Percolation on Hyperbolic Graphs

We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that pc<pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_c<p_u$$\end{document} for any such graph. Our proof also yields that the triangle condition ∇pc<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{p_c}<\infty $$\end{document} holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.


Introduction
In Questions of central interest concern the equality or inequality of these two values of p and the behaviour of percolation at and near p c and p u . These questions were traditionally studied primarily on Euclidean lattices such as the hypercubic lattice Z d , for which Aizenman et al. [AKN87] proved that Z d [p] has at most one infinite cluster almost surely for every p, and hence that p c (Z d ) = p u (Z d ) for every d ≥ 1. A very short proof of the same result was later obtained by Burton and Keane [BK89]. For further background on percolation, we refer the reader to [Dum17,Gri99,HH17,LP16].
is negatively curved at large scales. We prove our theorems under the additional assumption of unimodularity, the nonunimodular case having already been treated in [Hut17]. These results apply in particular to lattices in H d for d ≥ 2, for which Theorem 1.3 was previously known only for d = 2 and Theorem 1.4 is new for all d ≥ 2. Gromov hyperbolicity is invariant under rough isometry [Woe00, Theorem 22.2], and Theorem 1.4 gives the first family of examples of one-ended finitely generated groups all of whose Cayley graphs have mean-field critical exponents for percolation. Theorem 1.3. Let G be a connected, locally finite, nonamenable, Gromov hyperbolic, quasi-transitive graph. Then p c (G) < p u (G).
Here, a graph is said to be Gromov hyperbolic [Gro81,Gro87] if it satisfies the Rips thin triangles property, meaning that there exists a constant C such that for any three vertices u, v, w of G and any three geodesics [u, v], [v, w] and [w, u] between them, every point in the geodesic [u, v] is contained in the union of the C-neighbourhoods of the geodesics [v, w] and [w, u]. For example, every tree is hyperbolic, as it satisfies the Rips thin triangles property with constant C = 0. A finitely generated group is said to be (Gromov) hyperbolic if one (and hence all) of its Cayley graphs are Gromov hyperbolic. Every infinite, quasi-transitive Gromov hyperbolic graph is either nonamenable or rough-isometric to Z. Bonk and Schramm [BS00] proved that a bounded degree graph is Gromov hyperbolic if and only if it admits a rough-isometric embedding into real hyperbolic space H d for some d ≥ 1, a result that will be used extensively throughout this paper.
Many finitely generated groups and quasi-transitive graphs are Gromov hyperbolic. Examples include lattices in H d ; random groups below the collapse transition [AFL17,ALS14,DJKLMS16,Oll05,Zuk03]; small cancellation 1/6 groups [Gro87,§0.2A]; fundamental groups of compact Riemannian manifolds of negative sectional curvature [GH88, Chapter 3]; quasi-transitive graphs rough-isometric to simply connected Riemannian manifolds of sectional curvature bounded from above by a negative constant [GH88, Chapter 3]; quasi-transitive CAT(−k) graphs for k > 0 [GH88, Chapter 3]; and quasi-transitive, nonamenable, simply connected planar maps [FG17]. Many surveys and monographs on hyperbolic groups have been written, and we refer the reader to e.g. [Bow06,GH88,Gro87] for further background. The specific background on hyperbolic geometry needed for the proofs of this paper is reviewed in Section 3.
We remark that finitely generated hyperbolic groups are always finitely presented [GH88,Chapter 4], and it follows from the work of Babson and Benjamini [BB99] that their Cayley graphs have p u < 1 if and only if they are one-ended. Other properties of percolation on lattices in H d have been investigated in [Cza12,Cza18,Lal01].
Previous progress on Conjectures 1.1 and 1.2 can be briefly summarised as follows. First, several works [NP12,PS00,Sch01,Tho16] have established perturbative criteria under which p c < p u and ∇ pc < ∞. In these papers, the assumption of nonamenability is replaced by a stronger quantitative assumption, for example that the Cheeger constant is large, under which it can be shown that p c < p u and ∇ pc < ∞ by combinatorial methods. In particular, Pak and Smirnova-Nagnibeda [PS00] proved that every nonamenable finitely generated group has a Cayley graph for which p c < p u (see also [Tho15]). Papers that apply perturbative techniques to study specific examples, including some specific examples of hyperbolic lattices, include [Cza13,Tyk07,Yam17].
Let us now discuss non-perturbative results. Benjamini and Schramm [BS01] showed that p c < p u for every planar nonamenable quasi-transitive graph (see also [AHNR18]), generalizing earlier work of Lalley [Lal98]. The later work of Gaboriau [Gab05] and Lyons [Lyo00,Lyo13] used the ergodic-theoretic notion of cost to prove that p c < p u on any quasi-transitive graph admitting non-constant harmonic Dirichlet functions, a class that includes all those examples treated by [BS01,Lal98]. This property is invariant under rough isometry, and was until now the only condition (other than the conjecturally equivalent property of having cost > 1) under which a finitely generated group was known to have p c < p u for all of its Cayley graphs. In particular, this result applies to every quasi-transitive graph rough isometric to H 2 , but does not apply to higher dimensional hyperbolic lattices [LP16,Theorem 9.18]. Kozma [Koz11] proved that ∇ pc < ∞ for the product of two three-regular trees, the first time that the triangle condition had been established by non-perturbative methods in a non-trivial example. (This example was recently revisited in [Hut18b].) Schonmann [Sch02] proved, without verifying the triangle condition, that several mean-field exponents hold on every transitive nonamenable planar graph and every infinitely ended, unimodular transitive graph. Similar results for certain specific lattices in H 3 were proved by Madras and Wu [MT10]. Very recently, we established that p c < p u and ∇ pc < ∞ for every graph whose automorphism group has a quasi-transitive nonunimodular subgroup [Hut17]. This was until now the only setting in which both p c < p u and ∇ pc < ∞ were established under non-perturbative hypotheses. In a different direction, in [AH17] counterexamples were constructed to show that the natural generalization of Conjecture 1.1 to unimodular random rooted graphs is false.
The class of examples treated in this paper is mostly disjoint from the class treated in [Hut17], and the methods we employ here are also very different to those of that paper. Indeed, it follows from [DT19, Theorem H] that every hyperbolic group has a Cayley graph whose automorphism group is discrete, and consequently does not have any nonunimodular subgroups.
Theorem 1.4 has the following consequences regarding percolation at and near p c . These consequences were established for the hypercubic lattice in the papers [AB87, AN84, BA91, KN09, KN11,Ngu87]. A detailed overview of how to adapt 770 T. HUTCHCROFT GAFA these proofs to the general quasi-transitive setting is given in [Hut17,§7]. We write for an equality that holds up to multiplication by a function that is bounded between two positive constants in the vicinity of the relevant limit point.
Corollary 1.5 (Mean-field critical exponents). Let G be a connected, locally finite, nonamenable, Gromov hyperbolic, quasi-transitive graph. Then the following hold for each v ∈ V . (1.5) Here, we define the susceptibility χ p (v) to be the expected volume of the cluster at v, and define χ (k) p (v) to be the kth moment of the volume of the cluster at v. The implicit constants in (1.2) may depend on k. We denote the cluster at v by K v , writing |K v | for its volume and rad int (K v ) for its intrinsic radius, i.e., the maximum distance between v and some other point in K v as measured by the graph distance in G[p]. We write P p and E p for probabilities and expectations taken with respect to the law of G [p]. We remark that the susceptibility upper bound of (1.1) is proven as an intermediate step in the proofs of Theorems 1.3 and 1.4. Further applications of our techniques to the computation of critical exponents for percolation on hyperbolic graphs, including the computation of the extrinsic radius exponent, are given in the companion paper [Hut18a].
Finally, we remark that the following corollary of Theorem 1.3 is implied by the work of Lyons, Peres, and Schramm [LPS06]. We refer the reader to that paper and to [LP16,Chapter 11] for background on minimal spanning forests.
Corollary 1.6. Let G be a connected, locally finite, nonamenable, Gromov hyperbolic, quasi-transitive graph. Then the free and wired minimal spanning forests of G are distinct.

Organisation and overview.
The proof of our main theorems has two parts. The first part, which is contained in Section 2, applies to arbitrary quasitransitive graphs. In that part of the paper, we introduce a new critical parameter p 2→2 , defined to be the supremal value of p for which the matrix T p defined by T p (u, v) = τ p (u, v) is bounded as a linear operator from L 2 (V ) to L 2 (V ). We observe that p c ≤ p 2→2 ≤ p u and that ∇ p < ∞ for all p < p 2→2 , and derive a generally applicable necessary and sufficient condition for the strict inequality p c < p 2→2 . In particular, we show that a quasi-transitive graph has p c < p 2→2 if and only if lim sup where χ p = sup v∈V χ p (v). We also prove some related results concerning the similarly defined critical parameters p q→q for q ∈ [1, ∞]. Finally, we apply the results of [Hut17] to prove that p c < p 2→2 in the nonunimodular setting. The second part of the paper spans Sections 3-5, and is specific to the Gromov hyperbolic setting. In that part of the paper, following a review of relevant background and the proof of a few preliminary geometric facts in Section 3, we verify that (1.6) and (1.7) hold under the hypotheses of Theorems 1.3 and 1.4. The starting point for this analysis was the observation by Benjamini [Ben16] that in any nonamenable Gromov hyperbolic graph, a constant fraction of any finite set of vertices lie near the boundary of the convex hull of the set, a fact he deduced from related work of Benjamini and Eldan [BE12] (similar observations appeared earlier in [MT10]). In Section 5.1, we apply a variation on this observation to establish a differential inequality for the susceptibility which implies that (1.6) holds under the hypotheses of Theorems 1.3 and 1.4.
In Section 4, we apply the so-called Magic Lemma of Benjamini and Schramm [BS01, Lemma 2.3] to prove a refinement of this observation, which, roughly speaking, states that for any finite set of vertices in a Gromov hyperbolic graph, from the perspective of most vertices in the set, most of the set is contained in either one or two distant half-spaces. In Section 5.2, we apply this geometric fact to prove that (1.7) holds under the hypotheses of Theorems 1.3 and 1.4, completing the proof of our main theorems. To do this, we use a mixture of probabilistic and geometric techniques to show that a distant half-space can only contribute a small amount to the susceptibility, which yields (1.7) when combined with the aforementioned consequence of the Magic Lemma.
Finally, in Section 6 we conclude the paper with some remarks, conjectures, and open problems. In particular, we remark there that the proof given in [Hut18b] of the estimate known as Schramm's Lemma shows that the same estimate continues to hold at p 2→2 , and consequently that there cannot be a unique infinite cluster at p 2→2 .

An Operator-Theoretic Perspective on Percolation
In this section, we develop an 'operator-theoretic perspective' on percolation. In particular, we introduce a new critical parameter p 2→2 which satisfies p c ≤ p 2→2 ≤ p u and ∇ p < ∞ for all p < p 2→2 . This allows us to state Theorem 2.1, which strengthens both Theorems 1.3 and 1.4. We also give a sufficient condition for p c < p 2→2 which will be applied to Gromov hyperbolic graphs in the remainder of the paper. The approach taken in this section was inspired in part by Gady Kozma, who advocated the application of operator-theoretic techniques to percolation in [Koz11].
Let G = (V, E) be a connected, locally finite graph, and let R V be the space of real-valued functions on V .

and otherwise by
Now consider the matrix T p whose entries are given by the two-point function for every x, y ∈ V , the matrix T p is symmetric and its associated operator is self-adjoint 1 . The 1 → 1 and ∞ → ∞ norms of T p are given precisely by the susceptibility: It follows by sharpness of the phase transition [AB87, AV08, DT16] that if G is quasi-transitive then T p 1→1 < ∞ if and only if p < p c . On the other hand, we can also consider the q → q norm of T p for other q ∈ [1, ∞], and define p q→q to be the critical value associated to the finiteness of T p q→q , that is, Since T p is symmetric we have that for every q ∈ [1, ∞]. Moreover, it follows from the Riesz-Thorin Theorem that log T p 1/q→1/q is a convex function of q ∈ [0, 1]. Together, these facts imply that T p q→q is a decreasing function of q on [1, 2] and an increasing function of q on [2, ∞], so that p q→q is an increasing function of q on [1, 2] and a decreasing function of q on [2, ∞]. In particular, for every quasi-transitive graph G and q ∈ [1, ∞]. We will be particularly interested in the critical value p 2→2 (G), which by the above discussion is equal to sup q∈[1,∞] p q→q (G).
If G is quasi-transitive and G[p] has a unique infinite cluster, then the two-point function τ p (u, v) ≥ P p (u → ∞)P p (v → ∞) is bounded below by a positive constant, and it follows that p q→q (G) ≤ p u (G) whenever q ∈ [1, ∞] and G is infinite and quasi-transitive. (In Section 6, we prove the stronger statement that G[p 2→2 ] does GAFA PERCOLATION ON HYPERBOLIC GRAPHS 773 not have a unique infinite cluster a.s. when G is nonamenable and quasi-transitive.) Thus, the following theorem, which is proven in Section 5, strengthens Theorem 1.3. The difficult part of the theorem is that p c (G) < p 2→2 (G); we shall see in Proposition 2.3 that this implies that p c (G) < p q→q (G) for every q ∈ (1, ∞). The dependence of p q→q on q is further investigated in [Hut18a].
We now briefly discuss the relationship between the 2 → 2 norm and diagramatic sums. Recall that the nth polygon diagram at v is defined to be

) It follows by the Cauchy-Schwarz inequality and the symmetry of T p that
for every v ∈ V and n ≥ 1, so that in particular ∇ p < ∞ for all p < p 2→2 . The next proposition implies that T p 2→2 is in fact equal to the exponential growth rate of the polygon diagrams.
Proof. This is presumably a standard fact. It follows by the same proof as that of [LP16, Proposition 6.6], where the claim is stated in the special case that M is stochastic.
Our next goal is to prove the following; see [Hut18a] for a sharp quantitative version.
Proposition 2.3 will follow from a few simple lemmas, which will also be used in the proof of our criterion for p c < p 2→2 . The first two, Lemma 2.4 and Corollary 2.6, follow by similar reasoning to that used in [AB87], where similar inequalities are established for the susceptibility. See also [Hut17,§3].
Remark 2.5. With a little more care the 1/(1 − p 1 ) factor can be removed from the bracketed expression. This yields mild improvements to Corollary 2.6 and Proposition 2.7 below.
We briefly recall some background on correlation inequalities for percolation, referring the reader to [Gri99, §2.2 and §2.3] for more detail. An event A ⊆ {0, 1} E is said to be increasing if its indicator function is an increasing function of each bit. The Harris-FKG inequality states that increasing events are positively correlated under the product measure, that is, for every p ∈ [0, 1] and every two increasing events A , B ⊆ {0, 1} E . Given an increasing event A and a configuration ω ∈ A , a witness for A in ω is defined to be a set W ⊆ {e ∈ E : ω(e) = 1} such that 1(W ) ∈ A . For example, an open path connecting u to v is a witness for the event {u ↔ v} that u and v are connected in G[p]. Given two increasing events A and B, the disjoint occurrence A • B of A and B is defined to be the event that there exist disjoint witnesses for A and B. The van den Berg and Kesten inequality (a.k.a. the BK inequality) states that for any increasing events A , B depending on at most finitely many edges. In fact, the BK inequality applies to arbitrary product measures and does not require all edges to have the same probability of being open. It is usually unproblematic to apply the BK inequality to events depending on on infinitely many edges. For example, if A and B are increasing events for which every witness must have a finite subset that is also a witness (e.g., connection events) then we have that P p (A •B) ≤ P p (A )P p (B) by an obvious limiting argument; this applies every time we use the BK inequality in this paper.
Proof of Lemma 2.4. The lower bound is trivial. The upper bound follows by an argument very similar to that used in e.g. the proof of [Hut17, Proposition 1.12], which we include for completeness.
First sample G[p 1 ]. Independently, for each edge of G, add an additional blue edge in parallel to that edge with probability (p 2 − p 1 )/(1 − p 1 ). Thus, the subset of edges of G that are either open in G[p 1 ] or have a blue edge added in parallel to them is equal in distribution to G[p 2 ]. Denote the graph obtained by adding each of these blue edges to G[p 1 ] byG, so that τ p2 (u, v) is equal to the probability that u and v are connected inG. Letτ i (u, v) be the probability that u and v are connected by a simple path inG containing exactly i blue edges, and Considering the location of the last blue edge used in a simple path from u to v iñ G and applying the BK inequality yields that which is equivalent to the inequalitỹ for every i ≥ 0, and the claim follows.
Corollary 2.6. Let G be an infinite, connected, locally finite graph. Then for every 0 ≤ p < p q→q . In particular, T pq→q q→q = ∞.
Note that A q→q ≤ A 1→1 is bounded by the maximum degree of G.
Proof. If G has unbounded degrees then p q→q (G) = 0 for every q ∈ [1, ∞] and the claim is trivial, so suppose not. Applying Lemma 2.4 we have that , and consequently that T pq→q q→q = ∞ (the assumption that G is infinite handles the degenerate case p q→q = 1). Taking 0 ≤ p < p q→q and p = p q→q , we deduce that which implies the desired inequality.
We next state our sufficient condition for p c < p 2→2 . For each 0 ≤ p < p c , we define We interpret ι(T p ) as an isoperimetric constant that measures the extent to which percolation clusters are inclined to escape fixed finite sets: It is the Cheeger constant of the symmetric matrix χ −1 p T p , which is stochastic 2 when G is transitive and is substochastic when G is quasi-transitive.
Proposition 2.7. Let G be a connected, locally finite, quasi-transitive graph. Then In particular, if (2.3) holds then p c (G) < p u (G) and ∇ pc < ∞.
We will apply Proposition 2.7 by showing that the limit infimum in question is equal to zero under the hypotheses of Theorem 2.1. Note that if G is transitive with vertex degree k then A 2→2 = kρ(G), where ρ(G) is the spectral radius of the random walk on G.
Lemma 2.8. Let G be a connected, locally finite graph. Then Recall that a matrix is stochastic if it has non-negative entries and all of its row sums are 1, i.e., if it is the transition matrix of some Markov chain. A matrix is substochastic if it has non-negative entries and its row sums are at most 1, i.e., if it is the transition matrix of a Markov chain with killing. In the transitive case, the random walk associated to the stochastic matrix χ −1 p Tp can be interpreted as follows: At each step of the walk, we sample a size-biased percolation cluster at the vertex the walk currently occupies, independently of all previous steps, and then move to a vertex chosen uniformly at random from within this cluster. The quasi-transitive case is similar except that, at each step, the walk is killed with probability (χ p − χp(v))/χ p when it is at the vertex v.
Proof. The normalized matrixT p := χ −1 p T p is symmetric and substochastic, and ι(T p ) is its Cheeger constant. Thus, the claim follows from Cheeger's inequality, see e.g. [LP16, Theorem 6.7]. (Cheeger's inequality is usually stated for self-adjoint Markov operators but the proof is valid for self-adjoint sub-Markov operators, or, equivalently, symmetric substochastic matrices.) Proof of Proposition 2.7. The 'if' implication is immediate from Lemma 2.8 and Corollary 2.6. For the 'only if' implication, note that if p c < p 2→2 then ∇ pc < ∞ by (2.2), and consequently that there exists a constant C such that χ p ≤ C(p c − p) −1 for all p < p c , as discussed in the proof of Proposition 2.3. On the other hand, by Lemma 2.8 we must have that ι(T p ) → 1 as p → p c , and the claim follows.
Next, we prove that the following theorem can be deduced immediately from the results of [Hut17] and Proposition 2.2, so that it suffices for us to prove Theorem 2.1 in the unimodular case.
Theorem 2.9. Let G be a connected, locally finite graph, and suppose that Aut(G) has a quasi-transitive nonunimodular subgroup. Then p c (G) < p q→q (G) for every q ∈ (1, ∞).
Proof. By Proposition 2.3, it suffices to prove that p c (G) < p 2→2 (G). We use the notation of [Hut17]. Let Γ be a quasi-transitive nonunimodular subgroup of Aut(G). It follows from the proof of [Hut17, Lemma 7.1] that

Geometric Preliminaries
We now move away from the generalities of the previous section, and from now on will restrict attention to the Gromov hyperbolic setting. In this section, we provide geometric background on Gromov hyperbolic graphs that will be applied to study percolation in Sections 4 and 5. For more detailed background on various aspects of hyperbolic geometry, see e.g. [ This Riemannian metric is given explicitly by d H ((x 1 , y 1 ), (x 2 , y 2 )) = 2 log 1 and y 1 , y 2 ∈ (0, ∞) = (0, . . . , 0, 1) and γy = (0, . . . , 0, exp d H (x, y)).
Recall that a half-space in H d is a set of the form  H(a, b), which is obtained by reflecting a through the hyperplane ∂H.
Note that if H(a, b) ⊆ H d is a half-space with d(a, b) > r and c is the point on the geodesic between a and b that has d(a, c) = r, then the set x ∈ H d : d(x, a) ≤ d(x, b) + r is not itself a half-space, but is contained in the half- space H(c, b). This set also contains the r/2-neighbourhood of H(a, b), so that H(a, b) Similarly, if d is the point that lies on the infinite extension of the geodesic from b to a and has distance r from a and d(a, b) + r from b, then we have that H(b, a) H(d, b). 3.2 Hyperbolic graphs. Let G be a graph, and let x, y, w be vertices of G. (x, y)) .
Let δ ≥ 0. We say that G is δ-hyperbolic if the inequality holds for every w, x, y, z ∈ V , and that G is Gromov hyperbolic if it is δ-hyperbolic for some δ ≥ 0. (Note that while the use of the letter δ to describe this parameter is traditional, it should not generally be thought of as being small.) Roughly speaking, in a Gromov hyperbolic graph, the Gromov product (x | y) w measures the distance from w at which the geodesics from w to x and from w to y begin to diverge from each other. As mentioned in the introduction, Gromov hyperbolicity can be defined equivalently by the Rips thin triangle property.

The Bonk-Schramm theorem.
The following theorem of Bonk and Schramm [BS00] relates the abstract notion of Gromov hyperbolicity with the geometry of the concrete spaces H d . It will be the main tool by which we reason about the geometry of hyperbolic graphs in this paper. We must first introduce some definitions. Let f : for every x 1 , x 2 ∈ X. Note that this is a much stronger condition than being a rough isometry in the usual sense, and is particularly useful when discussing half-spaces.
Note that every closed convex subset X ⊆ H d is a Gromov hyperbolic, geodesic metric space when equipped with the restriction of the metric on H d .
Theorem 3.1 (Bonk and Schramm). Let G be a bounded degree, connected, Gromov hyperbolic graph. Then there exists d ≥ 1 such that G is roughly similar to a closed convex subset X ⊆ H d .

The Gromov boundary.
We now review the definition of the Gromov boundary. Proofs of the facts in this section can be found in [BS00] and references therein. Let G be a connected, locally finite, Gromov hyperbolic graph, and let w be a fixed vertex. A sequence of vertices v i i≥ in G is said to converge at infinity if lim n→∞ inf i,j≥n (v i | v j ) w = ∞. If u i i≥1 and v i i≥ are two sequences of vertices that both converge at infinity, we say that the two sequences are equivalent if (u i | v i ) w → ∞ as i → ∞. This defines an equivalence relation on the set of sequences that converge at infinity. We define the Gromov boundary δG of G to be the set of equivalence classes of this equivalence relation. Neither convergence at 780 T. HUTCHCROFT GAFA infinity or equivalence of convergent sequences depends on the choice of w. Given ξ ∈ δG and a sequence v i i≥1 in V , we write v i → ξ if v i i≥1 converges at infinity and is an element of the equivalence class ξ. Given two points ξ, ζ ∈ δG and v ∈ V we define We now define the topology on δG. If ε > 0 is sufficiently small, then there exists a metric d w,ε on V ∪ δG with the property that for every x, y ∈ V ∪ δG. We equip V ∪ δG with the topology induced by this metric, which does not depend on the choice of w or ε provided that ε is sufficiently small. If G has bounded degrees and φ : V → X is a rough similarity between G and a convex set X ⊆ H d , then φ extends to a unique continuous functionφ : V ∪ δG → X ∪ δX, and the restriction δφ ofφ to δG is a homeomorphism δφ : δG → δX.
We say that G is visible from infinity if there exists a constant C such that every vertex of G is at distance at most C from some doubly-infinite geodesic of G. Every nonamenable Gromov hyperbolic graph is visible from infinity, as is every infinite, quasi-transitive Gromov hyperbolic graph. If G is visible from infinity, then the space X in the Bonk-Schramm theorem is also visible from infinity (with the obvious extension of the definition), and is therefore easily seen to be roughly similar (with λ = 1) to the hyperbolic convex hull of its boundary δX ⊆ R d−1 ∪ {∞}. Thus, if G is visible from infinity, we can take the space X in the Bonk-Schramm Theorem to be equal to the convex hull of its boundary δX ⊆ R d−1 ∪ {∞}. E) is a Gromov hyperbolic graph, and consider the automorphism group Aut(G) of G. The the action of Aut(G) on V extends uniquely to a continuous action of Aut(G) on V ∪ δG [Woe00, Theorem 22.14]. The elements of Aut(G) can be classified as elliptic, parabolic, and hyperbolic as follows.

The action of automorphisms on the boundary. Now suppose that
1. We say that γ ∈ Aut(G) is elliptic if it fixes some finite set of vertices K ⊆ V . It is clear that these classes are mutually exclusive, and in fact we have the following.

Proposition 3.2. Let G be a Gromov hyperbolic graph. Then every γ ∈ Aut(G) is either elliptic, parabolic, or hyperbolic.
Proposition 3.2 is a direct analogue of the corresponding statement for isometries of H d , which is classical. See [KB02,§4] and references therein for a proof in the case that G is a Cayley graph of the hyperbolic group Γ, and [Woe93, Theorem 1 and Corollary 4] for a proof in full generality.
The following proposition extends this structure theory to groups of automorphisms. Let G be a Gromov hyperbolic graph and let Γ be a subgroup of Aut(G). The limit set L(Γ) of Γ is defined to be the set of accumulation points in δG of the orbit {γv : γ ∈ Γ} for some vertex v of G. The set of accumulation points does not depend on the choice of v. If Γ is quasi-transitive then L(Γ) = δG, and if G is infinite and quasi-transitive then |δG| ∈ {2, ∞}. A pair of boundary points (ξ, η) ∈ δG 2 is said to be a pole pair of Γ if there exists a hyperbolic element γ ∈ Γ with forward fixed point ξ and backward fixed point η.
The measure μ clearly assigns a positive mass to each element of O. For the remainder of the paper, we will use P and E to denote probabilities and expectations taken with respect to the law of a random root ρ drawn from this measure, use P p and E p to denote probabilities and expectations taken with respect to the joint law of the random root ρ and an independent percolation configuration G[p], and use P p and E p to denote probabilities and expectations taken with respect to the marginal law of the percolation configuration G [p].
A further important consequence of unimodularity is the following, which follows from [Woe00, Proposition 22.16] and Proposition 3.3.  H G (a, b), where a is the red square and b is the blue square, and the blue vertices represent the complement of H G (a, b). The three left-hand half-spaces are proper whereas the three right-hand half-spaces are not.
In fact, it is very unusual for Aut(G) to have the fixed set property when G is Gromov hyperbolic and quasi-transitive: This can occur only if G is rough-isometric to a tree [CCMT15,GH12].

Discrete half-spaces in hyperbolic graphs.
Let G = (V, E) be a connected, locally finite, Gromov hyperbolic graph. We say that a subset We say that a discrete half- space H G (a, b) is proper if there exist disjoint, non-empty open subsets U 1 and U 2 of δG such that U 1 is disjoint from the closure of H G (a, b) in V ∪ δG and U 2 is disjoint from the closure of H G (b, a) in V ∪ δG. See Figure 1 for examples of proper and non-proper discrete half-spaces. Now suppose that G is a bounded degree Gromov hyperbolic graph, and let Φ : V → X be a (λ, k)-rough similarity from G to a closed convex set X ⊆ H d . Then for each a, b ∈ V , the image ΦH G (a, b) of the discrete half-space H G (a, b) is contained in the set and contains the set If k > 0 these sets are not half-spaces in H d . However, it follows from the discussion in Section 3.1 that if d (Φ(a), Φ(b)) ≥ 2k and we consider the infinite geodesic in H d passing through Φ(a) and Φ(b), consider the pair of points on this geodesic at distance 2k from Φ(b), and take Φ(b) − to be the point of this pair closer to Φ(a) and Φ(b) + to be the point of the pair further from Φ(a), then we have that T. HUTCHCROFT GAFA Together with Lemma 3.8 below, which implies the corresponding statement for convex subsets of H d that are visible from infinity, this leads straightforwardly to the following basic fact about half-spaces in G.
Lemma 3.5. Let G be a bounded degree, Gromov hyperbolic graph that is visible from infinity. Then there exists a constant C such that if a, b ∈ V have d(a, b) ≥ C then the discrete half- space H G (a, b) is proper.
A further basic fact about half-spaces that will be important to us is given by the following lemma, which is a simple corollary of Proposition 3.3.
Lemma 3.6. Let G be a Gromov hyperbolic graph and let Γ be a quasi-transitive subgroup of Aut(G) that does not have the fixed set property. Then the following hold: Proof. We prove the first item, the second being similar. Let H be a proper discrete half-space of G, so that there exists a non-empty open set U in V ∪δG that is disjoint from the closure of H in V ∪δG. By Proposition 3.3, there exists a hyperbolic element γ ∈ Γ with forward and backward fixed points ξ and η both in U . Since γ n x → ξ as n → ∞ uniformly on compact subsets of V ∪ δG \ {η}, it follows that γ n H ⊆ U for sufficiently large n, and hence that γ n H and H are disjoint for all sufficiently large n.

Comparing continuum and discrete half-spaces.
In this section, we prove the following comparison between discrete and continuum half-spaces.
Lemma 3.7. Let G = (V, E) be a bounded degree, Gromov hyperbolic graph that is visible from infinity, and let Φ : V → X be a (λ, k)-rough similarity from G to some closed convex set X ⊆ H d that is equal to the convex hull of its boundary. Then there exists a constant C such that for every half-space H in We begin with the following simple geometric lemma.
Lemma 3.8. Let d ≥ 2 and let X be a closed convex subset of H d that is equal to the convex hull of its boundary. Then for every x, y ∈ X, there exists an infinite geodesic γ in X starting at x such that d(y, γ) ≤ log(1 + √ 2).
It is not hard to see by considering the case that X is an ideal triangle in H 2 that the constant log(1 + √ 2) cannot be improved. Proof. We work in the Poincaré half-space model. By applying an isometry of H d if necessary, we may assume that x = (0, . . . , 0, x d ) and y = (0, . . . , 0, 1) for some x d > 1. Since X is the convex hull of its boundary, there exist ξ, ζ ∈ R d−1 ∪ {∞} such that the hyperbolic geodesic between ξ and ζ passes through y. At least one of these points, say ξ, must lie in the closed unit disc in R d−1 , and the geodesic from x to ξ is necessarily contained in the closed cylinder lying above this unit disc. The Euclidean ball of radius 1 centred at (0, . . . , 0, √ 2) is tangent to this cylinder, and coincides with the hyperbolic ball of radius log(1 + √ 2) about y. The geodesic from x to ξ must pass through this ball, and the claim follows. See Figure 2 for an illustration. We will prove the claim with the constant C = log(17 + 12 √ 2), which is not optimal.
Proof. Let z be the point of ∂H ∩ X closest to x, and let y ∈ H d be the unique point with d(x, y) = 2d(x, z) = 2d(z, y). Note however that y need not be in X. It is clear that d(x, H(y, x) H(y, x). Indeed, by applying an isometry of H d if necessary, it suffices to consider the case that x = (0, . . . , 0, x d ), y = (0, . . . , 0, y d ), and z = (0, . . . , 0, z d ) with x d > z d > y d , so that ∂H(y, x) is represented by the Euclidean sphere that is orthogonal to R d−1 and has highest point z. In this case, the hyperbolic ball of radius d(x, z) around x is equal to a Euclidean ball B whose boundary sphere passes through z and has its center on the vertical axis, and hence is tangent to the sphere representing ∂H(y, x). If w ∈ H \ H(y, x), then the infinite geodesic passing through z and w has its highest point strictly higher than z. Thus, the tangent to the circle representing this geodesic has a positive vertical component at z, so that a point a small way along this geodesic from w to z is contained in the interior of the ball B. Since X ∩ H is convex and z was defined to be the closest point to x in H, we deduce that H ∩ X \ H(y, x) = ∅ as claimed. See Figure 2 for an illustration.
Unfortunately, we do not necessarily have that y ∈ X, and consequently are not yet done. Suppose that d(x, X ∩H) ≥ log(17+12 √ 2) and hence that d(x, H(y, x)) = d(x, z) ≥ log(17 + 12 √ 2). By Lemma 3.8, there exists z ∈ X such that z lies on an infinite geodesic in X starting at x, and d(z, z ) ≤ log(1 + √ 2), so that d(x, z ) ≥ log(17 + 12 √ 2) − log(1 + √ 2) = log(7 + 5 √ 2). Let y be the unique point in . The point y lies on the infinite geodesic from x passing through z , and so is in X by choice of z .
Thus, to complete the proof, it suffices to establish that the half-space H(y , x) contains the half-space H(y, x) and that d(x, H(y , x)) ≥ d(x, H(y, x))−log(7+5 √ 2). To see this, apply an isometry of H d so that x, y , z all lie on the vertical axis and z = (0, . . . , 0, 1), so that z lies in the Euclidean ball of radius 1 centred at (0, . . . , 0, √ 2) and x d ≥ 2. Let ξ ∈ R d−1 be the endpoint of the geodesic in H d starting at x and passing through z. Since x d > 1, ξ is in the ball of radius 1 + √ 2 about the origin in R d−1 . The half-space H(y, x) is represented by the Euclidean ball centred at ξ and whose boundary contains z. This ball has radius at most (1 + √ 2) √ 2 = 2+ √ 2, and we deduce that H(y, x) is contained in the Euclidean ball of radius 3 + 2 √ 2 about the origin in R d , which represents the half-space H(y , x) by choice of y . The distance d (x, H(y , x) Proof of Lemma 3.7. Let C be the constant from Lemma 3.9. Suppose that H is a Then it follows by Lemma 3.9 that there exists y ∈ X such that d( of y , the set and it follows by H(y, x).
3.9 Non-degeneracy. In our analysis of percolation, it will be convenient for us to place an additional geometric constraint of non-degeneracy on the space X in the Bonk-Schramm Theorem. In this subsection we introduce this property, show that is may always be assumed in the quasi-transitive setting, and then give an alternative characterisation of the property in Lemma 3.11. Let X be a convex subset of H d . We write B X (x, r) for the ball of radius r around x in X, and write B H d (x, r) for the ball of radius r around We say that X is non-degenerate if for every r < ∞ there exists R < ∞ such that for every x ∈ X and every hyperplane ∂H in H d , we have that In other words, X is non-degenerate if it does not contain arbitrarily large balls that are uniformly well-approximated by subsets of hyperplanes. If X is not nondegenerate we say that it is degenerate. Simple examples of degenerate X include geodesics between boundary points, and sets of the form ∂H ∪ K where ∂H ⊆ H d is a hyperplane and K ⊆ H d is compact.
Lemma 3.10. Let G be an infinite, quasi-transitive, Gromov hyperbolic graph. Then there exists a natural number d such that G is roughly similar to a non-degenerate closed convex subset X of H d that is the convex hull of its boundary.
Note that if G is nonamenable then we must have d ≥ 2 in this lemma.
Proof. Let d ≥ 1 be minimal such that there exists a rough similarity from G to some closed convex subset of H d , and let Φ : V → X be a (λ, k)-rough similarity from G to some closed convex subset X of H d for some λ ∈ (0, ∞) and k ∈ [0, ∞). As discussed in Section 3.3, we may assume that X is equal to the convex hull of its boundary. We claim that X must be non-degenerate. If d = 1 this holds trivially since a convex subset of H 1 is degenerate if and only if it is bounded.
Suppose then that d ≥ 2. If X is degenerate, then there exists r < ∞ such that for every R < ∞ there exists x R ∈ X and a hyperplane ∂H R in H d such 788 T. HUTCHCROFT GAFA Figure 4: Schematic illustration of the proof of Lemma 3.11. The point y must lie in the blue shaded region, which is equal to B \ C 1 . The limit point of the geodesic from x to y is bounded away from ∂H (left), and is within a bounded distance of x (right).
is compact, and so we may take a subsequential limit of the rough similarities γ xR • Φ • γ xR (all of which lie in this set) to obtain a function Φ : V → H d satisfying (3.4) and for which there exists a hyperplane ∂H in H d such that the entire set Φ V is contained in the (r + C)-neighbourhood of ∂H. Let Ψ(v) be the closest point in ∂H to Φ (v) for each v ∈ V . Then Ψ satisfies for every u, v ∈ V . If we identify ∂H with H d−1 and let Y be the closed convex hull of Ψ(v) in H d−1 , then Ψ is a rough similarity from G to Y . This contradicts the minimality of d.
The following characterisation of non-degeneracy will be particularly useful. Proof. We will prove the lemma in the case that ∞ is in the boundary of ∂H, so that ∂H is represented by a Euclidean hyperplane orthogonal to R d−1 . The proof of the general case is similar but the details are slightly more involved. In this case, the set of points of hyperbolic distance at most r from ∂H is an infinite conical prism (i.e., the product of a cone in R 2 with R d−2 ) for each r > 0. Let C 1 be the closed hyperbolic 1-neighbourhood of ∂H and let C 2 be the closed hyperbolic (1 + log(1 + √ 2))-neighbourhood of ∂H. Since X is non-degenerate, there exists a constant R such that for every x ∈ X ∩ ∂H, the hyperbolic ball of radius R around x contains some point y that is not in C 2 . Thus, by Lemma 3.8, there exists a point y that lies on an infinite geodesic in X starting from x, that is in the hyperbolic ball B of radius R = R + log(1 + √ 2) around x, and that is not in C 1 . Let ξ be the endpoint of this geodesic. The hyperbolic ball B is represented by the Euclidean ball that has its lowest point at the point (x 1 , x 2 , . . . , e −R x d ) and its highest point at the point (x 1 , x 2 , . . . , e R x d ). The geodesic from x to y is represented by a circle in R d that is orthogonal to R d−1 and intersects ∂H at an angle bounded away from 0 and π by a positive R-dependent constant. It follows that there exists an R-dependent constant C such that x − ξ ≤ Cx d and the Euclidean distance between ξ and ∂H is at least C −1 x d .

A Hyperbolic Magic Lemma
The goal of this section is to prove the following proposition, which will be of central importance to our analysis of percolation in Section 5. Intuitively, the proposition states that for every finite set of vertices A in a Gromov hyperbolic graph, from the perspective of a typical point of A, most of A is contained in either one or two distant half-spaces.
Proposition 4.1. Let G = (V, E) be a Gromov hyperbolic graph with degrees bounded by a constant M , and suppose that Φ : V → X is a (λ, k)-rough similarity from G to some closed convex set X ⊆ H d for some d ≥ 1, λ ∈ (0, ∞) and k ∈ [0, ∞). Then for every ε > 0 there exists a constant N (ε) = N M,λ,k,d (ε) such that for every finite set A ⊆ V there exists a subset A ⊆ A with the following properties: Remark 4.2. It is possible to deduce an intrinsic version of this result in which there is no embedding specified and the half-spaces are discrete.
The following corollary of Proposition 4.1 is very closely related to the fact concerning convex hulls of finite sets of vertices in nonamenable Gromov hyperbolic graphs discussed in Section 1.1. The two statements can be deduced from each other (in the connected case) by applying a suitable version of the Supporting Hyperplane Theorem. there exists a subset A ⊆ A with |A | ≥ |A|/2 such that for every u ∈ A , there We remark that Corollary 4.3 still holds without the hypothesis that A is connected, but with a longer proof. The proof of Corollary 4.3 is given at the end of this section.
We will deduce Proposition 4.1 from the following proposition, which establishes a similar statement for H d . We say that a set of points We will deduce Proposition 4.4 from the so-called Magic Lemma of Benjamini and Schramm [BS01, Lemma 2.3], which is a related statement for sets of points in Euclidean space. Benjamini and Schramm stated their lemma for R 2 , but the proof applies to R d for every d ≥ 1. (In fact, Gill [Gil14] proved that a version of the lemma holds for any doubling metric space.) Let A be a finite set of points in R d for some d ≥ 1. For each x ∈ A, the isolation radius ρ x of x is defined to be ρ x = inf{ x−y : y ∈ A\{x}}. Given x ∈ A, δ ∈ (0, 1) and s ≥ 2, we say that x is (δ, s)-supported 3 if inf y∈R d A ∩ B(x, δ −1 ρ x ) \ B(y, δρ x ) ≥ s.

GAFA PERCOLATION ON HYPERBOLIC GRAPHS 791
In other words, x is not (δ, s)-supported if there exists y ∈ R d such that all but at most s − 1 points of A are contained in either B(y, δρ x ) or R d \ B(x, δ −1 ρ x ).
Proposition 4.5 (Benjamini-Schramm Magic Lemma). Let d ≥ 1. Then for every δ ∈ (0, 1) there exists a constant C = C d (δ) such that for every finite set A ⊆ R d , at most C|A|/s of the points of A are (δ, s)-supported.
Intuitively, this lemma states that for any finite set A ⊆ R d , from the perspective of a typical point of A, most of the points of A are either far away from the point or are contained in a single small, nearby, high-density area. Similarly, for each x, y ∈ H d and δ > 0, we define H 2 (y, δx d ) to be the hyperbolic half-space that is defined to be represented by the Euclidean ball that is orthogonal to R d−1 and has its highest point at (y 1 , . . . , y d−1 , y d + δx d ), so that H 2 (y, δx d ) is the smallest half-space in H d that contains B(y, δx d ) ∩ H d . Observe that if y d ≤ 2δx d and δ ≤ 1/3 then the hyperbolic distance between x and H 2 (y, δx d ) is at least log 1/ √ 3δ. See Figure 6 for an illustration of these two half-spaces.
Note also that for every c-separated set A ⊆ H d , the number of points of A in a set B is bounded above by the ratio of the hyperbolic volume of the hyperbolic c/2neighbourhood of B to the hyperbolic volume of a hyperbolic ball of radius c/2. Let C 1 be the hyperbolic volume of the hyperbolic c/2-neighbourhood of the Euclidean ball B(x, x d /2), which does not depend on the choice of x ∈ H d , and let C 2 be the ratio of C 1 to the volume of a hyperbolic ball of radius c/2.
Let C d (δ) be the Magic Lemma constant, let s = C d (δ)ε −1 , and let N (ε) = N d,c (ε) = s + 1 + C 2 . Let A be the set of elements of A that are not (δ, s)-supported (with respect to the Euclidean metric), so that, by the Magic Lemma, |A | ≥ (1 − ε)|A|. Let x ∈ A . We claim that the following holds: There exists either a half-space H 1 or a pair of half-spaces H 1 , H 2 such that d(x, H i ) ≥ ε −1 and |A \ H i | ≤ N (ε). If ρ x ≥ δ −1/2 x d then we may simply take the single half-space H 1 = H 1 (x, ρ x ), since for this half-space we have d(x, H 1 ) ≥ ε −1 and |A \ H 1 | = 1, and hence that (4.1) holds with this choice of H 1 . Thus, it suffices to assume that c x d ≤ ρ x ≤ δ −1/2 x d . Let y ∈ R d be such that |A ∩ (B, xδ −1 ρ x ) \ B(y, δρ x )| ≤ s. Then the half-space , and one of the following holds: 1. If y d ≥ 2δρ x , then the hyperbolic c/2-neighbourhood of the Euclidean ball B(y, δρ x ) has volume at most C 1 , and hence |A ∩ B(y, δρ x )| is bounded above by C 2 . 2. If y d ≤ 2δρ x ≤ 2δ 1/2 x d , then the half-space H 2 = H 2 (y, δρ x ) has hyperbolic distance at least log 3 −1/2 δ −1/4 from x.
In the first case we take only the first half-space H 1 , whereas in the second case we take both H 1 and H 2 . In both cases we have that d(x, H i ) ≥ ε −1 and |A \ H i | ≤ N (ε) as claimed.
Proof of Proposition 4.1. Let ε > 0. Let the vertex degrees of G be bounded by M .
Since balls of radius r in G contain at most M r+1 vertices, it follows that, since Φ is a rough-similarity, there exists a constant C such that every unit ball in H d contains at most C points of Φ(V ). Let δ = min ε 3 + ε , ε C and let N (ε) = CN d,1 (δ).
Let A ⊂ V be finite, and let K ⊆ A be maximal such that Φ is injective on K and Φ(K) is 1-separated. That is, K is such that Φ is injective on K, Φ(K) is 1separated, and every point of Φ(A) is contained in the open 1-neighbourhood of Φ(K). For each v ∈ A, let v K (v) ∈ K be a vertex in K such that d(v K (v), v) < 1. Applying Proposition 4.4 to Φ(K), we deduce that there exists a subset K of K