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 $p_c<p_u$ for any such graph. Our proof also yields that the triangle condition $\nabla_{p_c}<\infty$ 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, Kesten, and Newman [2] 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 [20]. For further background on percolation, we refer the reader to [27,38,46,63].
The following conjecture was made in the highly influential paper of Benjamini and Schramm [15], who proposed a systematic study of percolation on quasi-transitive graphs, that is, graphs for which the action of the automorphism group on the vertex set has at most finitely many orbits. Theorem 1.4. Let G be a connected, locally finite, nonamenable, Gromov hyperbolic, quasitransitive graph. Then ∇ pc < ∞.
Here, a graph is said to be Gromov hyperbolic [39,40] 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 [18] 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 [7,8,26,70,83]; small cancellation 1/6 groups [40, §0.2A]; fundamental groups of compact Riemannian manifolds of negative sectional curvature [34,Chapter 3]; quasi-transitive graphs rough-isometric to simply connected Riemannian manifolds of sectional curvature bounded from above by a negative constant [34,Chapter 3]; quasitransitive CAT(−k) graphs for k > 0 [34,Chapter 3]; and quasi-transitive, nonamenable, simply connected planar maps [29]. Many surveys and monographs on hyperbolic groups have been written, and we refer the reader to e.g. [19,34,40] 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 [34,Chapter 4], and it follows from the work of Babson and Benjamini [10] 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 [22,24,59].
Previous progress on Conjectures 1.1 and 1.2 can be briefly summarised as follows. First, several works [68,71,74,77] 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 [71] proved that every nonamenable finitely generated group has a Cayley graph for which p c < p u (see also [76]). Papers that apply perturbative techniques to study specific examples, including some specific examples of hyperbolic lattices, include [23,79,82].
Let us now discuss non-perturbative results. Benjamini and Schramm [16] showed that p c < p u for every planar nonamenable quasi-transitive graph (see also [6]), generalizing earlier work of Lalley [58]. The later work of Gaboriau [31] and Lyons [61,62] 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 [16,58]. 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 [63,Theorem 9.18].
Kozma [54] 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 [50].) Schonmann [75] 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 [66]. Very recently, we established that p c < p u and ∇ pc < ∞ for every graph whose automorphism group has a quasi-transitive nonunimodular subgroup [48]. 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 [5] 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 [48], and the methods we employ here are also very different to those of that paper. Indeed, it follows from [25, 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 [1,3,11,56,57,69]. A detailed overview of how to adapt these proofs to the general quasi-transitive setting is given in [48, §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 .
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 [49].
Finally, we remark that the following corollary of Theorem 1.3 is implied by the work of Lyons, Peres, and Schramm [64]. We refer the reader to that paper and to [63,Chapter 11] for background on minimal spanning forests. Corollary 1.6. Let G be a connected, locally finite, nonamenable, Gromov hyperbolic, quasitransitive 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 quasi-transitive 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 [48] 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 [12] 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 [13] (similar observations appeared earlier in [66]). 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 [17, 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 [50] 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 [55].
Let G = (V, E) be a connected, locally finite graph, and let R V be the space of real-valued

and otherwise by
Now consider the matrix T p whose entries are given by the two-point function T p (u, v) = τ p (u, v). Since τ p (x, y) = τ p (y, x) 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 [1,9,28] 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 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 [49].
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 (Note that T n p is always well-defined as an element of [0, ∞] V 2 .) 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 [63,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 [49] for a sharp quantitative version.
Proposition 2.3. Let G be a connected, locally finite, quasi-transitive graph. If p c (G) < p 2→2 (G) then p c (G) < p q→q (G) for every q ∈ (1, ∞). 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 [1], where similar inequalities are established for the susceptibility. See also [48, §3].
Given two matrices S, T ∈ [−∞, ∞] V 2 , we write S T to mean that S(u, v) T (u, v) for every u, v ∈ V . It can easily be checked from the definitions that if S, T ∈ [0, ∞] V 2 are non-negative matrices with S T then S q→q ≤ T q→q for every q ∈ [1, ∞]. Let E → be the set of oriented edges of G. An oriented edge e has head e + and tail e − . Let A be the adjacency matrix of G, defined by letting A(u, v) = A(v, u) be the number of oriented edges with tail u and head v. Lemma 2.4. Let G be a connected, locally finite graph. Then 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 [38, §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 [48, 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 p 2 (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 letT i ∈ [0, ∞] V 2 be the matrix defined byT i (u, v) =τ i (u, v). ThenT 0 = T p 1 , and Considering the location of the last blue edge used in a simple path from u to v inG and applying the BK inequality yields that which is equivalent to the inequalityT Inducting over i yields thatT i 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 for every 0 ≤ p ≤ p ≤ 1. We deduce immediately that the set {p ∈ [0, 1] : T pq→q q→q < ∞} is open in [0, 1], 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 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. [63,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 [48] 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 quasitransitive nonunimodular subgroup. Then p c (G) < p q→q (G) for every q ∈ (1, ∞). 2 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. By Proposition 2.3, it suffices to prove that p c (G) < p 2→2 (G). We use the notation of [48]. Let Γ be a quasi-transitive nonunimodular subgroup of Aut(G). It follows from the proof of [

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. [4,19,34,40]. An overview of Gromov hyperbolicity particularly accessible to probabilists is given in [81, §20B and §22].

Hyperbolic space
Recall that for d ≥ 2, the hyperbolic d-space H d is defined to be the unique complete, connected, simply connected, d-dimensional Riemannian manifold of constant sectional curvature −1. Hyperbolic 1-space H 1 is defined to be isometric to R. Throughout this paper, we employ the Poincaré half-space model to identify equipped with the Riemannian metric given by the length element This Riemannian metric is given explicitly by for every x 1 , x 2 ∈ R d−1 and y 1 , y 2 ∈ (0, ∞). Geodesics in H d correspond to circles and lines in R d that are orthogonal to the boundary hyperplane R d−1 . The following operations all induce isometries of H d under this identification: Dilations of R d , isometries of R d fixing R d−1 , and inversions of R d ∪ {∞} through Euclidean spheres and hyperplanes that are orthogonal to R d−1 (equivalently, reflections through hyperplanes in H d ). In particular, it follows that for any two points x, y ∈ H d , there exists an isometry γ of H d such that γx = (0, . . . , 0, 1) and γy = (0, . . . , 0, exp d H (x, y)).
Recall half-space containing a, then there exists a unique b ∈ H d such that H = 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 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 Both claims can easily be verified by trigonometric calculations.

Hyperbolic graphs
Let G be a graph, and let x, y, w be vertices of G.
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 [18] 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 [18] and references therein. Let G be a connected, locally finite, Gromov hyperbolic graph, and let w be a fixed vertex.
and v i i≥ are two sequences of vertices that both converge at infinity, we say that the two sequences are 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 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 function φ : 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 ∪ {∞}.

The action of automorphisms on the boundary
Now suppose that G = (V, 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 [81,Theorem 22.14]. The elements of Aut(G) can be classified as elliptic, parabolic, and hyperbolic as follows.
1. We say that γ ∈ Aut(G) is elliptic if it fixes some finite set of vertices K ⊆ V .
2. We say that γ ∈ Aut(G) is parabolic if it fixes a unique boundary point ξ ∈ δG and 3. We say that γ ∈ Aut(G) is hyperbolic if it fixes exactly two points ξ, η of δG and We call ξ and η the forward and backward fixed points of γ respectively.
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 [52, §4] and references therein for a proof in the case that G is a Cayley graph of the hyperbolic group Γ, and [80, 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.

Unimodularity
We now briefly introduce unimodularity and the mass-transport principle, referring the reader to [63,Chapter 8] and [48, §2.1] for further background.
Let G = (V, E) be a connected, locally finite graph, and let Γ be a subgroup of Aut(G). We We say that G is unimodular if Aut(G) is unimodular. Every Cayley graph of a finitely generated group is unimodular.
If Γ is unimodular and quasi-transitive then it satisfies the mass-transport principle. Given a graph G and a subgroup Γ of Aut(G), we write [v] for the orbit of v under Γ, and let O be a set of orbit representatives of Γ, i.e., a subset of V with the property that for each u ∈ V there exists a unique v ∈ O with [u] = [v]. The mass-transport principle states that there exists a unique probability measure µ on O such that whenever ρ is a random root vertex on G with law µ and F : 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 [81,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.
2. G is nonamenable, |δG| = ∞, Γ does not have the fixed set property, and its set of pole pairs is dense in δG 2 .
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 [21,33].

Discrete half-spaces in hyperbolic graphs
Let G = (V, E) be a connected, locally finite, Gromov hyperbolic graph. We say that a subset 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 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: 1. For every proper discrete half-space H of G, there exists an automorphism γ ∈ Γ such that H and γH are disjoint.
2. For every pair of proper discrete half-spaces H 1 , H 2 of G and every finite set of vertices K of G, there exists an automorphism γ ∈ Γ such that γ( 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 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)) = d(x, H ∩ X). We claim that H ∩ 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 Thus, to complete the proof, it suffices to establish that the half-space H(y , x) contains the halfspace 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)) is equal to d(x, z )−log(3+2 Proof of Lemma 3.7. Let C be the constant from Lemma 3.9. Suppose that H is a half-space in H d with H ∩ X = ∅ and that v ∈ V is such that d(Φ(v), X ∩ H) ≥ C . Then it follows by Lemma 3.9 that there exists y ∈ X such that d(Φ(v), H(y, Φ(v))) = d(Φ(v), y)/2 ≥ d(x, H ∩ X) − C . Let C be a large constant to be chosen and let y lie on the geodesic from Φ(v) to y and satisfy d(Φ(v), y ) = d(Φ(v), y) − C . If C is sufficiently large, then whenever d(Φ(v), X ∩ H) ≥ 2C and y is within distance k of y , the set and it follows by choice of C that H G (u, v) ⊇ Φ −1 H = Φ −1 H(y, x).

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 x in H d , so that B X (x, r) = B H d (x, r) ∩ X. 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 non-degenerate 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 that B X (x R , R) ⊆ y∈∂H R B H d (y, r). Let v 0 be a fixed vertex of G. Since G is quasi-transitive, there exists a constant C such that for each x ∈ X, there exists γ x ∈ Aut(G) such that d(Φ(γ x (v 0 )), x) ≤ C. Let γ x be an isometry of H d mapping Φ(γ x (v 0 )) to (0, . . . , 0, 1). The set of functions φ : V → H d satisfying φ(v 0 ) = (0, . . . , 0, 1) and is compact, and so we may take a subsequential limit of the rough similarities γ x R • Φ • γ x R (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 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).
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.
Lemma 3.11. Let d ≥ 2, identify H d with R d + via the Poincaré half-space model, and let X ⊆ H d be a closed, convex, non-degenerate subset of H d that is equal to the convex hull of its boundary. There exists a constant C such that for every hyperplane ∂H ⊆ H d and every x ∈ ∂H ∩ X there exists ξ ∈ R d−1 such that ξ ∈ δX, x − ξ ≤ Cx d and the Euclidean distance between ξ and ∂H is at least C −1 x d .
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   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.
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 A ⊆ H d is c-separated if d(x, y) ≥ c for every two distinct x, y ∈ A.

2.
For every x ∈ A , there either exists 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).
We will deduce Proposition 4.4 from the so-called Magic Lemma of Benjamini and Schramm [17,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 [35] 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 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 ).
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 , . . . , 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/2-neighbourhood 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, ε > 0 and let A be a finite c-separated subset of H d . Since A is c-separated with respect to the hyperbolic metric, the Euclidean isolation radius ρ x is at least c x d for every x ∈ A, where 0 < c = 1 − e −c ≤ 1. Take δ = δ(ε, c) > 0 so that 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). (4.1) 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 halfspace 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 , 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 .
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 Figure 7: Illustration of the proof of Corollary 4.3. Left: If the boundary of X is contained in the union of two half-spaces that are both far away from Φ(v), then G has a large ball around Φ(v) that is rough-isometric to a line segment. Right: Illustration of the half-spaces that are discussed.
exists a constant C such that every unit ball in H d contains at most C points of Φ(V ). Let Let A ⊂ V be finite, and let K ⊆ A be maximal such that Φ is injective on K and Φ(K) is 1separated. That is, K is such that Φ is injective on K, Φ(K) is 1-separated, and every point of Applying Proposition 4.4 to Φ(K), we deduce that there exists a subset K of K such that |K | ≥ (1 − δ)|K| and for every v ∈ K , there either exists a half-space H 1,v or a pair of half-spaces It follows from the discussion in Section 3.1 that for each i, there exists a half-space and that for every v ∈ A , there exists either a half-space This completes the proof.
Proof of Corollary 4.3. Let G = (V, E) be a bounded degree, nonamenable, Gromov hyperbolic graph, and, applying the Bonk-Schramm Theorem, 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 for some d ≥ 1, λ ∈ (0, ∞), and k ∈ [0, ∞). First note that there exists a constant ε 0 ≤ 1 such that whenever v ∈ V and H 1 , H 2 are any two half-spaces in H d with d(Φ(v), H i ) ≥ ε −1 0 , then there exists ξ ∈ δX that is not in the closure of H i in R d−1 ∪{∞}. Indeed, if this were not the case then G would contain arbitrarily large regions roughly similar to line segments (with uniform constants), contradicting the nonamenability of G. See Figure 7 for an illustration. (This claim also follows, and is in fact equivalent to, the fact that the Gromov boundaries of nonamenable Gromov hyperbolic graphs are uniformly perfect [67].) Now suppose that H 1 = H(y 1 , Φ(v)) and H 2 = H(y 2 , Φ(v)) are half-spaces in H d such that d(Φ(v), H i ) ≥ 2ε −1 0 . Let y i be the point that lies on the geodesic from Φ(v) to y i and has distance 2ε −1 0 from Φ(v). Let H 1 = H(y 1 , Φ(v)) and H 2 = H(y 2 , Φ(v)), so that d(Φ(v), H i ) ≥ ε −1 0 . By the above discussion, there exists ξ ∈ δX that is not in the closure of H i in R d−1 ∪ {∞}. Let r 0 be a large constant to be chosen and let z(v, r) be the point that lies at distance r ≥ r 0 from Φ(v) along the geodesic from r to ξ. If r 0 is chosen to be sufficiently large (depending on the choice of ε 0 ), then the half-space H(z(v, r), Φ(v)) is disjoint from H i . (The constant r 0 can be chosen independently of ξ thanks to the 'buffer regions' H 1 and H 2 . Indeed, the worst case occurs when ξ lies in the boundary of H 1 or H 2 .) Now, let A ⊆ V be finite and connected, and apply Proposition 4.1 with ε = ε 0 /2 ≤ 1/2. Let A ⊆ A be as in the statement of that proposition. Then for every v ∈ A there exists either a half-space H 1 ⊆ H d or a pair of half-spaces H 1 , . Thus, applying the above discussion with this choice of half-spaces, there exists r 0 such that if r ≥ r 0 then the half-space H(z(v, r), Φ(v)) is disjoint from H i . On the other hand, since A is connected and Φ is a rough similarity, there exists a constant r 1 such that if for every r ≥ r 0 + N (ε)r 1 . Since z(v, r) ∈ X, the proof may easily be concluded by an application of Lemma 3.7.
Remark 4.6. Proposition 4.4 can also be used to prove that if G n n≥1 is a sequence of finite, δ-hyperbolic graphs with degrees bounded by M for some δ ≥ 0 and M < ∞ independent of n, then any subsequential Benjamini-Schramm limit of G n n≥1 has at most two points in its Gromov boundary almost surely. This application is similar in spirit to the original use of the Magic Lemma to study circle packings of Benjamini-Schramm limits of finite planar graphs [17].

Analysis of percolation
In this section, we complete the proofs of our main results, namely Theorems 1.3, 1.4 and 2.1. Using Proposition 2.7, it suffices to prove the following two propositions, which are proven in Section 5.1 and Section 5.2 respectively. Let us now briefly indicate why these two propositions imply our main theorems.
Proof of Theorems 1.3, 1.4 and 2.1 given Propositions 5.1 and 5.2. If G is nonunimodular then we have that p c (G) < p q→q (G) for every q ∈ (1, ∞) by Theorem 2.9. If G is unimodular, then combining Propositions 5.1 and 5.2 yields that, since p c (G) < 1, Applying Proposition 2.7 we deduce that p c (G) < p 2→2 (G). Applying Propositions 2.2 and 2.3, Theorems 1.3 and 1.4 follow immediately.
Let us note that the proof of Proposition 5.2 uses the full power of Proposition 4.1, whereas the proof of Proposition 5.1 uses only Corollary 4.3.

The susceptibility exponent
In this section we apply Corollary 4.3 to prove Proposition 5.1. The argument is similar to that appearing in [66, §5]. The key input is the following lemma, which is an easy consequence of Corollary 4.3. We write χ p = E p |K ρ | for the expected value of the cluster of the random root ρ as discussed in Section 3.6. Lemma 5.3. Let G be a unimodular, quasi-transitive, Gromov hyperbolic, nonamenable graph. Then there exist positive constants c and r such that for each 0 ≤ p < p c there exists v, u ∈ V with d(v, u) ≤ r such that H G (v, u) is proper and Proof of Lemma 5.3. For each v ∈ V , A ⊆ V , and r ≥ 1, define for every finite A ⊆ V . It follows from the mass-transport principle that for every p < p c . It follows trivially that there exists a vertex v ∈ V such that and hence that there exists u with d(u, v) ≤ r such that H G (v, u) is proper and Before beginning the proof of Proposition 5.1, let us recall the notion of Dini derivatives and some elementary facts about their calculus, and then state Russo's formula. Let f be a measurable function defined on [0, 1]. The lower right and upper right Dini derivatives of f are defined respectively to be d dp If f is decreasing then d dp + f (p) dp for every p 1 < p 2 . Finally, we have that, similarly to the usual chain rule, d dp See e.g. [51] for further background on Dini derivatives. We now recall Russo's formula. Recall that, given an increasing event A ⊆ {0, 1} E and a percolation configuration G[p], an edge e of G is said to be pivotal for the event A if A occurs in the configuration obtained from G[p] by changing the status of e to be open, but A does not occur in the configuration obtained from G[p] by changing the status of e to be closed. Suppose that A is an increasing event depending on only finitely many edges. Russo's formula [38,Theorem 2.25] states that P p (A) is differentiable and that d dp P p (A) = e∈E P p (e is pivotal for A) = 1 1 − p e∈E P p (e is closed and pivotal for A).
On the other hand, if A is an increasing event depending on infinitely many edges, then Russo's formula ceases to be an equality in general, and we obtain instead the lower right Dini derivative lower bound d dp + P p (A) ≥ e∈E P p (e is pivotal for A) = 1 1 − p e∈E P p (e is closed and pivotal for A).
Proof of Proposition 5.1. It suffices to show that there exists a positive constant C such that d dp + χ p ≥ C −1 χ 2 p (5.1) for all p c /2 ≤ p < p c . The conclusion of the theorem will then follow since we can integrate this differential inequality to obtain that for every p c /2 < p < p c . We conclude since We now begin to work towards a proof of a differential inequality of the form (5.1). It follows from Russo's formula that for every 0 < p < p c and v ∈ V . (In fact it is not difficult to show that χ p (v) is differentiable on (0, p c ) and that this inequality is an equality, but we will not need this.) Averaging over v and applying the mass-transport principle, we deduce that Let the constants r and c be as in Lemma 5.3. Since G is quasi-transitive, there are only finitely many isomorphism classes of pairs (v, u) with d(u, v) ≤ r. Thus, by Lemma 3.6, there exists a constant R such that for every v, u such that H G (v, u) is proper and d(u, v) ≤ r, there exists an automorphism γ v,u ∈ Aut(G) such that Now let 0 < p < p c and let v, u be as in Lemma 5.3. Write H = H G (v, u) and γ = γ v,u . Noting that the events {|K v | = n, K v ⊆ H} and {|K γv | = m, K γv ⊆ γH} depend on disjoint sets of edges and are therefore independent, we have that Let η be the geodesic from v to γv, so that η has length at most R. Let e be the first oriented edge of G that is crossed by η and is in ∂ E H. We claim that there exists a p-dependent constant c p , bounded on compact subsets of (0, 1), such that This follows by a standard finite energy argument: Given any percolation configuration in which K v ⊆ H and K γv ⊆ γH, we can modify the configuration, changing the status only of edges that are either contained in or incident to η, in such a way that the following hold: 1. The clusters of v and γv are both at least as large after the modification as they were originally.
2. In the modified configuration, v is connected to e − and γv is connected to e + , but e − is not connected to e + .
Indeed, to perform such a modification, decompose η = η 1 • e • η 2 into the pieces before and after crossing e, and make the following changes: 1. Open every edge of η 1 and every edge of η 2 , and close e.
2. Close every edge incident to (but not contained in) η 1 that does not already have both endpoints connected to v in the original configuration.
3. Close every edge incident to (but not contained in) η 2 that does not already have both endpoints connected to γv in the original configuration.
It is easily verified that this modification has the required properties, and we deduce the claimed inequality (5.4). Putting together (5.2), (5.3), and (5.4), and using the fact that p c (G) < 1, we deduce that there exists a constant C such that for all p c /2 ≤ p ≤ p c as desired.

The expected number of connections to a far hyperplane at criticality
In this section we prove Proposition 5.2. The rough idea of the proof is that, by Proposition 4.1, for any finite set A, most of A is contained in one or two far-away half-spaces, and these half-spaces do not contribute much to the susceptibility. In order to carry out this idea, we will prove the following estimate. The exact form of the bound in Lemma 5.4 is not important; we only need that the contribution to the susceptibility from a distant half-space is uniformly small.
Lemma 5.4. Let G = (V, E) be a unimodular, quasi-transitive Gromov hyperbolic graph, 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 ≥ 2, λ ∈ (0, ∞) and k ∈ [0, ∞). Suppose further that X is non-degenerate and is equal to the convex hull of its boundary. Then there exists a constant C such that for every vertex v, every half-space H ⊆ H d , and every 0 ≤ p < p c .
Before proving Lemma 5.4, let us see how it implies Proposition 5.2.
Proof of Proposition 5.2 given Lemma 5.4. By Lemma 3.10 there exists d ≥ 2, λ ∈ (0, ∞), k ∈ [0, ∞) and a (λ, k)-rough similarity Φ : V → X from G to some closed, convex set X ⊆ H d that is non-degenerate and equal to the convex hull of its boundary. For each ε > 0, let N (ε) be as in Proposition 4.1. For each 0 < p < p c , let ε p > 0 be infimal such that N (ε) ≤ (χ p ) 1/2 , so that ε p ↓ 0 as p ↑ p c . Thus, for each 0 < p < p c and each finite set A ⊆ V there exists a set A ⊆ A such that |A | ≥ (1 − 2ε p )|A| and for each u ∈ A there exists either a half-space H 1,u or a pair of half-spaces H 1,u , H 2,u in H d such that d(Φ(u), H i,u ) ≥ (2ε p ) −1 and |A \ Φ −1 H i,u | ≤ N (2ε p ) ≤ (χ p ) 1/2 . Thus, considering the various contributions to u,v∈A τ p (u, v) we have that u,v∈A and using Lemma 5.4 to bound the third term we deduce that u,v∈A where C is the constant from Lemma 5.4. It follows that which tends to zero as p ↑ p c .
We now begin to work towards the proof of Lemma 5.4, beginning with some general considerations. Let G be a quasi-transitive graph, and let v 0 be a fixed root vertex of G. We say that a (not necessarily finite) set K ⊆ V is r-roughly branching if it is empty or if there exists a subset K ⊆ K and a family of automorphisms {γ v : v ∈ K } ⊆ Aut(G) with the following properties: 1. K is contained in the r-neighbourhood of K in G.
3. For every k ≥ 1, if v 1 , . . . , v k and u 1 , . . . , u k are two distinct sequences of elements of K , then For example, suppose that T is a k-regular tree. Fix an orientation of T so that every vertex has one parent and k − 1 children and, for each vertex, fix an ordering of the children. For each two vertices u, v of T , there is a unique automorphism γ u,v of T that maps u to v and preserves all of the information regarding the orientation and the ordering of the children. If v 0 is a fixed root vertex of T and K is the set of descendants of v 0 that are n generations below v 0 , then K is 0-roughly branching, as can be seen by taking the automorphisms γ v = γ v 0 ,v . Proof. Let K and {γ v : v ∈ K } be as in the definition of K being r-roughly branching. Let k ≥ 1 and let (K ) k be the set of sequences of elements of K of length k. For each u = (u 1 , . . . , u k ) ∈ (K ) k , let γ(u) = γ u k • γ u k−1 • · · · • γ u 1 . For each 1 ≤ i ≤ k, let γ i (u) = γ u i • · · · • γ u 1 , and let γ 0 (u) be the identity automorphism. Then, by the Harris-FKG inequality, Since γ(u)v 0 = γ(u )v 0 for distinct u, u ∈ (K ) k , it follows that When p < p c , the left-hand side is finite and does not depend on k. Thus, we deduce that the right-hand side must be bounded in k, which implies that for every 0 < p < p c . Since τ p (u, v) is non-negative and left-continuous in p for every u, v ∈ V (see e.g. [47,Lemma 5]) the same inequality also holds at p = p c .
It follows that there exists a constant C such that for every 0 < p ≤ p c .
To apply Lemma 5.5 in our setting, we will show that certain sets arising as discrete approximations to hyperplanes in quasi-transitive Gromov hyperbolic graphs are roughly branching.
For the remainder of this section, we fix a unimodular, quasi-transitive, nonamenable, Gromov hyperbolic graph G with fixed root vertex v 0 , and take Φ : V → X be a (λ, k)-rough similarity from G to a closed, convex, non-degenerate set X ⊆ H d that is equal to the convex hull of its boundary for some λ ∈ (0, ∞), k ∈ [0, ∞) and d ≥ 2. Our next goal is to prove the following.
Lemma 5.6. There exist positive constants r, R, and C ≥ k such that whenever n ≥ 1 and Φ(v 0 ) = x 0 , x 1 , . . . , x n all lie on a geodesic in H d and have d(x 0 , x i ) = 4ri for every 1 ≤ i ≤ n, the set Figure 8: Schematic illustration of the some of the sets arising in the proof of Lemma 5.6.
The details regarding the choices of constants in the following proof can be dealt with via similar explicit computations to those performed in the proof of Lemma 3.9.
Proof. Since G is quasi-transitive, there exists C 1 such that the distance from u to the Aut(G)-orbit of v 0 is at most C 1 for every u ∈ V . Since Φ is a rough similarity, it follows that there exists a constant C 2 such that for every x ∈ X, the distance in H d between x and the image under Φ of the orbit of v 0 is at most C 2 . We will take the constant C in Lemma 5.6 to be this C 2 .
Let A i be the hyperbolic is also by choice of C 2 and by convexity, so we may assume without loss of generality that each of the sets K i is non-empty, reducing to a smaller value of n or to a triviality otherwise. Using the Poincaré half-space model and applying an isometry of H d if necessary, we may assume that x 0 = Φ(v 0 ) = (0, . . . , 0, 1) and x i = (0, . . . , 0, e −4ri ) for each i = 1, . . . , n, so that the hyperplane ∂H(x i , x i+1 ) is represented by the Euclidean sphere orthogonal to R d−1 that has its highest point at (0, . . . , 0, e −4ri−2 ). Let B i be the half-space in H d that has its highest point at (0, . . . , 0, e −ir−2r ). There exists a constant r 0 such that if r ≥ r 0 then A i ⊆ B 4i−1 \ B 4i+1 for each 1 ≤ i ≤ n, so that in particular the sets A i are disjoint. Since Φ −1 B 1 contains Φ −1 A 1 is therefore non-empty, we can apply Lemma 3.7 to B 1 to deduce that there exist constants r 1 and C 3 such that if r ≥ r 1 then there exists a proper discrete half-space Since X is non-degenerate and equal to the convex hull of its boundary, we can apply Lemma 3.11 to deduce that there exists a constant C 4 , that does not depend on r, such that for each x ∈ A i there exists ξ(x) ∈ R d−1 ∩ δX such that the Euclidean distance between x and ξ(x) is at most C 4 x d and the Euclidean distance between ξ(x) and A i is at least C −1 4 x d . It follows by a straightforward trigonometric calculation that there exist constants r 2 and C 5 such that if r ≥ r 2 and x ∈ A i satisfies x d ≤ C −1 5 e −4ri−2r then the geodesic from x to ξ(x) is contained in B 4i−1 \ B 4i+1 for every 1 ≤ i ≤ n and x ∈ A i . We now fix r = r 0 ∨ r 1 ∨ r 2 , and let A i be the set of x ∈ A i that are in the orbit of v 0 and satisfy x d ≤ C −1 5 e −4ri−2r . Let C 6 be a large constant to be chosen and let y(x) ∈ X be the point that lies on the geodesic from x to ξ(x) and has d H (x, y(x)) = C 6 . If C 6 is sufficiently large, then the half-space H(y(x), x) is contained in B 4i−2 \ B 4i+2 and has Euclidean distance at least C −1 4 x d /2 from A i for each 1 ≤ i ≤ n and x ∈ A i . Taking a suitably large constant r 3 , we can find for each . In particular, if v 1 ∈ A i and v 2 ∈ A j for i = j then the discrete half-spaces H G (u(v 1 ), v 1 ) and H G (u(v 2 ), v 2 ) are disjoint. See Figure 8 for an illustration.
If the constant C 7 is chosen to be sufficiently large then for every pair of distinct points x 1 , x 2 in A i with d(x 1 , x 2 ) ≥ C 7 , the hyperbolic half-spaces H(y(x 1 ), x 1 ) and H(y(x 2 ), x 2 ) are disjoint. We take K i to be a maximal subset of K i with the property that Φv ∈ A i for every v ∈ K i and such that ΦK i is C 7 -separated, and define K = n i=1 K i . Thus, we have that there exists a constant C 8 such that K is contained in the C 8 neighbourhood of K in G, and for any two v 1 , v 2 ∈ K , the half-spaces H G (u(v 1 ), v 1 ) and H G (u(v 2 ), v 2 ) are disjoint.
Recall that u 0 was defined so that d(v 0 , u 0 ) ≤ C 3 r and that H G (u 0 , v 0 ) is a proper discrete half-space containing Φ −1 B 1 . For each v ∈ V that is in the same orbit of Aut(G) as v 0 , let γ 1,v ∈ Aut(G) be such that γ 1,v v 0 = v. Applying Lemma 3.6, we have that for every v ∈ K there exists an automorphism γ 2,v ∈ Aut(G) such that γ 2,v γ 1,v Since there are only finitely many possibilities for the isomorphism class of the tuple (v, u(v), γ 1,v v 0 , γ 1,v u 0 ), we can choose the automorphisms γ 2,v in such a way that there exists a constant r 4 such that d( For each k ≥ 1 and each sequence v = (v 1 , v 2 , . . . , v k ) ∈ (K ) k , let γ(v) = γ v k • · · · • γ v 1 . To complete the proof, it suffices to prove that γ(v)v 0 = γ(v )v 0 for every k ≥ 1 and every two distinct v, v ∈ (K ) n . To do this, we will prove the following claim by induction on k ≥ 1: The sets γ(v) H G (u 0 , v 0 ) ∪ {v 0 } and γ(v ) H G (u 0 , v 0 ) ∪ {v 0 } are disjoint whenever v, v are distinct elements of (K ) k . (5.5) First observe that, by definition of γ v , we have that γ v H G (u 0 , v 0 ) ∪ {v 0 } ⊆ H G (u(v), v) for every v ∈ K , and since H G (u(v), v) ⊆ H G (u 0 , v 0 ), it follows by induction that for every k ≥ 1 and v ∈ (K ) k . Since H G (u(v), v) and H G (u(v ), v ) are disjoint whenever v, v ∈ K are distinct, this immediately establishes (5.5) in the base case k = 1. Now suppose that (5.5) has been established for sequences of length k, and suppose that v, v ∈ (K ) k+1 are distinct.
If v k+1 = v k+1 , then it follows from (5.6) that Proof of Lemma 5.4. Let C and r be as in Lemma 5.6. Let v 0 be a fixed root vertex of G. Since G is quasi-transitive, it suffices to prove the claim with constants that may a priori depend on the choice of v 0 . Let H be a half-space in H d , and let n = d(Φ(v 0 ), H)/4r . Let Φ(v 0 ) = x 0 , x 1 , . . . , x n ∈ H d lie on the geodesic from Φ(v 0 ) to H and have d(x 0 , x i ) = 4ri for each 1 ≤ i ≤ n. Let K i = {u ∈ V : d(u, ∂H(x i , x i+1 )) ≤ C}. By Lemma 5.6 and Lemma 5.5, there exists a constant C such that and hence that there exists some 1 ≤ i ≤ n such that E pc |K v 0 ∩ K i | ≤ C n −1 . Since any path from v 0 to Φ −1 H must pass through the set K i for every 1 ≤ i ≤ n, it follows by the BK inequality that as desired.

Closing discussion and open problems
It is natural to conjecture that the following strengthening of Conjectures 1.1 and 1.2 also holds.
By Proposition 2.3, this conjecture is equivalent to the statement that p c < p 2→2 under the same hypotheses. We remark that the perturbative proofs of p c < p u in [15,68,71,74] also implictly establish the stronger result p c < p 2→2 , so that in particular every nonamenable finitely generated group has a Cayley graph with p c < p 2→2 . By Theorem 2.9, to prove Conjecture 6.1 it would suffice to consider the unimodular case. The proofs of [16,31,58,61,62] establish p c < p u without establishing p c < p 2→2 . Further consequences of Conjecture 6.1 are investigated in [49].
The following conjecture would further strengthen Conjecture 6.1. The difficult part of the conjecture should be to prove that p q→q < p 2→2 for every q ∈ [1, 2). It can be deduced from the methods of [50] that Conjecture 6.2 holds for the product of finitely many trees each of which is regular of some degree ≥ 3. Conjecture 6.2. Let G be a connected, locally finite, quasi-transitive, nonamenable graph. Then p q→q (G) is a continuous, strictly increasing function of q on [1, 2].

Non-uniqueness at p 2→2 .
We remark that several estimates concerning critical percolation that are proven using supermultiplicativity arguments as in [47,50] can easily be adapted to establish related estimates at p 2→2 and p q→q . For example, the methods of [47] can easily be generalized to establish the following. Note that Proposition 6.3 also implies that p q→q (G) ≤ gr(G) −(q−1)/q < 1 for every quasitransitive nonamenable graph G and every q ∈ [2, ∞]. This bound and that of Proposition 6.3 are both attained for all q ∈ [2, ∞] when G is a k-regular tree for some k ≥ 3.
Finally, we remark that the methods of [50] can be used to show that the estimate known as Schramm's Lemma continues to apply at p 2→2 . The fact that this bound holds at p c was originally proven for unimodular transitive graphs by Schramm, see [54]. Proposition 6.4. Let G be a connected, locally finite, transitive, nonamenable graph, let X be simple random walk on G, and let ρ(G) < 1 be the spectral radius of G. Then E τ p (X 0 , X n ) ≤ ρ(G) n for every 0 ≤ p ≤ p 2→2 .
Proof. We omit some details since the proof is very similar to that of [50,Theorem 3.1]. First suppose that 0 ≤ p < p 2→2 . Let P be the Markov operator for simple random walk on G. We have by Cauchy-Schwarz that E τ p (X 0 , X n ) = T p 1 v , P n 1 v ≤ T 2 p 2→2 P 2n 2→2 .
This completes the proof in the case 0 ≤ p < p 2→2 . The case p = p 2→2 follows by the same left-continuity argument used in [50].
6.2 Some speculation Proposition 6.3 implies in particular that there cannot be a unique infinite cluster at p 2→2 on any quasi-transitive nonamenable graph, and hence that any quasi-transitive graph G that has a unique infinite cluster at p u must have p 2→2 < p u . This motivates the following problem.