5], U𝑑 is a closed subset of the space T𝑑 of all transitive graphs with degree 𝑑 endowed with the local topology. In particular (recalling that the local topology is metrisable), since T𝑑 is compact, so is U𝑑 . Proof of Lemma 8.3.6. Suppose for contradiction that the statement is false. Then we can find 𝜀 > 0 and 𝑛1 ≥ 3 such that for all 𝑁 ∈ N there exists 𝑡 𝑁 ≤ 𝜀1 and a finite transitive graph 𝐺 𝑁 with degree 𝑑 such that in 𝐺 𝑁 , the interval [𝑛1 , 𝑁] is orange at time 𝑡 𝑁 , but for every 𝑚 with [𝐿 (𝑚), 𝑚] ⊆ [𝑛1 , 𝑁], there exists 𝑛 ∈ [𝐿 (𝑚), 𝑚] ∩ L(𝐺 𝑁 ) with Δ𝑡𝐺𝑁𝑁+𝜀 (𝑛) > 𝜀. (We write Δ𝐺 , 𝑈 𝐺 , L(𝐺) to denote Δ, 𝑈, L defined with respect to a specific graph 𝐺.) By compactness, there exists an infinite subset M ⊆ N and a unimodular transitive graph 𝐺 such that 𝐺 𝑁 → 𝐺 as 𝑁 → ∞ with 𝑁 ∈ M. First consider the case that 𝐺 is finite. Then trivially, there exists 𝑛0 (𝐺) < ∞ such that for all 𝑛 ≥ 𝑛0 and for all 𝑠 ∈ R, we have 𝑈𝑠𝐺 (𝑛) = ⌊ 18 𝑛1/3 ⌋. In particular, lim𝑛→∞ sup𝑠∈R Δ𝐺 𝑠 (𝑛) = 0. So there exists 𝑚 with 𝐿 (𝑚) ≥ 𝑛1 such that sup sup Δ𝐺 𝑠 (𝑛) ≤ 𝜀. 𝑛∈[𝐿(𝑚),𝑚] 𝑠∈R 𝐺 𝑁 Pick 𝑁 ∈ M sufficiently large that 𝑁 ≥ 𝑚 and 𝐵𝐺 𝑚  𝐵𝑚 (or even that 𝐺  𝐺 𝑁 ). Then we have a contradiction because there exists 𝑛 ∈ [𝐿(𝑚), 𝑚] such that Δ𝑡𝐺𝑁 +𝜀 (𝑛) = Δ𝑡𝐺𝑁𝑁+𝜀 (𝑛) > 𝜀. So we may assume that 𝐺 is infinite. We claim that lim inf 𝜙(𝑡 𝑁 ) ≥ 𝑝 𝑐 (𝐺). 𝑁→∞ 𝑁 ∈M (8.3.4) Indeed, suppose that 𝑞 ∈ (0, 𝑝 𝑐 (𝐺)). By the sharpness of the phase transition for percolation on −𝑛/𝐶 for all 𝑛 ≥ 1. infinite transitive graphs, there exists 𝐶 (𝑞, 𝐺) < ∞ such that P𝐺 𝑞 (𝑜 ↔ 𝑆 𝑛 ) ≤ 𝐶𝑒 Pick 𝑚 ≥ 𝑛1 such that 𝐶𝑒 −𝑚/𝐶 < 𝛿(𝑚). Pick 𝑁0 ≥ 𝑚 such that for all 𝑁 ≥ 𝑁0 with 𝑁 ∈ M, we 𝐺 𝑁 have 𝐵𝐺 𝑚  𝐵𝑚 . Then for all 𝑁 ≥ 𝑁0 with 𝑁 ∈ N, 𝐺𝑁 𝐺 𝑁 min P𝐺 𝑞 (𝑜 ↔ 𝑢) ≤ P𝑞 (𝑜 ↔ 𝑆 𝑚 ) = P𝑞 (𝑜 ↔ 𝑆 𝑚 ) < 𝛿(𝑚), 𝐺 𝑢∈𝐵𝑚 𝑁 353 so 𝑚 is not orange for 𝐺 𝑁 at time 𝜙−1 (𝑞), and hence 𝑞 ≤ 𝜙(𝑡 𝑁 ). Since 𝑞 was arbitrary, this establishes eq. (8.3.4). Now by hypothesis, 𝑡 𝑁 ≤ 𝜀1 for every 𝑁 ≥ 1. So by eq. (8.3.4), we know that 𝑝 𝑐 (𝐺) ≤ 𝜙(1/𝜀) < 1. (We also know that 𝑝 𝑐 (𝐺) > 0 since this holds for every infinite transitive graphs.) By passing to a further subsequence, we may assume that for all 𝑁 ∈ M, 𝑡 𝑁 + 𝜀 ≥ 𝜙−1 ( 𝑝 𝑐 (𝐺)) + 𝜀 =: 𝑡. 2 Note that 𝜙(𝑡) > 𝑝 𝑐 (𝐺). So by Lemma 8.3.8, lim sup Δ𝐺 𝑠 (𝑛) 1L(𝐺) (𝑛) = 0. 𝑛→∞ 𝑠≥𝑡 Pick 𝑚 with 𝐿 (𝑚) ≥ 𝑛1 such that sup sup Δ𝐺 𝑠 (𝑛) 1L(𝐺) (𝑛) ≤ 𝜀. (8.3.5) 𝑛∈[𝐿(𝑚),𝑚] 𝑠≥𝑡 𝐺 𝑁 Pick 𝑁 ∈ M such that 𝑁 ≥ 𝑚 and 𝐵𝐺 𝑚  𝐵𝑚 . Then [𝐿(𝑚), 𝑚] ⊆ [𝑛1 , 𝑁], and the same inequality as eq. (8.3.5) holds with 𝐺 𝑁 in place of 𝐺. This contradicts the existence of 𝑛 ∈ [𝐿(𝑚), 𝑚] ∩L(𝐺 𝑁 ) satisfying Δ𝑡𝐺𝑁𝑁+𝜀 (𝑛) > 𝜀 because 𝑡 𝑁 + 𝜀 ≥ 𝑡. □ The obstacles are circles In this subsection we prove Lemma 8.3.7. Our argument is a finitary refinement of the argument in [EH23b, Section 5]. Our first step is to isolate the part of that previous argument that needs to be improved. For this we need to introduce the definition of plentiful tubes. Plentiful tubes Let 𝐺 be a transitive graph and fix a scale 𝑛 ≥ 1. We call the 𝑟-neighbourhood Ð 𝐵𝑟 (𝛾) := 𝑖 𝐵𝑟 (𝛾𝑖 ) of a path 𝛾 ∈ Γ a tube. Given constants 𝑘, 𝑟, 𝑙 ≥ 1, we say that 𝐺 has (𝑘, 𝑟, 𝑙)plentiful tubes at scale 𝑛 if the following always holds. Let 𝐴 and 𝐵 be sets of vertices such that ( 𝐴, 𝐵) = (𝑆𝑛 , 𝑆4𝑛 ) or such that 𝐴 and 𝐵 both contain paths from 𝑆𝑛 to 𝑆3𝑛 . Then there is a set Γ of paths from 𝐴 to 𝐵 such that |Γ| ≥ 𝑘, each path has length at most 𝑙, and 𝐵𝑟 (𝛾1 ) ∩ 𝐵𝑟 (𝛾2 ) = ∅ for all pairs of distinct paths 𝛾1 , 𝛾2 ∈ Γ. Note that the property of having (𝑘, 𝑟, 𝑙)-plentiful tubes gets stronger as we increase 𝑘 (the number of tubes), increase 𝑟 (the thickness of tubes), or decrease 𝑙 (the lengths of tubes). We will be concerned mainly with the following two-parameter subset of this three-parameter family of properties. Given constants 𝑐, 𝜆 > 0, we say that 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes at scale 𝑛 if 𝐺 has (𝑘, 𝑟, 𝑙)-plentiful tubes at scale 𝑛 with   𝑐𝜆 −𝜆/𝑐 𝜆/𝑐 (𝑘, 𝑟, 𝑙) := [log 𝑛] , 𝑛[log 𝑛] , 𝑛[log 𝑛] . 354 We think of 𝑐 as representing a fixed exchange rate for the tradeoff between asking for more tubes that are long and thin vs fewer tubes that are short and thick, which we can realise by varying 𝜆. Finally, recall from Section 8.3 that T(𝑐, 𝜆) is defined to be the set of all scales 𝑛 ≥ 3 such that 100 Gr(𝑛) ≤ 𝑒 (log 𝑛) (i.e. 𝑛 ∈ L) and there exists 𝑚 satisfying [𝑚, 𝑚 1+𝑐 ] ⊆ [𝑛1/3 , 𝑛1/(1+𝑐) ] such that 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes at every scale in [𝑚, 𝑚 1+𝑐 ]. Now suppose in the context of proving Lemma 8.3.7 that we have a large scale 𝑛 ∈ L with 𝑛 < 𝛾, and we want to build the required plentiful tubes to establish that 𝑛 ∈ T(𝑐, 𝜆). We split our argument into two cases, slow growth and fast growth, according to the rate of change of Gr near 𝑛, as measured by whether Gr(3𝑚)/Gr(𝑚) for 𝑚 ≈ 𝑛 exceeds some particular constant24 . The next lemma says that if 𝐺 has fast growth throughout a sufficiently large interval around scale 𝑛, then for some fixed exchange rate 𝑐, we have (𝑐, 𝜆)-polylog plentiful tubes for every choice of 𝜆 whenever 𝑛 is sufficiently large. This is [EH23b, Proposition 5.4], which was originally stated for infinite unimodular transitive graphs, but as we will justify in the appendix, exactly the same proof also works for finite transitive graphs. Lemma 8.3.11. Let 𝐺 be a unimodular transitive graph of degree 𝑑. Suppose that Gr(𝑚) ≤ 𝑒 (log 𝑚) 𝐷 and Gr(3𝑚) ≥ 35 Gr(𝑚) for every 𝑚 ∈ [𝑛1−𝜀 , 𝑛1+𝜀 ], where 𝜀, 𝐷, 𝑛 > 0. Then there is a constant 𝑐(𝑑, 𝐷, 𝜀) > 0 with the following property. For every 𝜆 ≥ 1, there exists 𝑛0 (𝑑, 𝐷, 𝜀, 𝜆) < ∞ such that if 𝑛 ≥ 𝑛0 then 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes at scale 𝑛. The next lemma says that if 𝐺 has slow growth at some scale 𝑛, then outside of a bounded number of small problematic intervals, 𝐺 has plentiful tubes with good constants (𝑘, 𝑟, 𝑙) unless 𝐺 is one-dimensional. This is equivalent to [EH23b, Proposition 5.3]. Lemma 8.3.12. Let 𝐺 be an infinite transitive graph of degree 𝑑. Suppose that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛), where 𝑛, 𝜅 > 0. There exists 𝐶 (𝑑, 𝜅) < ∞ such that the following holds if 𝑛 ≥ 𝐶: Ð There is a set 𝐴 ⊆ [1, ∞) with | 𝐴| ≤ 𝐶 such that for every 𝑘 ≥ 1 and every 𝑚 ∈ [𝐶 𝑘𝑛, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎], if 𝐺 does not have (𝐶 −1 𝑘, 𝐶 −1 𝑘 −1 𝑚, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes at scale 𝑚, then 𝐺 is one-dimensional. 24 In [EH23b] we considered ratios of triplings Gr(3𝑛)/Gr(𝑛) rather than of doublings Gr(2𝑛)/Gr(𝑛) because only the former was known at the time to be sufficient to invoke the structure theory of transitive graphs of polynomial growth. Tointon and Tessera have since proved that small doublings imply small triplings [TT23], so it is now possible to work with doublings Gr(2𝑛)/Gr(𝑛) instead, which is slightly more natural. However, since this does not significantly simplify our arguments, we have chosen to stay with triplings to avoid some repetition of work from [EH23b]. 355 We need to improve this lemma in two ways. First, we need the conclusion to be that “𝐺 looks one-dimensional from around scale 𝑚”, rather than just “𝐺 is one-dimensional”. Second, we need to allow 𝐺 to be finite. Here is the modified version of Lemma 8.3.12 that we will prove.25 Lemma 8.3.13. Let 𝐺 be a finite transitive graph of degree 𝑑. Suppose that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛), where 𝑛, 𝜅 > 0. There exists 𝐶 (𝑑, 𝜅) < ∞ such that the following holds if 𝑛 ≥ 𝐶: Ð There is a set 𝐴 ⊆ [1, ∞) with | 𝐴| ≤ 𝐶 such that for every 𝑘 ≥ 1 and every 𝑚 ∈ [𝐶 𝑘𝑛, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎], if 𝐺 does not have (𝐶 −1 𝑘, 𝐶 −1 𝑘 −1 𝑚, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes at scale 𝑚, then  distGH  𝜋 𝐶𝑚 𝐺, 𝑆 1 ≤ . diam 𝐺 diam 𝐺 In the next two subsections we will prove Lemma 8.3.13. Before that, let us quickly check that Lemmas 8.3.11 and 8.3.13 together do imply Lemma 8.3.7. This is essentially the same as the proof of [EH23b, Proposition 5.2] given [EH23b, Propositions 5.3 and 5.4]. Proof of Lemma 8.3.7 given Lemmas 8.3.11 and 8.3.13. Let 𝑐(𝑑) > 0 be a small positive constant to be determined. Let 𝜆 ≥ 1 be arbitrary. Suppose that 𝑛 ≥ 3 satisfies 𝑛 ∈ L\T(𝑐, 𝜆). We will freely (and implicitly) assume that 𝑛 is large with respect to 𝑑 and 𝜆. Our goal is to show that if 𝑐 is sufficiently small, it then necessarily follows that 𝛾 ≤ 𝑛. First consider the possibility that Gr(3𝑚) ≥ 35 Gr(𝑚) for all 𝑚 ∈ [𝑛1/3 , 𝑛1/2 ]. Let 𝜂 := 1/100, and let 𝑐 1 (𝑑, 101, 𝜂) > 0 be given by Lemma 8.3.11. Note that for all 𝑚 ∈ [𝑛1/3+𝜂 , (𝑛1/3+𝜂 ) 1+𝜂 ], we 100 101 have [𝑚 1−𝜂 , 𝑚 1+𝜂 ] ⊆ [𝑛1/3 , 𝑛1/2 ] and Gr(𝑚) ≤ Gr(𝑛) ≤ 𝑒 [log 𝑛] ≤ 𝑒 [log 𝑚] . So by construction of 𝑐 1 , we know that 𝐺 has (𝑐 1 , 𝜆)-polylog tubes at every scale 𝑚 ∈ [𝑛1/3+𝜂 , (𝑛1/3+𝜂 ) 1+𝜂 ]. In particular, if we pick 𝑐(𝑑) ≤ 𝑐 1 ∧ 𝜂, then 𝑛 ∈ T(𝑐, 𝜆) - a contradiction. So we may assume that there exists 𝑚 ∈ [𝑛1/3 , 𝑛1/2 ] such that Gr(3𝑚) ≤ 35 Gr(𝑚). Let 𝐶 (𝑑, 5) < ∞ be as given by Lemma 8.3.13. Without loss of generality, assume that 𝐶 is an integer and 𝐶 ≥ 2. Let 𝐴 ⊆ [1, ∞) with | 𝐴| ≤ 𝐶 be the set guaranteed to exist for our particular small-tripling scale 1 log 32 , and 𝑚, and apply the conclusion of Lemma 8.3.13 with 𝑘 := (log 𝑛) 𝜆 . Define 𝜀(𝑑) := 3𝐶 consider the sequence (𝑢𝑖 : 1 ≤ 1 ≤ 3𝐶) defined by   (1+𝜀) 𝑖 𝑢𝑖 := 𝑛1/2 . 25 Although we have chosen to write everything for finite graphs, our proof also yields the analogous finitary refinement of Lemma 8.3.12 when 𝐺 is infinite. 356 Note that thanks to our choice of 𝜀,  𝑢 3𝐶 = 𝑛 1/2  (1+𝜀) 3𝐶  ≤ 𝑛 1/2  (𝑒 𝜀 ) 3𝐶 = 𝑛3/4 . As we are assuming that 𝑛 is large with respect to 𝑑 and 𝜆, we also have that 𝐶 𝑘𝑚 ≤ 𝑢 1 , and for all 𝑎 ∈ 𝐴, the interval [𝑎, 2𝑘𝑎] contains at most one of the 𝑢𝑖 ’s. So by the pigeonhole principle, there Ð exists 𝑖 such that [𝑢𝑖 , 𝑢𝑖+1 ] ⊆ [𝐶 𝑘𝑚, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎]. By construction of 𝐴, we know that either (1) for every scale 𝑙 ∈ [𝑢𝑖 , 𝑢𝑖+1 ], the graph 𝐺 has (𝐶 −1 𝑘, 𝐶 −1 𝑘 −1 𝑚, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes at scale 1 , 𝜆)-polylog plentiful tubes at scale 𝑙, or (2) there exists 𝑙 ∈ [𝑢𝑖 , 𝑢𝑖+1 ] such 𝑙, and in particular ( 2𝐶 1 that 𝛾 ≤ 𝐶𝑙 and hence 𝛾 ≤ 𝑛. If (1) holds, then by picking 𝑐(𝑑) ≤ 𝜀 ∧ 2𝐶 , we can guarantee that 𝑛 ∈ T(𝑐, 𝜆) - a contradiction. So (2) holds, i.e. 𝛾 ≤ 𝑛 as required. □ Cutsets and cycles In this subsection we reduce Lemma 8.3.13 to Lemma 8.3.16, which is a less technical statement about cutsets and cycles. We will prove Lemma 8.3.16 in the next section. 𝐵 Cutsets Let 𝐴, 𝐵, 𝐶 be sets of vertices in a graph 𝐺. We write 𝐴 ← → 𝐶 to mean that there exists 𝐵 𝑛 a finite path (𝛾 𝑘 ) 𝑘=0 such that 𝛾0 ∈ 𝐴; 𝛾1 , . . . , 𝛾𝑛−1 ∈ 𝐵; and 𝛾𝑛 ∈ 𝐶, and we write 𝐴 ← → ∞ to mean that there exists an infinite self-avoiding path (𝛾 𝑘 ) ∞ 𝑘=0 such that 𝛾0 ∈ 𝐴 and 𝛾1 , 𝛾2 , . . . ∈ 𝐵. 𝐵 We write ̸← → to denote the negations of these properties. Now we say that 𝐵 is an ( 𝐴, 𝐶)-cutset to 𝐵𝐶 mean that 𝐴 ̸←→ 𝐶, and we say that 𝐵 is a minimal ( 𝐴, 𝐶)-cutset if no proper subset of 𝐵 is also an ( 𝐴, 𝐶)-cutset. We extend all of these definitions in the obvious way to allow 𝐴 or 𝐶 to be vertices rather than set of vertices. Now suppose that 𝐺 is an infinite transitive graph. Of course the spheres 𝑆𝑛 for 𝑛 ∈ N are all (𝑜, ∞)-cutsets, but interestingly, they are not always minimal (𝑜, ∞)-cutsets because some transitive graphs contain dead-ends, i.e. a vertex that is at least as far from 𝑜 as all of its neighbours. The exposed sphere 𝑆𝑛∞ is defined to be the unique minimal (𝑜, ∞)-cutset contained in the usual sphere 𝑆𝑛 , which is given concretely by 𝐵𝑛𝑐 𝑆𝑛∞ = {𝑢 ∈ 𝑆𝑛 : 𝑢 ← → ∞}. Thanks to the following result of Funar, Giannoudovardi, and Otera [FGO15, Proposition 5], exposed spheres also admit the following finitary characterisation: 𝑆𝑛∞ is the unique minimal (𝑜, 𝑆2𝑛+1 )-cutset contained in 𝑆𝑛 . We have included the short and elegant proof from their paper for the reader to appreciate that it does not adapt well to finite graphs. Specifically, it does not yield Lemma 8.3.19, which is what we will need. We like to call this the inflexible geodesic argument. 357 𝑐 satisfies Lemma 8.3.14. Let 𝐺 be an infinite transitive graph. Let 𝑟 ∈ N. Then every vertex 𝑢 ∈ 𝐵2𝑟 𝐵𝑟𝑐 → ∞. 𝑢← Proof of Lemma 8.3.14: The inflexible geodesic argument. This proof uses the well-known fact that every infinite transitive graph contains a bi-infinite geodesic 𝛾 = (𝛾𝑛 )𝑛∈Z . Here is a sketch of 2𝑁 for every 𝑁 ∈ N. By transitivity, how to prove this: There exist geodesic segments 𝛾 𝑁 = (𝛾𝑛𝑁 )𝑛=0 we can pick these with 𝛾 𝑁𝑁 = 𝑜 for every 𝑁. Then 𝛾 is any local limit of these geodesic segments rooted at 𝑜, which exists by compactness. 𝑐 . Let 𝛾 = (𝛾 ) Now fix 𝑢 ∈ 𝐵2𝑟 𝑛 𝑛∈Z be a bi-infinite geodesic with 𝛾0 = 𝑢. Suppose for contradiction that there exist 𝑠, 𝑡 ∈ N such that 𝛾−𝑠 , 𝛾𝑡 ∈ 𝐵𝑟 . Note that dist(𝛾−𝑠 , 𝛾𝑡 ) ≤ diam 𝐵𝑟 ≤ 2𝑟. Since dist(𝑜, 𝑢) > 2𝑟, we have 𝛾−𝑠 , 𝛾𝑡 ∉ 𝐵𝑟 (𝑢). Since 𝛾 is a path, it follows that 𝑠, 𝑡 > 𝑟, and in particular, 𝑠 + 𝑡 > 2𝑟. On the other hand, since 𝛾 is a geodesic, 𝑠 + 𝑡 = dist(𝛾−𝑠 , 𝛾𝑡 ). Therefore 2𝑟 < 𝑠 + 𝑡 ≤ 2𝑟, a contradiction. So either {𝛾𝑛 : 𝑛 ≥ 0} or {𝛾𝑛 : 𝑛 ≤ 0} is disjoint from 𝐵𝑟 and therefore forms a 𝐵𝑟𝑐 path witnessing that 𝑢 ← → ∞. □ The usual definition of the exposed sphere is clearly inappropriate when working with finite transitive graphs. We propose that the exposed sphere in a finite transitive graphs should instead be defined according to this alternative finitary characterisation. Since Lemma 8.3.14 only applies to infinite transitive graphs, there is no reason for now that the reader should believe us that this is a good definition. We will fix this later by proving a finite graph analogue of Lemma 8.3.14, namely Lemma 8.3.19. As with (usual) spheres and balls, we extend the definition of exposed spheres to non-integer 𝑛 by setting 𝑆𝑛∞ := 𝑆 ∞ . ⌊𝑛⌋ Definition 8.3.15. Let 𝐺 be a transitive graph, which may be finite or infinite. Let 𝑛 ∈ N. We define the exposed sphere 𝑆𝑛∞ to be the unique minimal (𝑜, 𝑆2𝑛+1 )-cutset contained in 𝑆𝑛 , or equivalently, 𝐵 𝑛𝑐 𝑆𝑛∞ := {𝑢 ∈ 𝑆𝑛 : 𝑢 ← → 𝑆2𝑛+1 }. Cycles Let 𝐺 be a graph. Recall that we identify spanning subgraphs of 𝐺 with functions 𝐸 → {0, 1}. Pointwise addition and scalar multiplication of these functions makes the set of all spanning subgraphs into a (Z/2Z)-vector space. Recall that a cycle is finite path that starts and ends at the same vertex and visits no other vertex more than once. We identify cycles (ignoring orientation) with spanning subgraphs and hence with elements of this (Z/2Z)-vector space. Now let 𝛿(𝐺) be the minimal 𝑛 ∈ N such that every cycle can be expressed as the linear combination of cycles having (extrinsic) diameter ≤ 𝑛, if such an 𝑛 exists, and set 𝛿(𝐺) := +∞ otherwise. It 358 is natural to ask whether cycles with diameter ≤ 𝛿(𝐺) also generate every bi-infinite geodesic 𝛾 ∞ of cycles each having diameter ≤ 𝛿(𝐺) such that in the sense that there is a sequence (𝛾𝑛 )𝑛=1 𝛾𝑛 → 𝛾 pointwise as 𝑛 → ∞. Notice that this is equivalent to 𝐺 being one-ended. Benjamini and Babson [BB99b] (see also the proof by Timar [Tim07]) proved that if 𝐺 is one-ended, then for all 𝑢 ∈ 𝑉 and 𝑣 ∈ 𝑉 ∪ {∞}, every minimal (𝑢, 𝑣)-cutset 𝐴 is 𝛿(𝐺)-connected in the sense that dist𝐺 ( 𝐴1 , 𝐴2 ) ≤ 𝛿(𝐺) for every non-trivial partition 𝐴 = 𝐴1 ⊔ 𝐴2 . We will use this in the next section. We claim that Lemma 8.3.13 can be reduced to the following statement about cutsets and cycles by applying the structure theory of groups and transitive graphs of polynomial growth. Since this step is essentially identical to the proof of [EH23b, Proposition 5.3], we have chosen to defer the details to the appendix. For the same reason, we will not give an overview of the rich theory of polynomial growth or even the definition of a virtually nilpotent group. The relevant background can be found in [EH23b; EH23d; TT21a]. Lemma 8.3.16. Let 𝑟, 𝑛 ≥ 1. Let 𝐺 be a finite transitive graph such that 𝑆𝑛∞ is not 𝑟-connected. Let 𝐻 be a (finite or infinite) transitive graph with 𝛿(𝐻) ≤ 𝑟 that does not have infinitely many ends. If 𝐻  𝐵𝐺 , then 𝐵50𝑛 50𝑛   𝜋 200𝑛 𝐺, 𝑆 1 ≤ . distGH diam 𝐺 diam 𝐺 Solving the reduced problem In this section we prove Lemma 8.3.16. Benjamini and Babson [BB99b] tell us that if in an infinite transitive graph 𝐻, the exposed sphere 𝑆𝑛∞ is not 𝛿(𝐻) connected26 , then 𝐻 must not be one-ended. If 𝐻 is not infinitely-ended either, then 𝐻 must in fact be two-ended and hence one-dimensional. This is how the argument (implicitly) went in [EH23b]. To make this more finitary, let us start by noting that the proof that 𝐻 not one-ended actually also tells us that this is witnessed by 𝑆𝑛∞ itself in the sense that 𝐻\𝑆𝑛∞ has multiple infinite components. Equivalently (by Lemma 8.3.14), the exposed sphere 𝑆𝑛∞ disconnects27 𝑆2𝑛+1 . (This alternative phrasing has the benefit that it also makes sense when 𝐻 is finite.) Indeed, this follows from the next lemma with ( 𝐴, 𝐵) := ({𝑜}, 𝑆2𝑛+1 ). This is also an instance of Benjamini-Babson, just phrased slightly differently in terms of sets of vertices, vertex cutsets, and (extrinsic) diameter rather than length of generating cycles. For completeness, 1 ⊔ 𝐴2 with dist( 𝐴1 , 𝐴2 ) > 𝛿(𝐻) 27 We say that a set of vertices 𝐴 disconnects another set of vertices 𝐵 if there exist vertices 𝑏 , 𝑏 ∈ 𝐵 such that 1 2 𝐴𝑐 𝑏 1 ̸← → 𝑏2 . 26 meaning that there is a non-trivial partition 𝛿(𝐻) = 𝐴 359 we have written Timar’s proof of Benjamini-Babson with the necessary tiny adjustments in the appendix. Lemma 8.3.17. Let 𝐺 be a graph. Let 𝐴 and 𝐵 be sets of vertices. Let Π be a minimal ( 𝐴, 𝐵)-cutset that does not disconnect 𝐴 or 𝐵. Then Π is 𝛿(𝐺)-connected. The following elementary lemma lets us conclude from this that 𝐻 must begin to look onedimensional already from scale 𝑛. We say that a path 𝛾 = (𝛾𝑡 : 𝑡 ∈ 𝐼) is 𝑛-dense if sup𝑣∈𝑉 dist𝐺 (𝑣, 𝛾) ≤ 𝑛, where dist𝐺 (𝑣, 𝛾) := dist𝐺 (𝑣, {𝛾𝑡 : 𝑡 ∈ 𝐼}). Lemma 8.3.18. Let 𝐺 be an infinite transitive graph. Let 𝑛 ≥ 1. If 𝐺 is two-ended and 𝐺\𝐵𝑛 has two infinite components, then 𝐺 contains an 𝑛-dense bi-infinite geodesic. Proof. Let 𝐴 and 𝐵 be the two infinite components of 𝐺\𝐵𝑛 . For each integer 𝑁 ≥ 𝑛 + 1, let 𝛾 𝑁 = (𝛾𝑡𝑁 : −𝑎 𝑁 ≤ 𝑡 ≤ 𝑏 𝑁 ) be a shortest path among those that start in 𝑆 𝑁 ∩ 𝐴 and end in 𝑆 𝑁 ∩ 𝐵, indexed such that 𝛾0𝑁 ∈ 𝐵𝑛 . By compactness, there exists a bi-infinite geodesic 𝛾 = (𝛾𝑡 : 𝑡 ∈ Z) and a subsequence (𝛾 𝑁 : 𝑁 ∈ M) such that for every 𝑡 ∈ Z, we have 𝛾𝑡 = 𝛾𝑡𝑁 for all sufficiently large 𝑁 ∈ M. As in the inflexible geodesic argument used to prove Lemma 8.3.14 (i.e. by the triangle 𝑐 in between two visits to 𝐵 . It follows that there exists 𝑡 inequality), a geodesic can never visit 𝐵2𝑛 𝑛 0 − + such that 𝛾 := (𝛾−𝑡 : 𝑡 ≥ 𝑡0 ) is entirely contained in 𝐴, and 𝛾 := (𝛾𝑡 : 𝑡 ≥ 𝑡0 ) is entirely contained in 𝐵. Suppose for contradiction that 𝛾 is not 𝑛-dense. Pick 𝑢 ∈ 𝑉 with dist(𝑢, 𝛾) > 𝑛. Since 𝐵𝑛 (𝑢) does not intersect 𝛾, the path 𝛾 must be entirely contained in one of the two infinite components of 𝐺\𝐵𝑛 (𝑢), say 𝐶. Since 𝐵𝑛 (𝑜) disconnects 𝛾 − from 𝛾 + , there are at least two infinite components in 𝐶\𝐵𝑛 (𝑜). So there are at least three infinite components in 𝐺\(𝐵𝑛 (𝑜) ∪ 𝐵𝑛 (𝑢)), contradicting the fact that 𝐺 is two-ended. □ What happens if instead 𝐻 is finite? Lemma 8.3.17 still tells us that if 𝑆𝑛∞ is not 𝛿(𝐻) connected, then 𝑆𝑛∞ disconnects 𝑆2𝑛+1 . When 𝐻 was infinite, this had a nice interpretation in terms of ends because we could go back to the original infinitary definition of 𝑆𝑛∞ as a minimal (𝑜, ∞)-cutset. The problem when 𝐻 is finite is that we are stuck with our artificial finitary definition of 𝑆𝑛∞ as a minimal (𝑜, 𝑆2𝑛+1 )-cutset. The next lemma justifies our definition by establishing that 𝑆𝑛∞ is 𝑐 . Thanks to this lemma, it is simply automatically a minimal (𝑜, 𝑢)-cutset for every vertex 𝑢 ∈ 𝐵2𝑛 impossible that 𝑆𝑛∞ is not 𝛿(𝐻)-connected when 𝐻 is finite. The analogous statement for one-ended infinite transitive graphs follows from Lemma 8.3.1428 . In 28 This was the motivation for [FGO15] to prove Lemma 8.3.14. 360 this sense, Lemma 8.3.19 lets us treat finite transitive graphs as if they were infinite transitive graphs that are one-ended. Note that a naive finite-graph adaptation of the inflexibe geodesic argument used to prove Lemma 8.3.14 would not yield Lemma 8.3.19. (It would just say that every vertex in 𝑆2𝑛+1 belongs to a cluster in 𝐺\𝐵𝑛 of large diameter.) Our argment also yields a new proof of Lemma 8.3.14. 𝑐 . Lemma 8.3.19. Let 𝐺 be a finite transitive graph. Let 𝑟 ∈ N. Then 𝐵𝑟 does not disconnect 𝐵2𝑟 𝑐 . (For Lemma 8.3.19 Proof of Lemma 8.3.14 and Lemma 8.3.19. Suppose that 𝐵𝑟 does disconnect 𝐵2𝑟 we assume this for sake of contradiction, whereas for Lemma 8.3.14, we may assume this otherwise 𝑐 . It suffices to prove that the conclusion is trivial.) Let 𝐶 be a component of 𝐺\𝐵𝑟 intersecting 𝐵2𝑟 𝐶 is infinite. (For Lemma 8.3.19, this establishes the required contradiction because 𝐺 is finite, whereas for Lemma 8.3.14, this is the desired conclusion.) Suppose for contradiction that 𝐶 is finite. Then we can pick a vertex 𝑢 ∈ 𝐶 maximising dist(𝑜, 𝑢). Since dist(𝑜, 𝑢) ≥ 2𝑟 + 1 and (by transitivity) 𝐵𝑟 (𝑢) disconnects 𝐵2𝑟 (𝑢) 𝑐 , there exists a vertex 𝐵𝑟 (𝑢) 𝑐 𝑣 ∈ 𝐵2𝑟 (𝑢) 𝑐 such that 𝑜 ← ̸ −−−→ 𝑣. Since 𝐵𝑟 (𝑢) ∩ 𝐵𝑟 (𝑜) = ∅ and the subgraph induced by 𝐵𝑟 (𝑜) is 𝐵𝑟 (𝑢) 𝑐 𝐵𝑟 (𝑜) 𝑐 connected, 𝐵𝑟 (𝑜) ← ̸ −−−→ 𝑣. Since 𝐺 is connected, 𝑣 ←−−−→ 𝐵𝑟 (𝑢). Since 𝐵𝑟 (𝑢) ∩ 𝐵𝑟 (𝑜) = ∅ and 𝐵𝑟 (𝑜) 𝑐 the subgraph induced by 𝐵𝑟 (𝑢) is connected, 𝑣 ←−−−→ 𝑢, i.e. 𝑣 ∈ 𝐶. However, since every path from 𝑜 to 𝑣 must visit 𝐵𝑟 (𝑢), dist(𝑜, 𝑣) ≥ dist(𝑜, 𝐵𝑟 (𝑢)) + dist(𝐵𝑟 (𝑢), 𝑣) ≥ (dist(𝑜, 𝑢) − 𝑟) + (dist(𝑢, 𝑣) − 𝑟) ≥ dist(𝑜, 𝑢) + 1, contradicting the maximality of dist(𝑜, 𝑢). □ By applying our work up to this point, under the hypothesis of Lemma 8.3.16, we can prove that the graph 𝐻 must be infinite and begin to look one-dimensional from around scale 𝑛. By the next lemma, it follows that 𝐺 looks like a circle from around scale 𝑛. Lemma 8.3.20. Let 𝐺 and 𝐻 be transitive graphs. Suppose that 𝐺 is finite whereas 𝐻 contains an 𝐺  𝐵 𝐻 then 𝑛-dense bi-infinite geodesic for some 𝑛 ≥ 1. If 𝐵50𝑛 50𝑛  distGH  200𝑛 𝜋 1 𝐺, 𝑆 ≤ . diam 𝐺 diam 𝐺 Proof. Let 𝛾 = (𝛾𝑡 )𝑡∈Z be an 𝑛-dense bi-infinite geodesic in 𝐻. Without loss of generality, assume that 𝛾0 = 𝑜 𝐻 . We will break our proof into a sequence of small claims. 361 𝐻 disconnects 𝑆 ∞,𝐻 . Claim. 𝐵2𝑛 2𝑛 𝑘 be an arbitrary path from 𝜁 = 𝛾 Proof of claim. Let 𝜁 = (𝜁𝑡 )𝑡=0 0 −2𝑛 to 𝜁 𝑘 = 𝛾2𝑛 . Since 𝛾 is 𝑛-dense, for all 𝑡 ∈ {0, . . . , 𝑘 }, there exists 𝑔𝑡 ∈ Z such that dist(𝜁𝑡 , 𝛾𝑔𝑡 ) ≤ 𝑛. We can of course require that 𝑔0 := −2𝑛 and 𝑔 𝑘 := 2𝑛. Since 𝛾 is geodesic, for all 𝑡 ∈ {0, . . . , 𝑘 − 1},
|𝑔𝑡+1 − 𝑔𝑡 | = dist 𝛾𝑔𝑡+1 , 𝛾𝑔𝑡

≤ dist 𝛾𝑔𝑡 , 𝜁𝑡 + dist (𝜁𝑡 , 𝜁𝑡+1 ) + dist 𝜁𝑡+1 , 𝛾𝑔𝑡+1 ≤ 𝑛 + 1 + 𝑛 = 2𝑛 + 1. In particular, since 𝑔0 ≤ −(𝑛 + 1) but 𝑔 𝑘 ≥ 𝑛 + 1, there must exist 𝑡 ∈ {1, . . . , 𝑘 − 1} such that 𝐻 . Since 𝜁 was arbitrary, this establishes that 𝐵 𝐻 −𝑛 ≤ 𝑔𝑡 ≤ 𝑛. Then 𝛾𝑔𝑡 ∈ 𝐵𝑛𝐻 and hence 𝜁𝑡 ∈ 𝐵2𝑛 2𝑛 disconnects 𝛾−2𝑛 from 𝛾2𝑛 . Since 𝛾 is a geodesic, 𝛾𝑠 ∈ 𝑆 𝑠𝐻 for all 𝑠 ∈ Z. So the path (𝛾𝑠 : 𝑠 ≥ 2𝑛) ∞,𝐻 ∞,𝐻 𝐻 disconnects 𝑆 ∞,𝐻 . witnesses the fact that 𝛾2𝑛 ∈ 𝑆2𝑛 . Similarly, 𝛾−2𝑛 ∈ 𝑆2𝑛 . So 𝐵2𝑛 □ 2𝑛 ∞,𝐻 𝐻 is an ( 𝐴, 𝐵)-cutset. Now suppose that there Fix a non-trivial partition 𝑆2𝑛 = 𝐴 ⊔ 𝐵 such that 𝐵2𝑛 𝐺 → 𝐵 𝐻 . Note that 𝜓 induces a bijection 𝑆 ∞,𝐺 ↔ 𝑆 ∞,𝐻 . In is a graph isomorphism 𝜓 : 𝐵50𝑛 50𝑛 2𝑛 2𝑛 ∞,𝐺 𝐺 does not disconnect 𝜓( 𝐴) particular, 𝑆2𝑛 = 𝜓( 𝐴) ⊔ 𝜓(𝐵). By definition of exposed sphere, 𝐵2𝑛 𝐺 ) 𝑐 . So by Lemma 8.3.19, 𝐵𝐺 is not a (𝜓( 𝐴), 𝜓(𝐵))-cutset. Consider a shortest or 𝜓(𝐵) from (𝐵4𝑛 2𝑛 path from 𝜓( 𝐴) to 𝜓(𝐵) that witnesses this, then connect the start and end of this path to 𝑜𝐺 by geodesics. Let 𝜆 = (𝜆 𝑘 ) 𝑘∈Z𝑙 be the resulting cycle, labelled such that 𝜆0 = 𝑜𝐺 . We will write |𝑠| 𝑙 for the distance from 𝑠 to 0 in the cycle graph Z𝑙 . The next three claims establish that 𝜆 is roughly dense and geodesic. 𝐺 ) = |𝑠| − 2𝑛 Claim. For all 𝑠 ∈ Z𝑙 , if |𝑠| 𝑙 > 2𝑛 then dist𝐺 (𝜆 𝑠 , 𝐵2𝑛 𝑙 𝐻 is an ( 𝐴, 𝐵)-cutset, the segment (𝜆 : |𝑡| > 2𝑛) Proof of claim. Fix 𝑠 ∈ Z𝑙 with |𝑠| 𝑙 > 2𝑛. Since 𝐵2𝑛 𝑡 𝑙 𝐺 𝑐 must intersect (𝐵4𝑛 ) (it must exit the ball 𝐵50𝑛 , on which 𝐺 and 𝐻 are isomorphic), but by 𝐺 . So every path from 𝛾 to 𝐵𝐺 must intersect 𝑆 ∞,𝐺 . construction, this segment does not intersect 𝐵2𝑛 𝑠 2𝑛 2𝑛 In particular, by minimality in the construction of 𝜆, 𝐺 dist𝐺 (𝛾𝑠 , 𝐵2𝑛 ) = dist𝐺 (𝛾𝑠 , 𝜓( 𝐴)) ∧ dist𝐺 (𝛾𝑠 , 𝜓(𝐵)) = |𝑠| 𝑙 − 2𝑛. Claim. For all 𝑠, 𝑡 ∈ Z𝑙 , we have |𝑠 − 𝑡| 𝑙 − 4𝑛 ≤ dist𝐺 (𝜆 𝑠 , 𝜆𝑡 ) ≤ |𝑠 − 𝑡| 𝑙 . 362 □ Proof of claim. The second inequality is trivial, and the first inequality is trivial when |𝑠| 𝑙 ∨|𝑡| 𝑙 ≤ 2𝑛. By our previous claim, if |𝑠| 𝑙 > 2𝑛 and |𝑡| ≤ 2𝑛, then 𝐺 dist𝐺 (𝜆 𝑠 , 𝜆𝑡 ) ≥ dist𝐺 (𝜆 𝑠 , 𝐵2𝑛 ) = |𝑠| 𝑙 − 2𝑛 ≥ |𝑠 − 𝑡| 𝑙 − 4𝑛. Similarly, the first inequality also holds if instead |𝑡| 𝑙 > 2𝑛 and |𝑠| ≤ 2𝑛. So let us consider 𝑠 and 𝑡 𝐺 , then by our satisfying |𝑠| 𝑙 ∧ |𝑡| 𝑙 > 2𝑛, and fix an arbitrary path 𝜂 from 𝜆 𝑠 to 𝜆𝑡 . If 𝜂 intersects 𝐵2𝑛 previous claim, 𝜂 has length at least 𝐺 𝐺 dist𝐺 (𝜆 𝑠 , 𝐵2𝑛 ) + dist𝐺 (𝜆𝑡 , 𝐵2𝑛 ) = (|𝑠| 𝑙 − 2𝑛) + (|𝑡| 𝑙 − 2𝑛) ≥ |𝑠 − 𝑡| 𝑙 − 4𝑛. 𝐺 , then by minimality in the construction of 𝜆, the length of 𝜂 is at least If 𝜂 does not intersect 𝐵2𝑛 |𝑠 − 𝑡| 𝑙 . Either way, dist𝐺 (𝜆 𝑠 , 𝜆𝑡 ) ≥ |𝑠 − 𝑡| 𝑙 − 4𝑛. □ Claim. 𝜆 is 10𝑛-dense Proof of claim. Suppose for contradiction that 𝑢 is a vertex with dist𝐺 (𝑢, 𝜆) > 10𝑛. Let 𝑣 be a vertex in 𝜆 that is closest to 𝑢. Let 𝑧 be a vertex in 𝑆10𝑛 (𝑣) that lies along a geodesic from 𝑢 to 𝐺 (on which 𝐺 and 𝐻 are isomorphic), we know that 𝜆 has 𝑣. Since 𝜆 visits 𝑜𝐺 but must exit 𝐵50𝑛 𝐺 (𝑣) (extrinsic) diameter > 50𝑛. By our previous claim, it follows that 𝜆 visits vertices 𝑥 and 𝑦 in 𝑆10𝑛 𝐺 (𝑣) such that satisfying dist𝐺 (𝑥, 𝑦) ≥ 2 · 10𝑛 − 4𝑛 ≥ 10𝑛. Now 𝑥, 𝑦, 𝑧 are three vertices in 𝑆10𝑛 𝐺  𝐵 𝐻 , three the distance between any pair is at least 10𝑛. By transitivity and the fact that 𝐵50𝑛 50𝑛 𝐻 such vertices can also be found in 𝑆10𝑛 (𝑜), say 𝑣 1 , 𝑣 2 , 𝑣 3 . Since 𝛾 is 𝑛-dense, there exist integers 𝑘 1 , 𝑘 2 , 𝑘 3 such that dist𝐻 (𝑣 𝑖 , 𝛾 𝑘 𝑖 ) ≤ 𝑛 for each 𝑖 ∈ {1, 2, 3}. Notice that since 𝛾0 = 𝑜 𝐻 and 𝛾 is geodesic, |𝑘 𝑖 | ∈ [9𝑛, 11𝑛] for all 𝑖. So by the pigeonhole principle, either [−9𝑛, −11𝑛] or [9𝑛, 11𝑛] contains 𝑘 𝑖 for at least two distinct values of 𝑖. On the other hand, for all 𝑖 ≠ 𝑗, since 𝛾 is geodesic, 𝑘 𝑖 − 𝑘 𝑗 = dist𝐻 (𝛾 𝑘 𝑖 , 𝛾 𝑘 𝑗 ) ≥ dist𝐻 (𝑣 𝑖 , 𝑣 𝑗 ) − 2𝑛 ≥ 8𝑛. So an interval of width 2𝑛 can never contain 𝑘 𝑖 for at least two distinct values of 𝑖, a contradiction. □ Thanks to the previous two claims, the map Z𝑙 → 𝐺 sending 𝑡 ↦→ 𝜆𝑡 is a (1, 10𝑛)-quasi-isometry. So (by exercise 5.10 (b) in [Pet23], for example), distGH (Z𝑙 , 𝐺) ≤ 10𝑛. By the obvious 1-dense isometric embedding of Z𝑙 into 2𝜋𝑙 𝑆 1 , we know that distGH (Z𝑙 , 2𝜋𝑙 𝑆 1 ) ≤ 1. Let 𝐷 := diam 𝐺. By the previous two claims 𝐷 − 2𝑙 ≤ 20𝑛. So by considering the identity map from 𝐺 to itself,   2 2 1 1 distGH 𝐺, 𝐺 ≤ sup dist𝐺 (𝑢, 𝑣) − dist𝐺 (𝑢, 𝑣) 𝐷 𝑙 𝑙 𝑢,𝑣∈𝑉 (𝐺) 𝐷 ≤𝐷· 1 2 40𝑛 − ≤ . 𝐷 𝑙 𝑙 363 Putting these bounds together, 𝜋 distGH 𝐺, 𝑆 𝐷 1        2𝜋 𝜋 2𝜋 2𝜋 2𝜋 1 ≤ distGH 𝐺, 𝐺 + distGH 𝐺, Z𝑙 + distGH Z𝑙 , 𝑆 𝐷 𝑙 𝑙 𝑙 𝑙 2𝜋 40𝑛 2𝜋 + · 10𝑛 + ·1 ≤𝜋· 𝑙 𝑙 𝑙 200𝑛 2 1 20𝑛/𝐷 ≤ = 100𝑛 · ≤ 100𝑛 · =5· . 𝑙 𝑙 𝐷 − 20𝑛 1 − 20𝑛/𝐷 (8.3.6) We may assume that 𝐷 ≥ 40𝑛, other the conclusion of the lemma holds trivially because distGH ( 𝐴, 𝐵) ≤ 1 for all non-empty compact metric spaces 𝐴 and 𝐵 each having diameter at 1 𝑥 most 1. In particular, 20𝑛 𝐷 ≤ 2 . Since 1−𝑥 ≤ 2𝑥 for all 𝑥 ∈ [0, 1/2], it follows from eq. (8.3.6) that
□ distGH 𝐷𝜋 𝐺, 𝑆 1 ≤ 5 · 2 · 20𝑛/𝐷 = 200𝑛/𝐷 as required. We now combine these lemmas to formalise this sketch of a proof of Lemma 8.3.16, thereby concluding our proof of Proposition 8.3.1. 𝐺  𝐵 𝐻 . Note that 𝑆 ∞,𝐺 is trivially 2𝑛-connected. So Proof of Lemma 8.3.16. Suppose that 𝐵50𝑛 𝑛 50𝑛 ∞ 𝑟 ≤ 2𝑛. In particular, in any transitive graph, the statement “𝑆𝑛 is not 𝑟-connected” is determined by the subgraph induced by 𝐵50𝑛 . So 𝑆𝑛∞,𝐻 is not 𝑟-connected either. By definition, 𝑆𝑛∞,𝐻 is a 𝐻 )-cutset. So by Lemma 8.3.17, since 𝛿(𝐻) ≤ 𝑟, the exposed sphere 𝑆 ∞,𝐻 must minimal (𝑜 𝐻 , 𝑆2𝑛+1 𝑛 𝐻 𝐻 𝐻 𝑐 disconnect 𝑆2𝑛+1 . In particular, 𝐵𝑛 disconnects (𝐵2𝑛 ) . So by Lemma 8.3.19, 𝐻 is infinite, and by Lemma 8.3.14, 𝐻\𝐵𝑛𝐻 contains at least two infinite components. Since 𝐻 has at most finitely many ends, 𝐻\𝐵𝑛𝐻 must contain exactly two infinite components and 𝐻 must be exactly two-ended. So by Lemma 8.3.18, 𝐻 contains an 𝑛-dense bi-infinite geodesic. The conclusion follows by Lemma 8.3.20. □ 8.4 Global connections → unique large cluster In this section we apply the methods of [EH21a]. It will be convenient to adopt the following notation from that paper: given a set of vertices 𝐴 in a graph 𝐺, we define its density to be 𝐴| ∥ 𝐴∥ := |𝑉|(𝐺)| . In [EH21a], together with Hutchcroft, we showed that the supercritical giant cluster for percolation on bounded-degree finite transitive graphs is always unique with high probability. More precisely, for every infinite set G of finite transitive graphs with bounded degrees, for every supercritical sequence of parameters 𝑝, and for every constant 𝜀 > 0, the density of the second largest cluster ∥𝐾2 ∥ satisfies lim P 𝑝 (∥𝐾2 ∥ ≥ 𝜀) = 0. 364 The following proposition contains a quantitative version of this statement that is useful even if we slightly weaken the hypothesis that 𝑝 is supercritical. We think of this as saying that if at some parameter 𝑝 we have a point-to-point lower bound that is only slightly worse than constant as |𝑉 | → ∞, then after passing to 𝑝 + 𝜀, we can still pretend that we are actually in the supercritical phase and still prove that the second largest cluster is typically much smaller than the largest cluster. Note that this largest cluster is not necessarily a giant cluster because we are not (a priori) really in the supercritical phase.29 Proposition 8.4.1. Let 𝐺 be a finite transitive graph with degree 𝑑. Define 𝛿 := (log |𝑉 |) −1/20 . There exists 𝐶 (𝑑) < ∞ such that if |𝑉 | ≥ 𝐶, then for all 𝑝, 𝑞 ∈ (0, 1) with 𝑞 − 𝑝 ≥ 𝛿,   2 min P 𝑝 (𝑢 ↔ 𝑣) ≥ 2𝛿 =⇒ P𝑞 ∥𝐾1 ∥ ≥ 𝛿 and ∥𝐾2 ∥ ≤ 𝛿 ≥ 1 − 𝛿4 . 𝑢,𝑣∈𝑉 In Section 8.4 we will explain why this proposition is implied by the sandcastles30 argument of [EH21a]. In fact, [EH21a] already explicitly contains a very similar quantitative statement, namely [EH21a, Theorem 1.5]. Unfortunately this statement is not quantitatively strong enough for our purposes. One could alternatively prove a version of Proposition 8.4.1 by applying the ghost-field technology developed in [EH23b, Section 4]. (See the discussion at the end of [EH23b, Section 7.1].) This approach would be less elementary and less generalisable31 but quantitatively stronger. Proof via sandcastles Let 𝐺 be a finite transitive graph. In [EH21a] we made the definition of “supercritical sequence” finitary as follows. Given a constant 𝜀 > 0, we say that a parameter 𝑝 ∈ (0, 1) is 𝜀-supercritical if P (1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀 and |𝑉 | ≥ 2𝜀 −3 , the latter being a technical condition that the reader may like to ignore. Note that a sequence of parameters is supercritical if and only if there exists a constant 𝜀 > 0 such that all but finitely many of the parameters are 𝜀-supercritical. On the other hand, in the present paper the more relevant finitary notion of supercriticality concerns point-to-point connection probabilities, i.e. min P (1−𝜀) 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜀. 𝑢,𝑣 29 This is reminiscent of [EH23b, Section 6]. There we used the hypothesis of a point-to-point lower bound on a large scale to enable us to run arguments from [CMT22], which were ostensibly about supercritical percolation, to study subcritical percolation. 30 We thank Coales for suggesting this name. 31 The ghost-field arguments ultimately rely on two-arm bounds, which are not elementary and which break down when working with graphs with rapidly diverging vertex degrees. 365 These properties are equivalent up to changing the constant 𝜀. Indeed, for every parameter 𝑝 ∈ (0, 1) and every constant 𝜀 > 0 satisfying the technical condition |𝑉 | ≥ 2𝜀 −3 , min P 𝑝 (𝑢 ↔ 𝑣) ≥ 2𝜀 =⇒ P 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀 =⇒ 5 −18 min P 𝑝 (𝑢 ↔ 𝑣) ≥ 𝑒 −10 𝜀 . (8.4.1) The first implication is an easy application of Markov’s inequality, and the second implication is [EH21a, Theorem 2.1]. A version of the second implication assuming an upper bound on the degree of 𝐺 is originally due to Schramm. Notice that the second implication quantitatively loses much more than the first. In this sense, we can think of the hypothesis “min𝑢,𝑣 P (1−𝜀) 𝑝 (𝑢 ↔ 𝑣) ≥ 𝜀” as being quantitatively much stronger than the hypothesis “P (1−𝜀) 𝑝 (∥𝐾1 ∥ ≥ 𝜀) ≥ 𝜀”. 𝑢,𝑣 𝑢,𝑣 Below is [EH21a, Theorem 1.5], which contains a finitary uniqueness statement similar to Proposi−18 tion 8.4.1. Unfortunately, the terrible 𝑒 −𝐶𝜀 dependence on 𝜀 is not good enough for our purposes. Fortunately, it turns out that in the proof of this theorem, the source of this poor dependence is a conversion from the hypothesis of a giant cluster bound (implicit in 𝑝 being 𝜀-supercritical) into a point-to-point bound, i.e. an application of the second implication in eq. (8.4.1). This saves us because in the present setting we actually start with the “stronger” hypothesis of a point-to-point bound. Theorem 8.4.2. Let 𝐺 be a finite transitive graph with degree 𝑑. There exists 𝐶 (𝑑) < ∞ such that for every 𝜀 > 0, every 𝜀-supercritical parameter 𝑝, and every 𝜆 ≥ 1,  1/2 !  −18 1 log 𝑑 ≤ . P 𝑝 ∥𝐾2 ∥ ≥ 𝜆𝑒𝐶𝜀 log |𝑉 | 𝜆 A key ingredient in the sandcastles argument of [EH21a] is the sharp density property, which measures the extent to which the events {∥𝐾1 ∥ ≥ 𝛼} for each 𝛼 ∈ (0, 1) have uniformly-in-𝛼 sharp thresholds. Let Δ : (0, 1) → (0, 1/2] be a decreasing function. For all 𝛼, 𝛿 ∈ (0, 1), let 𝑝 𝑐 (𝛼, 𝛿) ∈ (0, 1) be the parameter satisfying P 𝑝 𝑐 (𝛼,𝛿) (∥𝐾1 ∥ ≥ 𝛼) = 𝛿, which is unique by the strict monotonicity of this probability with respect to 𝑝. We say that 𝐺 has the Δ-sharp density property if for all 𝛼 ∈ (0, 1) and 𝛿 ∈ [Δ(𝛼), 1/2], 𝑝 𝑐 (𝛼, 1 − 𝛿) ≤ 𝑒𝛿 . 𝑝 𝑐 (𝛼, 𝛿) The following lemma establishes a sharp density property for graphs with bounded degrees. This is [EH21a, Proposition 3.2] and is an easy consequence of Talagrand’s well-known sharp threshold theorem [Tal94]. 366 Lemma 8.4.3. Let 𝐺 be a finite transitive graph with degree 𝑑. There exists 𝐶 (𝑑) < ∞ such that 𝐺 has the Δ-sharp density property for the function Δ : (0, 1) → (0, 1/2] given by  𝐶  1 ∧  1/2 Δ(𝛼) := 2 (log|𝑉 |)  1 2 if 𝛼 ≥  2 |𝑉 |  1/3 otherwise. The sandcastles argument combines a sharp density property with a point-to-point bound to establish the uniqueness of the largest cluster. Here is the technical output of that argument. Lemma 8.4.4. Let 𝐺 be a finite transitive graph. Let 𝜀 ∈ (0, 1) and suppose that 𝑝 ∈ (0, 1) is 𝜀-supercritical. Suppose that 𝐺 satisfies the Δ-sharp density property for some decreasing function Δ : (0, 1) → (0, 1/2]. Then for all 𝜆 ≥ 1,    200Δ(𝜀) 25 𝜀 + 2 P 𝑝 ∥𝐾2 ∥ ≥ 𝜆 ≤ , 3 𝜆 𝜀 𝜏 𝜀 𝜏 |𝑉 | where 𝜏 := min P (1−𝜀) 𝑝 (𝑢 ↔ 𝑣). 𝑢,𝑣 Proof. [EH21a, Theorem 3.3] is the same statement but where 𝜏 is instead defined to be 5 −18 𝜏 := 𝑒 −10 𝜀 , which is the function appearing in eq. (8.4.1). We claim that the proof of [EH21a, Theorem 3.3] actually also establishes Lemma 8.4.4. First note that in the statement of [EH21a, Lemma 3.5], we can require that 𝑞 ∈ ((1 − 𝜀) 𝑝, 𝑝) rather than just 𝑞 ∈ ( 𝑝 𝑐 (𝜀, 𝜀), 𝑝). Indeed, the exact same proof works, using 𝑞 𝑗 := 𝑒 𝑗Δ(𝜀) (1 − 𝜀) 𝑝 instead of 𝑞 𝑗 := 𝑒 𝑗Δ(𝜀) 𝑝 𝑐 (𝜀, 𝜀), because (1 − 𝜀) 𝑝 ≥ 𝑝 𝑐 (𝜀, 𝜀). So in the proof of [EH21a, Theorem 3.3], we may assume that the parameter called 𝑞, which is provided by [EH21a, Lemma 3.5], satisfies 𝑞 ≥ (1 − 𝜀) 𝑝. In particular, when we later apply 5 −18 [EH21a, Theorem 2.1] to lower bound min𝑢,𝑣 P𝑞 (𝑢 ↔ 𝑣) by 𝑒 −10 𝜀 , we could instead simply lower bound min𝑢,𝑣 P𝑞 (𝑢 ↔ 𝑣) by min𝑢,𝑣 P (1−𝜀) 𝑝 (𝑢 ↔ 𝑣). Running the rest of the proof of [EH21a, Theorem 3.3] exactly as written, except for the new definition “𝜏 := min𝑢,𝑣 P (1−𝜀) 𝑝 (𝑢 ↔ 𝑣)” in 5 −18 place of “𝜏 := 𝑒 −10 𝜀 ”, yields the desired conclusion. □ The uniqueness part of Proposition 8.4.1 will follow from Lemmas 8.4.3 and 8.4.4. The existence part will follow from the following well-known and (again) easy consequence of Talagrand’s sharp threshold theorem [Tal94]. 367 Lemma 8.4.5. Let 𝐺 be a finite transitive graph. Let 𝐴 be a non-trivial increasing event that is invariant under all graph automorphisms of 𝐺. Let 0 < 𝑝 1 < 𝑝 2 < 1 and set 𝛿 := 𝑝 2 − 𝑝 1 . There exists a universal constant 𝑐 > 0 such that P 𝑝1 ( 𝐴) ≤ 1 |𝑉 | 𝑐𝛿 or 𝑃 𝑝2 ( 𝐴) ≥ 1 − 1 . |𝑉 | 𝑐𝛿 Proof. For every edge 𝑒, let Orb(𝑒) denote the orbit of 𝑒 under the action of the automorphism group of 𝐺. By [EH21a, Theorem 3.10], there is a universal constant 𝑐 1 > 0 such that for all 𝑝 ∈ (0, 1), the function 𝑓 ( 𝑝) := P 𝑝 ( 𝐴) satisfies   𝑐1 ′ · 𝑓 ( 𝑝) (1 − 𝑓 ( 𝑝)) · log 2 min |Orb(𝑒)| . 𝑓 ( 𝑝) ≥ 2 𝑒∈𝐸 𝑝(1 − 𝑝) log 𝑝(1−𝑝) Since 𝐺 is (vertex-)transitive, |Orb(𝑒)| ≥ |𝑉2 | for every 𝑒 ∈ 𝐸. Also, by calculus, sup 𝑝∈(0,1) 𝑝(1 − 2 < ∞. Therefore, there is another universal constant 𝑐 > 0 such that for all 𝑝 ∈ (0, 1), 𝑝) log 𝑝(1−𝑝)  ′ 𝑓 𝑓′ log = ≥ 2𝑐 log |𝑉 | . 1− 𝑓 𝑓 (1 − 𝑓 ) The result follows by integrating this differential inequality. □ Proof of Proposition 8.4.1. Suppose that 𝑞, 𝑝 ∈ (0, 1) satisfy 𝑞 − 𝑝 ≥ 𝛿 and min𝑢,𝑣 P 𝑝 (𝑢 ↔ 𝑣) ≥ 2𝛿. We will assume throughout this proof that |𝑉 | is as large as we like with respect to 𝑑. Let us start with the existence of a large cluster. Let 𝑐 > 0 be the universal constant from Lemma 8.4.5. By the first implication in eq. (8.4.1), we know that P 𝑝 (∥𝐾1 ∥ ≥ 𝛿) ≥ 𝛿. Since |𝑉 | is large, 1 𝛿 = 𝑒 − 20 log log|𝑉 | ≥ 𝑒 −𝑐(log|𝑉 |) (log|𝑉 |) −1/20 = |𝑉 | −𝑐𝛿 . (8.4.2) So by Lemma 8.4.5 with 𝐴 := {∥𝐾1 ∥ ≥ 𝛿}, since 𝛿2 ≥ |𝑉 | −𝑐𝛿 (by a calculation like eq. (8.4.2)), 4 P𝑞 (∥𝐾1 ∥ ≥ 𝛿) ≥ 1 − 1 |𝑉 | ≥ 1− 𝑐𝛿 𝛿4 . 2 (8.4.3) We now turn to the uniqueness of the largest cluster. The parameter 𝑞 is (𝛿/2)-supercritical because |𝑉 | ≥ 2(𝛿/2) −3 and P (1−𝛿/2) ( 𝑝+𝛿) (∥𝐾1 ∥ ≥ 𝛿) ≥ P 𝑝 (∥𝐾1 ∥ ≥ 𝛿) ≥ 𝛿. By Lemma 8.4.3, since 𝛿/2 ≥ (2/|𝑉 |) 1/3 , there is a constant 𝐶1 (𝑑) < ∞ such that 𝐺 has the Δ-sharp density property for some Δ satisfying Δ(𝛿/2) ≤ 𝐶1 . (log |𝑉 |) 1/2 368 So by Lemma 8.4.4, there is a constant 𝐶2 (𝑑) < ∞ such that for every 𝜆 ≥ 1,  P𝑞  200 𝐶1     ª 𝛿/2 1/2 25 𝐶2 𝜆 |) (log|𝑉 ©  ® ≤ ∥𝐾 ∥ ∥𝐾2 ∥ ≥ 4 ≤ P . ≥ 𝜆 + ­ 𝑞 2   3 2 𝜆 𝛿 (log |𝑉 |) 1/2  (𝛿/2) · (2𝛿) (𝛿/2) · (2𝛿) · |𝑉 |  «  ¬ 𝐶2 𝜆 By picking 𝜆 such that 𝛿4 (log|𝑉 = 𝛿2 (which satisfies 𝜆 ≥ 1 when |𝑉 | is large), it follows that |) 1/2 P𝑞 (∥𝐾2 ∥ ≥ 𝛿2 ) ≤ 𝛿4 𝐶2 𝐶2 𝛿5 ≤ . = 2 2 2𝛿5 (log |𝑉 |) 1/2 The conclusion follows by combining eqs. (8.4.3) and (8.4.4) with a union bound. 8.5 (8.4.4) □ Unique large cluster → giant cluster In this section we apply the methods of [Eas23]. We will again use the notation ∥𝐾1 ∥ , ∥𝐾2 ∥ introduced in Section 8.4. Let G be an infinite set of finite transitive graphs with possibly unbounded degrees. Recall that G is said to have a percolation threshold if there is a fixed sequence 𝑝 𝑐 : G → (0, 1) such that for every sequence 𝑝 : G → (0, 1), if lim sup 𝑝/𝑝 𝑐 < 1 then lim P 𝑝 (∥𝐾1 ∥ ≥ 𝜀) = 0 for all 𝜀 > 0, and if lim inf 𝑝/𝑝 𝑐 > 1 then lim P 𝑝 (∥𝐾1 ∥ ≥ 𝜀) = 1 for some 𝜀 > 0. In [Eas23] we showed that G has a percolation threshold unless and only unless G contains a very particular family of pathological sequences of dense graphs. This might appear to be simply a matter of proving that some nice event has a sharp threshold, perhaps by a simple application of Lemma 8.4.5 in the bounded-degree case. The subtle problem is that “{𝐾1 is a giant}” is not an event. Really the challenge is to prove that multiple events of the form {∥𝐾1 ∥ ≥ 𝛼}, for different choices of 𝛼, all have sharp thresholds that in fact coincide with each other. The bulk of our proof consisted in proving that if the supercritical giant cluster for G is unique (as given by [EH21a]), then we can embed this fact into Vanneuville’s new proof of the sharpness of the phase transition for infinite transitive graphs [Van23; Van24] to deduce a kind of mean-field lower bound for the supercritical giant cluster density. This mean-field-like lower bound implies that for every 𝛿 > 0 and sequence 𝑝, if there is a giant whose density exceeds some constant 𝛼 > 0 at 𝑝, i.e. lim P 𝑝 (∥𝐾1 ∥ ≥ 𝛼) = 1, then there is a giant whose density exceeds some constant 𝑐(𝛿) > 0 at (1 + 𝛿) 𝑝, i.e. lim P (1+𝛿) 𝑝 (∥𝐾1 ∥ ≥ 𝑐(𝛿)) = 1, where, crucially, 𝑐(𝛿) is independent of 𝛼. By a diagonalisation argument, it is clear that 𝛼 can be allowed to decay slowly rather than remain constant. However, in general, the slowest allowable rate of decay can be arbitrarily slow32 . This is why there was no discussion of rates of convergence in [EH23b]. Luckily, this is not the case in our restricted setting where graphs have bounded degrees. 32 Consider sequences that approximate sequences that do not have percolation thresholds. 369 In the following subsection we simply note that the argument in [EH23b] is fully quantitative in the sense that 𝛼 can decay at any particular rate provided that we supply a sufficiently strong bound on the uniqueness of the largest (possibly non-giant) cluster. This is the content of the following proposition. 𝜀 ). There is a Proposition 8.5.1. Let 𝐺 be a finite transitive graph. Let 𝑝, 𝜀 ∈ (0, 1) and 𝛼 ∈ (0, 10 𝑐𝜀 1 universal constant 𝑐 > 0 such that the following holds whenever |𝑉 | ≥ 𝑐𝜀 . If P 𝑝−𝜀 (∥𝐾𝑜 ∥ ≥ 𝛼) ≥ 𝛼 and  𝛼 P 𝑝 ∥𝐾1 ∥ ≥ 𝛼 and ∥𝐾2 ∥ < > 1 − 𝛼2 , 2 then P 𝑝+𝜀 (∥𝐾1 ∥ ≥ 𝑐𝜀) ≥ 1 − 1 . |𝑉 | 𝑐𝜀 Proof via coupled explorations At the heart of Vanneuville’s new proof of the sharpness of the phase transition for infinite transitive graphs is a stochastic comparison lemma. This says that starting with percolation of some parameter 𝑝, decreasing from 𝑝 to 𝑝 − 𝜀 for a certain 𝜀 > 0 has more of an effect than conditioning on a certain disconnection event 𝐴, roughly in the sense that P 𝑝−𝜀 (𝜔 = · ) ≤st P 𝑝 (𝜔 = · | 𝐴), where ≤st denotes stochastic dominance with respect to the usual partial ordering {0, 1} 𝐸 . This is proved by coupling two explorations of the cluster at the origin, sampled according to each of the two laws. In [Eas23] we modified Vanneuville’s argument to prove the following lemma ([Eas23, Lemma 8]). Note that here the stochastic dominance only holds approximately, i.e. only on the complement of an event with small probability. Lemma 8.5.2. Let 𝐺 be a finite transitive graph. Let 𝑝, 𝛼 ∈ (0, 1). Define 𝜃 := E 𝑝 ∥𝐾1 ∥ ,  𝛼 , ℎ := P 𝑝 ∥𝐾1 ∥ < 𝛼 or ∥𝐾2 ∥ ≥ 2 𝛿 := 2ℎ1/2 , 1−𝜃−ℎ and assume that 𝜃 + ℎ < 1 (so that 𝛿 is well-defined and positive). Then there is an event 𝐴 with P 𝑝 ( 𝐴 | ∥𝐾𝑜 ∥ < 𝛼) ≤ ℎ1/2 such that P (1−𝜃−𝛿) 𝑝 (𝜔 = · ) ≤st P 𝑝 (𝜔 ∪ 1 𝐴 = · | ∥𝐾𝑜 ∥ < 𝛼), where 1 𝐴 denotes the random configuration with every edge open on 𝐴 and every edge closed on 𝐴𝑐 . 370 To prove Proposition 8.5.1, we will simply combine this lemma together with Lemma 8.4.5, which was a standard application of Russo’s formula and Talagrand’s inequality. Proof of Proposition 8.5.1. Define 𝜃 and ℎ as in Lemma 8.5.2. Suppose that P 𝑝−𝜀 (∥𝐾𝑜 ∥ ≥ 𝛼) ≥ 𝛼
and P 𝑝 ∥𝐾1 ∥ ≥ 𝛼 and ∥𝐾2 ∥ < 𝛼2 > 1 − 𝛼2 , i.e. ℎ < 𝛼2 . First consider the case that 𝜃 + ℎ ≥ 𝜀2 . 𝜀 Then by hypothesis and the fact that 𝛼 ≤ 10 , 𝜀 𝜀 𝜀 𝜃 ≥ − ℎ ≥ − 𝛼2 ≥ . (8.5.1) 2 2 4 Now consider the case that 𝜃 + ℎ < 𝜀2 . Define 𝛿 as in Lemma 8.5.2. By Lemma 8.5.2, there is an event 𝐴 such that P (1−𝜃−𝛿) 𝑝 (∥𝐾𝑜 ∥ ≥ 𝛼) ≤ P 𝑝 ( 𝐴 | ∥𝐾𝑜 ∥ < 𝛼) ≤ ℎ1/2 . On the other hand, by our hypotheses, ℎ1/2 < 𝛼 ≤ P 𝑝−𝜀 (∥𝐾𝑜 ∥ ≥ 𝛼). So by monotonicity, we must have (1 − 𝜃 − 𝛿) 𝑝 ≤ 𝑝 − 𝜀. In particular, 𝜃 + 𝛿 ≥ 𝜀. We can upper bound 𝛿 by 𝜀 2 · 10 2𝛼 2𝜀 2ℎ1/2 ≤ , 𝛿= 𝜀 ≤ 𝜀 ≤ 1 − (𝜃 + ℎ) 1− 2 1− 2 5 where the last inequality used the fact that 𝜀 ∈ (0, 1). Therefore, again, 𝜃 ≥ 𝜀 − 𝛿 ≥ 𝜀4 , as in eq. (8.5.1). Let 𝑐 > 0 be the constant from Lemma 8.4.5. Without loss of generality, assume that 𝑐 < 81 .
1 . By Markov’s inequality, P 𝑝 ∥𝐾1 ∥ ≥ 𝜀8 ≥ 𝜀8 because 𝜃 ≥ 𝜀4 . Therefore, Suppose that |𝑉 | 𝑐𝜀 ≥ 𝑐𝜀  𝜀 𝜀 1 P 𝑝 (∥𝐾1 ∥ ≥ 𝑐𝜀) ≥ P 𝑝 ∥𝐾1 ∥ ≥ ≥ > 𝑐𝜀 ≥ . |𝑉 | 𝑐𝜀 8 8 So by applying Lemma 8.4.5, P 𝑝+𝜀 (∥𝐾1 ∥ ≥ 𝑐𝜀) ≥ 1 − |𝑉 | −𝑐𝜀 , as required. 8.6 □ Proof of Theorem 8.1.1 Let G be an infinite set of finite transitive graphs with bounded degrees. Suppose that for all but at most finitely many 𝐺 ∈ G,  𝜋  𝑒 (log diam 𝐺) 1/9 1 distGH 𝐺, 𝑆 > . (8.6.1) diam 𝐺 diam 𝐺 Our goal is to prove that both statements (1) and (2) are true. By Proposition 8.2.9, statement (1) implies statement (2). So it suffices to prove statement (1), i.e. percolation on G has a sharp phase transition. We will assume without loss of generality that there exists 𝑑 ∈ N such that every 𝐺 ∈ G has degree exactly 𝑑. We will again adopt the notation ∥𝐾1 ∥ , ∥𝐾2 ∥ from Section 8.4. 371 Claim 8.6.1. For every constant 𝜀 > 0, there exist constants 𝑐(𝜀) > 0 and 𝜇(𝑑, 𝜀) < ∞ such that for every infinite subset H ⊆ G and every sequence 𝑝 : H → (0, 1), lim inf P 𝑝 (|𝐾1 | ≥ 𝜇 log |𝑉 |) > 0 =⇒ 𝐺∈H lim P 𝑝+4𝜀 (∥𝐾1 ∥ ≥ 𝑐) = 1. 𝐺∈H Before proving this claim, let us explain how to conclude from it. For each 𝐺 ∈ G, pick a parameter 𝑞(𝐺) ∈ (0, 1) satisfying P𝐺 (|𝐾1 | ≥ |𝑉 | 2/3 ) = 21 . We will prove that percolation on G has 𝑞(𝐺) a sharp phase transition with percolation threshold given by 𝑞 : G → (0, 1). First notice that 1 > 0. Indeed, this follows from the proof of [EH21a, Lemma 2.8], but let us explain lim inf 𝑞 ≥ 2𝑑 the elementary argument here for completeness. For every 𝐺 ∈ G and 𝑛 ≥ 1, there are at most 𝑑 𝑛 self-avoiding paths starting from 𝑜. So by a union bound, every 𝐺 ∈ G satisfies ∞ ∑︁ 𝑑𝑛 = 2. E 1 |𝐾𝑜 | ≤ 2𝑑 (2𝑑) 𝑛 𝑛=0 On the other hand, by transitivity, every 𝐺 ∈ G satisfies     E 1 |𝐾𝑜 | ≥ |𝑉 | 2/3 P 1 |𝐾𝑜 | ≥ |𝑉 | 2/3 ≥ |𝑉 | 1/3 P 1 |𝐾1 | ≥ |𝑉 | 2/3 . 2𝑑 2𝑑 2𝑑 Therefore for all but finitely many 𝐺 ∈ G,   1 P 1 |𝐾1 | ≥ |𝑉 | 2/3 ≤ 2 |𝑉 | −1/3 < , 2𝑑 2 1 and hence by monotonicity, 𝑞(𝐺) ≥ 2𝑑 . 1 Now fix a constant 𝜀 > 0. Since lim inf 𝑞 ≥ 2𝑑 > 0, there exists a constant 𝛿(𝜀, 𝑑) > 0 such that (1 − 𝜀)𝑞 ≤ 𝑞 − 𝛿 and 𝑞 + 𝛿 ≤ (1 + 𝜀)𝑞 for all but finitely many 𝐺 ∈ G. Let 𝑐 (𝛿/4) > 0 and 𝜇(𝑑, 𝛿/4) < ∞ be the constants provided by the claim. For all but finitely many 𝐺 ∈ G, we have 𝜇 log |𝑉 | < |𝑉 | 2/3 . So by applying the claim with “H ” being the whole of G and “𝑝” being 𝑞, lim P𝑞+𝛿 (∥𝐾1 ∥ ≥ 𝑐) = 1. 𝐺∈G On the other hand, for all but finitely many 𝐺 ∈ G, we have 𝑐 |𝑉 | > |𝑉 | 2/3 . So by applying the claim (contrapositively) with “𝑝” being 𝑞 − 𝛿, for every infinite subset H ⊆ G, lim inf P𝑞−𝛿 (|𝐾1 | ≥ 𝜇 log |𝑉 |) = 0. 𝐺∈H Equivalently, for every infinite subset H ⊆ G there exists a further infinite subset H ′ ⊆ H such that lim𝐺∈H ′ P𝑞−𝛿 (|𝐾1 | ≥ 𝜇 log |𝑉 |) = 0. Therefore, lim P𝑞−𝛿 (|𝐾1 | ≥ 𝜇 log |𝑉 |) = 0. 𝐺∈G 372 Since 𝜀 > 0 was arbitrary, this establishes that percolation on G has a sharp phase transition. All that remains is to verify the claim. Proof Fix 𝜀 > 0. Let 𝑐 1 (𝑑, 𝜀) > 0 be the constant from Proposition 8.2.1. Let   of claim. 2 𝜀 < ∞ be the constant from Proposition 8.3.1 (with “𝜂” set to 𝜀 2 /20). Let 𝑐 2 > 0 be the 𝜆 𝑑, 𝜀, 20   𝜆 1 universal constant from Proposition 8.5.1. We will prove that the claim holds with 𝜇 := exp 𝑐1 + 𝑐2 1 and 𝑐 := 𝑐 2 𝜀. Let H ⊆ G be an infinite subset, and let 𝑝 : H → (0, 1) be a sequence satisfying 𝜂 := lim inf P 𝑝 (|𝐾1 | ≥ 𝜇 log |𝑉 |) > 0. 𝐺∈H We say that a statement 𝐴 holds for almost every 𝐺 to mean that the set {𝐺 ∈ H : 𝐴 does not hold for 𝐺} is finite. For almost every 𝐺, P 𝑝 (|𝐾1 | ≥ 𝜇 log |𝑉 |) ≥ 1 𝜂 ≥ . 2 𝑐 1 |𝑉 | 𝑐1 So by Proposition 8.2.1, noting that 𝑐 1 log 𝜇 − 𝑐11 = 𝜆, min P 𝑝+𝜀 (𝑜 ↔ 𝑢) ≥ 𝑢∈𝐵𝜆 𝜀2 . 20 For each 𝐺 ∈ H , define 𝛾(𝐺) and 𝛾 + (𝐺) as in Proposition 8.3.1. Then by Proposition 8.3.1, thanks to our choice of 𝜆, for almost every 𝐺, + 1/2 min P 𝑝+2𝜀 (𝑜 ↔ 𝑢) ≥ 𝑒 −(log log 𝛾 ) . (8.6.2) 𝑢∈𝐵 𝛾 + Consider a particular 𝐺 ∈ H satisfying eq. (8.6.1). Then 𝛾(𝐺) > 𝑒 (log diam 𝐺) , and by applying 9 the monotone function 𝑥 ↦→ 𝑒 (log 𝑥) to both sides, 𝛾 + (𝐺) > diam 𝐺. In particular, 𝐵𝐺 is the 𝛾 + (𝐺)   1 1 1 whole vertex set 𝑉 (𝐺). We trivially have distGH diam ≤ 1, because both metric spaces 𝐺 𝐺, 𝜋 𝑆 1/9 9 involved have diameter ≤ 1. So conversely, 𝛾(𝐺) ≤ 𝜋 diam 𝐺, and hence 𝛾 + (𝐺) ≤ 𝑒 (log(𝜋 diamG)) . By applying these upper and lower bounds on 𝛾 + (𝐺) to eq. (8.6.2), we deduce that for almost every 𝐺, 1/2 1/2 min P 𝑝+2𝜀 (𝑢 ↔ 𝑣) ≥ 𝑒 −3(log log(𝜋 diam 𝐺)) ≥ 𝑒 −3(log log(𝜋|𝑉 |)) , 𝑢,𝑣∈𝑉 where the second inequality follows from the trivial bound |𝑉 | ≥ diam 𝐺. For each 𝐺 ∈ H , define 𝛿(𝐺) := (log |𝑉 |) −1/20 . For every sufficiently large positive real 𝑥, 1/2 1 2(log 𝑥) −1/20 = 2𝑒 − 20 log log 𝑥 ≤ 𝑒 −3(log log(𝜋𝑥)) . 373 Therefore for almost every 𝐺, min P 𝑝+2𝜀 (𝑢 ↔ 𝑣) ≥ 2𝛿. 𝑢,𝑣∈𝑉 (8.6.3) By applying Proposition 8.4.1, it follows that for almost every 𝐺, (since 𝛿 ≤ 𝜀)   P 𝑝+3𝜀 ∥𝐾1 ∥ ≥ 𝛿 and ∥𝐾2 ∥ ≤ 𝛿2 ≥ 1 − 𝛿4 . 𝜀 For almost every 𝐺, we have 𝛿2 < 2𝛿 , 𝛿4 < 𝛿2 , 𝛿 ∈ (0, 10 ), |𝑉 | 𝑐2 𝜀 ≥ 𝑐12 𝜀 , and by applying Markov’s inequality to eq. (8.6.3), P 𝑝+2𝜀 (∥𝐾𝑜 ∥ ≥ 𝛿) ≥ 𝛿. So by Proposition 8.5.1, for almost every 𝐺, P 𝑝+4𝜀 (∥𝐾1 ∥ ≥ 𝑐 2 𝜀) ≥ 1 − In particular, lim𝐺∈H P 𝑝+4𝜀 (∥𝐾1 ∥ ≥ 𝑐 2 𝜀) = 1, as claimed. 374 1 . |𝑉 | 𝑐2 𝜀 □ Appendix: Details for some claims in Section 8.3 In this appendix, we will explain how some of the lemmas in Section 8.3 can be established by minor modifications of existing arguments. Lemma (Lemma 8.3.3). There exists 𝑛0 (𝑑) < ∞ such that for all 𝑛 ≥ 𝑛0 , 𝑠([𝑅(𝑛), 𝑅 2 (𝑛)] is orange) ≤ 𝑠(𝑛 is green) + 𝛿(𝑛). The following argument is essentially contained in the proof of [EH23b, Proposition 4.5]. Proof. Let 𝑛 ≥ 3 be some scale. Throughout this proof, we will assume that 𝑛 is large with respect to 𝑑. Note that the result is trivial if 𝑛 > diam 𝐺, because in that case 𝐵𝑚 = 𝐵𝑛 for all 𝑚 ∈ [𝑅(𝑛), 𝑅 2 (𝑛)]. So let us assume to the contrary that 𝑆𝑛 ≠ ∅. First consider the case that 𝑛 ∈ L.
Then at any time 𝑡 when 𝑛 is green, we know that 𝜅 𝜙(𝑡) 𝑅 2 (𝑛), 𝑛 ≥ 𝛿(𝑅(𝑛)), which implies that [𝑅(𝑛), 𝑅 2 (𝑛)] is already orange at time 𝑡. So let us assume to the contrary that 𝑛 ∉ L. Define 100 ℎ := 𝑒 −(log 𝑛) , which therefore satisfies ℎ ≥ Gr(𝑛) −1 . Pick 𝑝 1 ∈ (0, 1) such that 𝑛 is green at time 𝜙−1 ( 𝑝 1 ). Note that 𝑝 1 ≥ 1/𝑑 because by a union bound, using that 𝑆𝑛 ≠ ∅ and that 𝑛 is large with respect to 𝑑,  𝑛 1 𝑛−1 < 𝛿(𝑛). min P1/𝑑 (𝑜 ↔ 𝑢) ≤ P1/𝑑 (𝑜 ↔ 𝑆𝑛 ) ≤ 𝑑 (𝑑 − 1) · 𝑢∈𝐵 𝑛 𝑑 Define 𝑝 2 := 𝜙(𝜙−1 ( 𝑝 1 ) + 𝛿(𝑛)). In the language of [EH23b, Section 3], the quantity “𝛿( 𝑝 1 , 𝑝 2 )” is equal to 𝛿(𝑛) by construction. Let 𝑢 ∈ 𝐵 𝑅2 (𝑛) be arbitrary, and let 𝑜 = 𝑢 0 , 𝑢 1 , . . . , 𝑢 𝑘 = 𝑢 be a path with 𝑘 ≤ 𝑅 2 (𝑛). Let 𝑐 1 (1), ℎ0 (𝑑, 1), 𝑐 2 , 𝑐 3 > 0 be the constants from [EH23b, Proposition 4.1] with 𝐷 := 1. Since 𝑛 is large with respect to 𝑑, we have ℎ ≤ ℎ0 , 𝛿(𝑛) ≤ 1, 3 100 −3(log log 𝑛) 1/2 𝑐3 𝑐3 𝑐3 ≤ , = 2 ℎ𝑐1 𝛿(𝑛) = 𝑒 −𝑐1 (log 𝑛) 𝑒 ≤ 81 (log 𝑛) 𝑒 + 1 𝑅 (𝑛) + 1 𝑘 + 1 and for all 𝑖 ∈ {0, . . . , 𝑘 − 1}, by Harris’ inequality,  min P 𝑝1 (𝑥 ↔ 𝑦) : 𝑥, 𝑦 ∈ 𝐵𝑛 (𝑢𝑖 ) ∪ 𝐵𝑛 (𝑢𝑖+1 ) ≥ 𝛿(𝑛) · 𝑝 1 · 𝛿(𝑛) 1/2 1 ≥ 𝑒 −2(log log 𝑛) 𝑑 ≥ 4𝑒 −𝑐1 (log 𝑛) 100 𝑒 −4(log log 𝑛) 1/2 4 = 4ℎ𝑐1 𝛿(𝑛) . So by [EH23b, Proposition 4.1], where the sets “𝐴1 , . . . , 𝐴𝑛 ” are the balls 𝐵𝑛 (𝑢 0 ), . . . , 𝐵𝑛 (𝑢 𝑘 ), P 𝑝2 (𝑜 ↔ 𝑢) ≥ 𝑐 2 𝛿(𝑛) 2 ≥ 𝛿(𝑅(𝑛)). Since 𝑢 ∈ 𝐵 𝑅2 (𝑛) was arbitrary, it follows that [𝑅(𝑛), 𝑅 2 (𝑛)] is orange at time 𝜙−1 ( 𝑝 2 ), as required. □ 375 Lemma (Lemma 8.3.4). For all 𝑐 > 0 there exist 𝜆(𝑑, 𝑐), 𝑛0 (𝑑, 𝑐), 𝐾 (𝑑, 𝑐) < ∞ such that the following holds for all 𝑛 ≥ 𝑛0 with 𝑛 ∈ T(𝑐, 𝜆). For all 𝑡 ∈ R, if 𝑛 is orange at time 𝑡 and 𝐾Δ𝑡 (𝑛) ≤ 1 then 𝑠(𝑛 is green) ≤ 𝑡 + 𝐾Δ𝑡 (𝑛). This is implicit in the proof of [EH23b, Proposition 6.1] Proof. In [EH23b], we made the following definitions: given 𝑑 ≥ 1, we wrote U𝑑∗ for the set of all infinite non-one-dimensional unimodular transitive graphs with degree 𝑑, and given 𝐷 ≥ 1 and a 𝐷 transitive graph 𝐺, we wrote L (𝐺, 𝐷) for the set of all scales 𝑛 ≥ 1 such that Gr(𝑚) ≤ 𝑒 (log 𝑚) for all 𝑚 ∈ [𝑛1/3 , 𝑛]. Let us now introduce the following variants of these definitions: given 𝑑 ≥ 1, write W𝑑 for the set of all (possibly finite) unimodular transitive graphs with degree 𝑑, and given 𝐷, 𝜆 ≥ 1, 𝑐 > 0, and a transitive graph 𝐺, write T (𝐺, 𝐷, 𝜆, 𝑐) for the set all of scales 𝑛 ∈ L (𝐺, 𝐷) with 𝑛 ≤ diam 𝐺 such that 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes throughout an interval of the form [𝑚 1 , 𝑚 2 ] with 𝑚 2 ≥ 𝑚 11+𝑐 satisfying [𝑚 1 , 𝑚 2 ] ⊆ [𝑛1/3 , 𝑛1/(1+𝑐) ]. Let [EH23b, Proposition* 6.1] be the result of modifying the statement of [EH23b, Proposition 6.1] as follows: 1. Weaken the hypothesis that 𝐺 ∈ U𝑑∗ to the hypothesis that 𝐺 ∈ W𝑑 . 2. Strengthen the hypothesis that 𝑛 ∈ L (𝐺, 𝐷) to the hypothesis that 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷). Note that 𝑝 𝑐 (𝐺) in this statement refers to the usual percolation threshold for an infinite cluster, so in particular, 𝑝 𝑐 (𝐺) := 1 if 𝐺 is finite. The same proof works because the hypothesis that 𝐺 was infinite and non-one-dimensional was only used to invoke [EH23b, Proposition 5.2] to establish that there is a constant 𝑐 1 (𝑑, 𝐷) > 0 such that for all 𝜆, whenever 𝑛 is large with respect to 𝑑, 𝐷, 𝜆, if 𝑛 ∈ L (𝐺, 𝐷) then automatically 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 𝑐 1 ). We are just circumventing this application of [EH23b, Proposition 5.2]. Specifically, we can prove [EH23b, Proposition* 6.1] by modifying the proof of [EH23b, Proposition 6.1] as follows: 1. Strengthen the condition 𝑛 ∈ L (𝐺, 𝐷) to 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷) in the definition of A. 2. Rather than define 𝑐 1 and 𝑁 to be the constants guaranteed to exist by [EH23b, Proposition 5.2], set 𝑐 1 := 1/𝐷 and 𝑁 := 3. 3. Restrict the domain of the definition of P (𝑛) from all 𝑛 ∈ L (𝐺, 𝐷) to all 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷). 376 4. Include the hypothesis 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷) in the statement of [EH23b, Lemma 6.8].33 Taking [EH23b, Proposition* 6.1] for granted, let us now explain how to prove Lemma 8.3.4. Recall that 𝐺 is a finite transitive graph with degree 𝑑. Let 𝑐 > 0 be given, and define 𝐷 := 101 ∨ (1/𝑐). Let 𝜆 0 (𝑑, 𝐷) and 𝑐 1 (𝑑, 𝐷) (called “𝑐(𝑑, 𝐷)”) be the constants provided by [EH23b, Proposition* 6.1]. Define 𝜆 := 𝜆 0 ∨ (100/𝑐 1 ). Now let 𝐾1 (𝑑, 𝐷, 𝜆) and 𝑛0 (𝑑, 𝐷, 𝜆) be the corresponding constants provided by [EH23b, Proposition* 6.1]. Define 𝐾 := 𝐾11/4 . By the same argument as in our proof of Lemma 8.3.3 above, there exists 𝑛1 (𝑑) < ∞ such that for all 𝑛1 ≤ 𝑛 ≤ diam 𝐺 and 101 𝑡 ∈ R, if 𝑛 is orange at time 𝑡 then 𝜙(𝑡) ≥ 1/𝑑. Set 𝑛2 := 𝑛0 ∨ 𝑛1 ∨ 𝑒 3 . We claim that 𝜆, 𝑛2 , 𝐾 have the properties required of the constants called “𝜆, 𝑛0 , 𝐾” in the statement of Lemma 8.3.4. Indeed, suppose that 𝑡 ∈ R and 𝑛 ≥ 𝑛2 with 𝑛 ∈ T(𝑐, 𝜆) are such that 𝑛 is orange at time 𝑡 and 𝐾Δ𝑡 (𝑛) ≤ 1. Now apply [EH23b, Proposition* 6.1] with the variables called “𝐾, 𝑛, 𝑏, 𝑝 1 , 𝑝 2 ” in that statement set to our variables 𝐾1 , 𝑛, 𝑈𝑡 (𝑛), 𝜙(𝑡), 𝜙(𝑡 + 𝐾Δ𝑡 (𝑛)). The only hypothesis that is not immediately obvious is that 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷). To see this, first note that since 𝑛 ∈ L and 101 𝑛 ≥ 𝑒 3 , every 𝑚 ∈ [𝑛1/3 , 𝑛] satisfies Gr(𝑚) ≤ Gr(𝑛) ≤ 𝑒 (log 𝑛) 100 ≤ 𝑒 (log(𝑛 1/3 )) 101 ≤ 𝑒 (log 𝑚) 101 ≤ 𝑒 (log 𝑚) . 𝐷 So 𝑛 ∈ L (𝐺, 𝐷). Second, we may assume that 𝑛 ≤ diam 𝐺, otherwise the conclusion of Lemma 8.3.4 is trivial. Finally, since 𝑛 ∈ T(𝑐, 𝜆) and 1/𝐷 < 𝑐, and the property of having “(𝑥, 𝜆)-polylog plentiful tubes” at a given scale gets weaker as we decrease 𝑥, it follows that 𝑛 ∈ T (𝐺, 𝐷, 𝜆, 1/𝐷). Therefore, by applying [EH23b, Proposition* 6.1], we deduce that   𝑐1 𝜆 1/2 𝜅 𝜙(𝑡+𝐾Δ𝑡 (𝑛)) 𝑒 (log 𝑛) , 𝑛 ≥ 𝑒 −3(log log 𝑛) . In particular, since 𝑐 1𝜆 ≥ 100 ≥ 81,   𝜅 𝜙(𝑡+𝐾Δ𝑡 (𝑛)) 𝑅 2 (𝑛), 𝑛 ≥ 𝛿(𝑅(𝑛)). So 𝑠(𝑛 is green) ≤ 𝑡 + 𝐾Δ𝑡 (𝑛), as required. □ Lemma (Lemma 8.3.5). There exists 𝑛0 (𝑑) < ∞ such that the following holds for all 𝑛 ∈ L with 𝑛 ≥ 𝑛0 . For all 𝑡 ∈ R, if 𝐿 (𝑛) is green at time 𝑡 then Δ𝑡 (𝑛) ≤ 1 . log log 𝑛 33 While writing this paper, we noticed the following typo: 𝑛 ∈ L (𝐺, 𝐷). 377 [EH23b, Lemma 6.8] is missing the hypothesis that This proof is implicit in [EH23b, Section 6.3]. Proof. Suppose that 𝑛 ∈ L. Throughout this proof we will implicitly assume that 𝑛 is large with respect to 𝑑. Let 𝑡 ∈ R and assume that 𝐿 (𝑛) is green at time 𝑡. We may assume that ⌊𝑛1/3 ⌋ ≤ diam 𝐺, otherwise we trivially have 𝑈𝑡 (𝑛) = ⌊ 18 𝑛1/3 ⌋ and hence (since 𝑛 is assumed large) Δ𝑡 (𝑛) ≤ (log 𝑛) −1/5 . We split the proof into two cases according to whether 𝐿 (𝑛) ∈ L. First suppose that 𝐿 (𝑛) ∉ L. By the same argument as in our proof of Lemma 8.3.3 above, since 𝐿 (𝑛) ≤ diam 𝐺 and 𝐿 (𝑛) is green at time 𝑡 (and since 𝑛 is assumed large), 𝜙(𝑡) ≥ 1/𝑑. So by [EH23b, Corollary 2.4], there exist constants 𝑐(𝑑) > 0 and 𝐶 (𝑑), 𝑛0 (𝑑) < ∞ such that for all 𝑚 ≥ 𝑛0 (𝑑),  1/3  log Gr(𝑚) P𝜙(𝑡) (Piv[𝑐 log 𝑚, 𝑚]) ≤ 𝐶 . 𝑚 In particular, since 4𝐿 (𝑛) ≤ 𝑐 log(𝑛1/3 ) and Gr(𝑛1/3 ) ≤ Gr(𝑛) ≤ 𝑒 (log 𝑛) , 100   P𝜙(𝑡) Piv 4𝐿 (𝑛), 𝑛 1/3   (log 𝑛) 100 ≤𝐶 𝑛1/3  1/3 ≤ 1 . log 𝑛 Since we also clearly have 𝐿(𝑛) ≤ 81 𝑛1/3 , it follows that 𝑈𝑡 (𝑛) ≥ ⌊𝐿 (𝑛)⌋. Since 𝐿 (𝑛) ∉ L, this 100 implies that Gr(𝑈𝑡 (𝑛)) ≥ 𝑒 (log 𝐿(𝑛)) . So  log log 𝑛 Δ𝑡 (𝑛) ≤ (log 𝑛) ∧ (log 𝐿 (𝑛)) 100  1/4 ≤ 1 . log log 𝑛 
 Next suppose that 𝐿(𝑛) ∈ L. Define 𝑏 := 15 𝑅 ◦ 𝐿(𝑛) ∧ Gr−1 𝑅 −1 (𝑛) . By [EH23b, Lemma 2.3] (i.e. [CMT22, Lemma 6.2]), using the fact that 5𝑏 ≤ 21 𝑛1/3 ,   P𝜙(𝑡) Piv 4𝑏, 𝑛 1/3   ≤ P𝜙(𝑡)  1 Piv 1, 𝑛1/3 2  · |𝑆4𝑏 | 2 Gr(5𝑏) . 𝐵5𝑏 min𝑥,𝑦∈𝑆4𝑏 P𝜙(𝑡) (𝑥 ←−→ 𝑦) By [EH23b, Lem 2.1] (i.e. essentially [CMT22, Proposition 4.1]), there is a constant 𝐶 (𝑑) < ∞ such that   1/3 1 1/3    © log Gr 2 𝑛 ª 1 ® . P𝜙(𝑡) Piv 1, 𝑛1/3 ≤ 𝐶 ­­ ® 1 1/3 2 𝑛 2 « ¬ 1 1/3 By hypothesis, 𝑛 ∈ L. So we can upper bound log Gr( 2 𝑛 ) ≤ log Gr(𝑛) ≤ (log 𝑛) 100 . Since 𝐿(𝑛) is green at time 𝑡 but 𝐿 (𝑛) ∈ L, then 𝜅 𝜙(𝑡) (𝑅 2 ◦ 𝐿 (𝑛), 𝐿(𝑛)) ≥ 𝛿(𝑅 ◦ 𝐿(𝑛)). Note that 8𝑏 ≤ 𝑅 2 ◦ 𝐿(𝑛) 𝐵5𝑏 and (using that 𝐿(𝑛) ∈ L), 𝐿 (𝑛) ≤ 𝑏. Therefore, min𝑥,𝑦∈𝑆4𝑏 P𝜙(𝑡) (𝑥 ←−→ 𝑦) ≥ 𝛿(𝑅 ◦ 𝐿(𝑛)), since we 378 can connect any 𝑥, 𝑦 ∈ 𝑆4𝑏 by a path contained in 𝐵4𝑏 of length at most 8𝑏, and the 𝑏-thickened tube around this path is entirely contained in 𝐵5𝑏 . Finally, we can upper bound |𝑆4𝑏 | ≤ Gr(5𝑏) ≤ 𝑅 −1 (𝑛) by definition of 𝑏. Therefore,   P𝜙(𝑡) Piv 4𝑏, 𝑛  1/3 ≤𝐶 (log 𝑛) 100 1 1/3 2𝑛 ! 1/3
3 𝑅 −1 (𝑛) 1 ≤ . 𝛿(𝑅 ◦ 𝐿 (𝑛)) log 𝑛 Notice that by our choice of 𝑏, we have 𝑏 ≤ 81 𝑛1/3 and         1/5 1 1 1 −1  −1  1 −1 Gr(𝑏) ≥ Gr 𝑅 ◦ 𝐿 (𝑛) ∧ Gr Gr 𝑅 (𝑛) ≥ 𝑅 ◦ 𝐿 (𝑛) ∧ 𝑅 (𝑛) = 𝑅 ◦ 𝐿 (𝑛). 5 5 5 5 So 1/4 ª © log log 𝑛 ®  Δ𝑡 (𝑛) ≤ ­­  ® 1 (log 𝑛) ∧ log 5 𝑅 ◦ 𝐿(𝑛) ¬ « ≤ 1 . log log 𝑛 □ Lemma (Lemma 8.3.11). Let 𝐺 be a unimodular transitive graph of degree 𝑑. Suppose that Gr(𝑚) ≤ 𝑒 (log 𝑚) 𝐷 and Gr(3𝑚) ≥ 35 Gr(𝑚) for every 𝑚 ∈ [𝑛1−𝜀 , 𝑛1+𝜀 ], where 𝜀, 𝐷, 𝑛 > 0. Then there is a constant 𝑐(𝑑, 𝐷, 𝜀) > 0 with the following property. For every 𝜆 ≥ 1, there exists 𝑛0 (𝑑, 𝐷, 𝜀, 𝜆) < ∞ such that if 𝑛 ≥ 𝑛0 then 𝐺 has (𝑐, 𝜆)-polylog plentiful tubes at scale 𝑛. [EH23b, Lemma 5.4] is the same statement but with the additional hypothesis that 𝐺 is infinite. We claim that this additional hypothesis is unnecessary. Proof. [EH23b, Lemma 5.4] is the ultimate conclusion of [EH23b, Section 5.2]. The first result in [EH23b, Section 5.2] that requires 𝐺 to be infinite is [EH23b, Lemma 5.16]. By inspecting the proof of [EH23b, Lemma 5.16], we see that this hypothesis is only used in order to apply the elementary bound Gr(3𝑚𝑛) ≥ 𝑛 Gr(𝑚) for all 𝑚, 𝑛 ≥ 1. In fact, in the language of that proof, since we may assume that the constant 𝑐 > 0 satisfies 𝑐 ≤ 1/10, say, then the proof only invokes this 1 1/2 1 1/2 elementary bound for 𝑚, 𝑛 satisfying 3𝑚𝑛 ≤ 10 𝑡 . Now this holds whenever diam 𝐺 ≥ 10 𝑡 . So [EH23b, Lemma 5.16] holds with the hypothesis “𝐺 is infinite” replaced by the weaker hypothesis 1 1/2 “diam 𝐺 ≥ 10 𝑡 ”. When [EH23b, Lemma 5.16] is applied to establish [EH23b, Lemma 5.17], 1 1/2 the hypothesis “diam 𝐺 ≥ 10 𝑡 ” is already implied by the other hypothesis of [EH23b, Lemma 379 5.17] that Gr(3𝑚) ≥ 3𝜅 Gr(𝑚) for all 𝑛 ≤ 𝑚 ≤ 12 𝑡 1/2 (and the fact that conclusion of [EH23b, Lemma 5.17] is trivial if there is no integer in [𝑛, 12 𝑡 1/2 ]). So in the statement of [EH23b, Lemma 5.17], we can simply drop the hypothesis that 𝐺 is infinite. We can also drop the hypothesis that 𝐺 is infinite in [EH23b, Lemmas 5.18 and 5.20] because [EH23b, Lemma 5.18] is deduced from [EH23b, Lemma 5.17], and [EH23b, Lemma 5.20] is deduced from [EH23b, Lemma 5.18]. [EH23b, Lemma 5.19] already does not require 𝐺 to be infinite. The ultimate proof of [EH23b, Lemma 5.4] only required 𝐺 to be infinite in order to invoke [EH23b, Lemma 5.20] and (in the radial case) to know that 𝑆𝑛 ≠ ∅. The hypothesis that 𝑆𝑛 ≠ ∅ is anyway implied by the fact that Gr(3𝑚) ≥ Gr(𝑚) for some 𝑚 ∈ [𝑛, 𝑛1+𝜀 ], and as we explained, we can drop the hypothesis that 𝐺 is infinite in [EH23b, Lemma 5.20]. Therefore we can drop the hypothesis that 𝐺 is infinite in the statement of [EH23b, Lemma 5.4] too. □ The next claim we will justify is that Lemma 8.3.16 implies Lemma 8.3.13. Here are the statements of these results. Lemma (Lemma 8.3.16). Let 𝑟, 𝑛 ≥ 1. Let 𝐺 be a finite transitive graph such that 𝑆𝑛∞ is not 𝑟-connected. Let 𝐻 be a (finite or infinite) transitive graph with 𝛿(𝐻) ≤ 𝑟 that does not have 𝐻  𝐵𝐺 , then infinitely many ends. If 𝐵50𝑛 50𝑛  distGH  200𝑛 𝜋 𝐺, 𝑆 1 ≤ . diam 𝐺 diam 𝐺 Lemma (Lemma 8.3.13). Let 𝐺 be an finite transitive graph of degree 𝑑. Suppose that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛), where 𝑛, 𝜅 > 0. There exists 𝐶 (𝑑, 𝜅) < ∞ such that the following holds if 𝑛 ≥ 𝐶: Ð There is a set 𝐴 ⊆ [1, ∞) with | 𝐴| ≤ 𝐶 such that for every 𝑘 ≥ 1 and every 𝑚 ∈ [𝐶 𝑘𝑛, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎], if 𝐺 does not have (𝐶 −1 𝑘, 𝐶 −1 𝑘 −1 𝑚, 𝐶 𝑘 𝐶 𝑚)-plentiful tubes at scale 𝑚, then  distGH  𝜋 𝐶𝑚 𝐺, 𝑆 1 ≤ . diam 𝐺 diam 𝐺 The proof that Lemma 8.3.16 implies Lemma 8.3.13 is essentially the same as the proof of [EH23b, Proposition 5.3] (i.e. Lemma 8.3.12), except that 𝐺 is now assumed to be a finite transitive graph rather than a non-one-dimensional infinite transitive graph. For this reason, the following proof is terse. The argument relies on the structure theory of transitive graphs of polynomial growth. See the proof of [EH23b, Proposition 5.3] for more details and [EH23b, Section 5.1] for more background. 380 Proof of Lemma 8.3.13 given Lemma 8.3.16. Fix 𝜅 > 0. Suppose that Gr(3𝑛) ≤ 3𝜅 Gr(𝑛) for some 𝑛 ≥ 1. We will implicitly assume that 𝑛 is large with respect to 𝑑 and 𝜅. Let 𝐻 ≤ Aut(𝐺), 𝑆 ⊆ Γ := Aut(𝐺)/𝐻, and 𝐶1 (𝐾) < ∞ be as given by [EH23b, Theorem 5.5] (which is taken from [TT21a]) with 𝐾 := 3𝜅 . Let 𝐺 ′ := Cay(Γ, 𝑆). For each 𝑘 ∈ N, let 𝑅 𝑘 be the set of all relations in Γ having word length at most 𝑘, let ⟨⟨𝑅 𝑘 ⟩⟩ be the normal subgroup of the free group on 𝑆 generated by 𝑅 𝑘 , and let 𝐺 ′𝑘 := Cay(⟨𝑆 | 𝑅 𝑘 ⟩, 𝑆). By items 7 and 8 of [EH23b, Theorem 5.5], Gr′ (3𝑛) ≤ 𝐶12 (3 + 𝐶1 ) 𝐶1 . Gr′ (𝑛) In particular, by [EH23b, Theorem 5.5] again (and using that 𝑛 is large), every transitive graph whose 3𝑛-ball is isomorphic to the 3𝑛-ball in 𝐺 ′ is necessarily finite or infinite with polynomial growth. In particular, such graphs have at most finitely many ends. Now by [EH23d, Theorem 1.1], there exists 𝐶2 (𝜅, 𝑑) < ∞ such that  𝑖 ∈ N : 𝑖 ≥ log2 𝑛 and ⟨⟨𝑅2𝑖+1 ⟩⟩ ≠ ⟨⟨𝑅2𝑖 ⟩⟩ ≤ 𝐶2 .  Let 𝐴 := 2𝑖 : 𝑖 ∈ N and 𝑖 ≥ log2 𝑛 and ⟨⟨𝑅2𝑖+10 ⟩⟩ ≠ ⟨⟨𝑅2𝑖 ⟩⟩ , and note that | 𝐴| ≤ 10𝐶2 . Let 𝑘 ≥ 1 Ð and 𝑚 ∈ [2𝑘𝑛, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎] be arbitrary. By construction of 𝐴 (and Lemma 5.6]),  [EH23b,  𝑚 ′ ′ ′ the balls of radius (say) 50𝑛 in 𝐺 𝑚 and 𝐺 are isomorphic. Note that 𝛿 𝐺 𝑚 ≤ 𝑘 , and since the 𝑘 𝑘 3𝑛-ball in 𝐺 ′𝑚 is isomorphic to the 3𝑛-ball in 𝐺 ′, the graph 𝐺 ′𝑚 has at most finitely many ends. 𝑘 𝑘 Consider an arbitrary pair 𝑚 1 , 𝑚 2 ∈ N satisfying 𝑚𝑘 ≤ 𝑚 1 ≤ 𝑚 2 ≤ 3𝑚. By Lemma 8.3.16 applied ∞ (𝐺 ′) is ⌈ 𝑚 ⌉-connected, with the pair “(𝐺, 𝐻)” equal to (𝐺 ′, 𝐺 ′𝑚 ), either (1) the exposed sphere 𝑆 𝑚 𝑘 2 𝑘 or (2)  𝜋  200𝑚 2 ′ 1 distGH ≤ 𝐺 , 𝑆 . (8.6.4) diam 𝐺 ′ diam 𝐺 ′ In case (1), we deduce by the proof of [CMT22, Lemma 2.7] (which was behind [EH23b, ∞ (𝐺 ′) there exists a path from 𝑢 to 𝑣 in 𝐺 ′ that is contained Lemma 5.8]) that for all 𝑢, 𝑣 ∈ 𝑆 𝑚 2 Ð in 𝑥∈𝑆∞𝑚 (𝐺 ′ ) 𝐵2𝑚1 (𝑥) and has length at most 3𝑚 1 Gr(3𝑚 2 )/Gr(𝑚 1 ). Now consider case (2). The 2 existence of a (1, 𝐶1 𝑛)-quasi-isometry from 𝐺 to 𝐺 ′ implies that |diam 𝐺 − diam 𝐺 ′ | ≤ 3𝐶1 𝑛 and distGH (𝐺, 𝐺 ′) ≤ 𝐶1 𝑛. (For the latter, see exercise 5.10 (b) in [Pet23], for example.) We may assume without loss of generality that diam 𝐺 ≥ 100𝐶1 𝑛, say, otherwise our claim is trivial. By 𝐶3 𝑚 𝜋 1 combining these simple bounds with eq. (8.6.4), we deduce that distGH ( diam 𝐺 𝐺, 𝑆 ) ≤ diam 𝐺 for some constant 𝐶3 (𝜅, 𝑑) < ∞. We now run the rest of the proof of [EH23b, Proposition 5.3], after the application of [EH23b, Lemma 5.8], as it is written. This establishes that there is a constant 𝐶4 (𝜅, 𝑑) < ∞ such that   Ð 12 for all 𝑘 ≥ 1 and 𝑚 ∈ [𝐶4 𝑘𝑛, ∞)\ 𝑎∈𝐴 [𝑎, 2𝑘𝑎], either (A) there exists 𝑚 2 ∈ 10 9 𝑚, 9 𝑚 such 381 ∞ (𝐺 ′) is not ⌈ 𝑚 ⌉-connected, or (B) 𝐺 has (𝐶 −1 𝑘, 𝐶 −1 𝑘 −1 , 𝐶 𝑘 𝐶4 𝑚)-plentiful tubes at scale that 𝑆 𝑚 4 𝑘 4 4 2 𝑚. (Technically, as written, the radial case of the proof of [EH23b, Proposition 5.3] invokes the existence of a bi-infinite geodesic in 𝐺. All that is really required is a geodesic of length ≥ 24𝑚. So it suffices to know that diam 𝐺 ≥ 24𝑚, say, which we may anyway assume without loss of generality otherwise the conclusion holds trivially.) By above, if case (A) holds, then case (2) 𝐶3 𝑚 𝜋 1 □ holds, and hence distGH ( diam 𝐺 𝐺, 𝑆 ) ≤ diam 𝐺 . Therefore the set of scales 𝐴 is as required. The next claim we will justify is that Timar’s proof [Tim07] of Benjamini-Babson [BB99b] yields the following statement, which is phrased slightly differently to usual, in terms of sets of vertices, vertex cutsets, and (extrinsic) diameter rather than length of generating cycles. Lemma (Lemma 8.3.17). Let 𝐺 be a graph. Let 𝐴 and 𝐵 be sets of vertices. Let Π be a minimal ( 𝐴, 𝐵)-cutset that does not disconnect 𝐴 or 𝐵. Then Π is 𝛿(𝐺)-connected. Proof. Suppose that Π = Π1 ⊔ Π2 is a non-trivial partition of Π. By minimality of Π, there exist paths 𝛾1 avoiding Π2 and 𝛾2 avoiding Π1 that both start in 𝐴 and end in 𝐵. Let 𝛾 𝐴 be a path from the startpoint of 𝛾1 to the startpoint of 𝛾2 that avoids Π, and let 𝛾 𝐵 be a path from the endpoint of 𝛾1 to the endpoint of 𝛾2 that avoids Π. Let {𝐶𝑖 : 𝑖 ∈ 𝐼} be a set of cycles of diameter ≤ 𝛿(𝐺) such Í that 𝛾1 + 𝛾2 + 𝛾 𝐴 + 𝛾 𝐵 = 𝑖∈𝐼 𝐶𝑖 . 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