The $L^2$ boundedness condition in nonamenable percolation

Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities $T_{p_c}(u,v)=\mathbb{P}_{p_c}(u\leftrightarrow v)$ is bounded as an operator $T_{p_c}:L^2(V)\to L^2(V)$ and proved that this conjecture holds for several classes of graphs. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour. In this paper we further investigate the consequences of the $L^2$ boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all $p<p_{2\to 2}$; ii) If $p_c<p_{2\to 2}$, then the critical exponent governing the extrinsic diameter of a critical cluster is $1$; iii) Below $p_{2\to 2}$, percolation is"ballistic"in the sense that the intrinsic distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If $p_c<p_{2\to 2}$, then $\|T_{p_c}\|_{q\to q} \asymp (q-1)^{-1}$ and $p_{q\to q}-p_c \asymp q-1$ as $q\downarrow 1$. v) If $p_c<p_{2\to 2}$, then various 'multiple-arm' events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order $(p-p_c)^3$ as $p \downarrow p_c$. All of these results are new even in the Gromov hyperbolic case. Finally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.


Introduction
Let G = (V, E) be a connected, locally finite graph and let p ∈ [0, 1]. In Bernoulli bond percolation, we independently declare each edge of G to either be open or closed, with probability p of being open, and let G[p] be the subgraph of G formed by deleting every closed edge and retaining every open edge. The connected components of G[p] are referred to as clusters. We are particularly interested in phase transitions, which occur when qualitative features of G[p] change abruptly as p is varied through some critical value. The two most interesting such phase transitions occur at the critical probability, which is defined to be p c = p c (G) = inf p ∈ [0, 1] : G[p] has an infinite cluster a.s. , and the uniqueness threshold, which is defined to be p u = p u (G) = inf p ∈ [0, 1] : G[p] has a unique infinite cluster a.s. .
Many of the most interesting questions in percolation theory concern the behaviour of percolation at and near these critical values.
Historically, percolation was studied primarily on Euclidean lattices such as the hypercubic lattice Z d . In this case, Aizenman, Kesten, and Newman [2] proved that there is at most one infinite cluster and hence that p c = p u . An alternative proof of the same fact was later found by Burton and Keane [10]. In their celebrated paper [8], Benjamini and Schramm proposed a systematic study of percolation on general quasi-transitive graphs, that is, graphs for which the action of the automorphism group on the vertex set has at most finitely many orbits. (In other words, graphs for which 'there are only finitely many different types of vertices'.) They made the following conjecture. Here, a graph G = (V, E) is said to be nonamenable if there exists a positive constant c such that |∂ E K| ≥ c v∈K deg(v) for every finite set of vertices K in G, where ∂ E K denotes the set of edges with one endpoint in K and one in V \ K. Both proofs of uniqueness in Z d can be generalized to show that p c (G) = p u (G) for every amenable quasi-transitive graph, so that only the 'if' direction of Conjecture 1.1 is open. While various special cases of Conjecture 1.1 have now been verified [9,15,22,24,31,34,39,42], it remains completely open in general. We refer the reader to [11,16,20] for background on percolation, and to [17] and the introduction of [24] for a survey of progress on Conjecture 1.1 and related problems. Conjecture 1.1 is closely related to understanding critical exponents in percolation on nonamenable graphs. It is strongly believed that critical percolation on nonamenable transitive graphs has mean-field behaviour. Roughly speaking, this means that critical percolation on a nonamenable transitive graph should behave similarly to critical branching random walk on the same graph. In particular, one expects that various critical exponents exist and take the same value as on a 3-regular tree, so that for example we expect that where v is a vertex, K v is the cluster of v, and we write ≍ to denote an equality that holds up to positive multiplicative constants in the vicinity of the relevant limit point. A well-known signifier of mean-field behaviour is the triangle condition, which is said to hold if for every vertex v of G. The triangle condition was introduced by Aizenman and Newman [3], and has subsequently been shown to imply that various exponents exist and take their mean-field values [3,6,22,29,30,38]. In particular, if G is a connected, locally finite, quasi-transitive graph that satisfies the triangle condition then the estimates (1.1), (1. 2) hold on every quasi-transitive graph [1,3].) See [16,Chapter 10] and [22,Section 7] for an overview of the relevant literature. The following conjecture is widely believed.
As with Conjecture 1.1, this conjecture is known to hold in a variety of special cases [9,22,24,27,31,39,42] but remains completely open in general. Some much weaker unconditional results on the critical behaviour of percolation in the nonamenable setting have however been recently been obtained in [23]. In the Euclidean context, the triangle condition is believed to hold on Z d if and only if d > 6, and was proven to hold for large d in the milestone work of Hara and Slade [18]. The record is now held by Fitzner and van der Hofstad [14], who proved that it holds for all d ≥ 11. It is reasonable to conjecture that the triangle condition holds on every transitive graph of at least seven dimensional volume growth; nonamenability should be greatly surplus to requirements.
In our recent work [24], we made the following strong quantitative conjecture which implies both Conjectures 1.1 and 1.2. Let τ p (u, v) be the probability that u and v are connected in G[p], which is known as the two-point function. Given a non-negative matrix M ∈ [0, ∞] V 2 indexed by the vertices of G and p, q ∈ [1, ∞], we define M p→q : , the norm of M as an operator from L p (V ) to L q (V ). Let T p ∈ [0, ∞] V 2 be the two-point matrix defined by T p (u, v) = τ p (u, v), and define the critical values It is easily seen that T p 1→1 = T p ∞→∞ = sup v∈V u∈V τ p (v, u) = sup v∈V E p |K v |, and therefore by sharpness of the phase transition [1,12] that p c (G) = p 1→1 (G) = p ∞→∞ (G) for every quasitransitive graph G.
It is shown in [24] that Conjecture 1.3 is implied by the weaker statement that T pc 2→2 < ∞, and implies the stronger statement that p c (G) < p q→q (G) for every q ∈ (1, ∞); a quantitative version of this fact is given in Theorem 4.1.
As a heuristic justification of Conjecture 1.3, we note that if G is a quasi-transitive graph then the the two-point function of the critical branching random walk on G (which coincides with the simple random walk Greens function) is bounded as an operator on L 2 (V ) if and only if G is nonamenable [44,Theorem 10.3]. Thus, Conjecture 1.3 is predicted by the general philosophy that, in the high-dimensional context, critical percolation should behave similarly to critical branching random walk. Conjecture 1.3 is known to hold in most of the special cases in which either Conjecture 1.1 or Conjecture 1.2 had been established. Indeed, the proofs that p c < p u holds under various perturbative assumptions such as small spectral radius [39], large Cheeger constant [42], or high girth [37] also implicitly yield the stronger claim that p c < p 2→2 under the same assumptions. In particular, it can be deduced from the work of Pak and Smirnova-Nagnibeda [39] that every finitely generated nonamenable group has a Cayley graph for which p c < p 2→2 , which lends very strong evidence that the conjecture is true in general. As for non-perturbative results, it is now know that Conjecture 1.3 holds under the additional assumption that G is either Gromov hyperbolic [24] or has a quasi-transitive nonunimodular subgroup of automorphisms [22]. The latter condition holds, for example, if G = T k × H is the Cartesian product of a regular tree of degree k ≥ 3 with a quasi-transitive graph H. A notable exception is given by groups of cost > 1, which are known to have p c < p u [15,34] but are not known to have p c < p 2→2 or to satisfy the triangle condition at p c . (As a modest first step in this direction, one could try to prove p c < p 2→2 for infinitely-ended transitive graphs.) In this paper, we further explore the consequences of Conjecture 1.3. We find in particular that Conjecture 1.3 has various very strong consequences that are not known to follow from Conjecture 1.1 or Conjecture 1.2 alone. Our results can briefly be summarized as follows: 1. Theorem 2.2: If p < p 2→2 , then the two-point function τ p (u, v) is exponentially small in the graph distance d(u, v). Thus, if p c < p 2→2 then this property holds for both critical and slightly supercritical percolation.
2. Theorem 3.1: If p c < p 2→2 , then the critical exponent governing the extrinsic radius of clusters is 1. That is, the probability that the cluster of the origin in G[p c ] reaches distance at least n in G is of order n −1 . This is not known to follow from the triangle condition even under the assumption of nonamenability.
3. Theorem 3.3: If p < p 2→2 , then the clusters of G[p] are 'ballistic' in the following sense: For any two vertices u and v of G, conditioned on u and v being connected in G[p], the intrinsic distance d int (u, v) between u and v in G[p] is exponentially unlikely to be much longer than d(u, v). Moreover, for fixed u, the random variable sup{d int (u, v)/d(u, v) : v ↔ u} has an exponential tail.
4. Theorem 4.1: If p c < p 2→2 , then we have the asymptotic estimates T pc q→q ≍ (q − 1) −1 and p q→q − p c ≍ q − 1 as q ↓ 1. This can be used to deduce various strong quantitative estimates on critical and slightly supercritical percolation.
5. Theorems 5.1 and 5.2: If p c < p 2→2 then the probabilities of various 'multiple arm' events are of the same order as the upper bound given by the BK inequality. In particular, we deduce that if G is transitive then the probability that the origin is a trifurcation point is of order (p − p c ) 3 as p ↓ p c (Corollary 5.6).
Finally, we apply some of the results above together with duality arguments to study the critical behaviour of percolation at the uniqueness threshold p u on nonamenable, quasi-transitive, simply connected planar maps (e.g. tesselations of the hyperbolic plane), which are always Gromov hyperbolic [13] and hence have p c < p 2→2 by the results of [24]. In particular, we prove the following.
6. If G is a nonamenable quasi-transitive simply connected planar map, then the probability that two neighbouring vertices are connected in G[p u ] but have intrinsic distance at least n is of order n −1 (excluding pairs of vertices for which there are local obstructions that make the probability trivially equal to zero).
We have organized the paper textbook-style into several short sections, each of which contains a complete treatment of one or more of the topics above. We will assume that the reader is familiar with Fekete's subadditive lemma, the Harris-FKG inequality, the BK inequality, Reimer's inequality, and Russo's formula, referring them to [16,Chapter 2] otherwise.

Exponential decay of the two-point function
In this section we study the consequences of the p c < p 2→2 condition for the decay of the twopoint function. Let G be a connected, locally finite graph. We say that G has exponential connectivity decay at p if the quantity is positive. That is, ξ p is maximal such that for every ε > 0 there exists C p,ε < ∞ such that for all vertices u and v in G. Following [42], we define the exponential connectivity decay threshold to be p exp = p exp (G) = sup p ∈ [0, 1] : G has exponential connectivity decay at p .
It is clear that p exp ≤ p u when G is quasi-transitive, and it is known that this inequality is strict in some examples, such as the free product Z 2 * (Z/2Z), and saturated in others, such as trees. It is also classical that p c (G) ≤ p exp (G) for every quasi-transitive graph G, see [16,Chapter 6].
The following conjecture is widely believed among experts but does not seem to have appeared in print. See [42, Open Problems 3.1 and 3.2] for further related problems. Perturbative proofs of p c < p u typically also implicitly yield the stronger result p c < p exp , and the previously mentioned papers [8,37,39,42] all yield that p c < p exp for the graphs they consider. The most notable non-perturbative result is due to Schonmann [43], who proved that p exp = p u for transitive, one-ended, nonamenable, planar graphs. Besides this, Schramm proved that for any transitive, unimodular, nonamenable graph, the critical two-point function is exponentially small in the distance for some pairs of vertices, namely, the endpoints of a random walk have this property with high probability. See [27] for Schramm's proof and [21,24,25] for related results.
Our first result verifies Conjecture 2.1 under the assumption that p c < p 2→2 . In particular, applying the results of [22,24], we deduce that the conjecture holds under the additional assumption that the graph in question is either Gromov hyperbolic or has a quasi-transitive nonunimodular subgroup of automorphisms. Let us mention again that Cayley graphs of groups of cost > 1 are known to have p c < p u [15,34] but are not known to have p c < p 2→2 or p c < p exp .
The proof is an 'L q version' of the usual proof that there is exponential connectivity decay when the susceptibility is finite [16, Theorem 6.1]. Let G be a connected, locally finite graph, let p ∈ [0, 1] and n ≥ 0, and consider the symmetric matrices C p,n , S p,n ∈ [0, ∞] V 2 defined by 1 Theorem 2.2 will be deduced from the following proposition.
We begin with the following simple consequence of the BK inequality. Given two matrices Lemma 2.5. Let G be a connected, locally finite graph. Then for every p ∈ [0, 1] and n, m ≥ 0.
Proof. Suppose that u, v ∈ V have d(u, v) ≥ n + m. On the event that u is connected to v in G[p], there must exist w with d(u, w) = n such that the disjoint occurrence {u ↔ w} • {w ↔ v} occurs, and such a w must have d(w, v) ≥ m by the triangle inequality. Indeed, simply take some simple path from u to v in G[p], and take w to be the first vertex with d(u, w) = n that this path visits. Summing over the possible choices of w and applying the BK inequality yields the claim.
Proof. Since S p,n C p,n for every n ≥ 0, Lemma 2.5 implies that C p,n q→q satisfies the submultiplicative inequality C p,n+m q→q ≤ C p,n q→q C p,m q→q , and applying Fekete's Lemma we deduce that is well-defined when 0 < p < p q→q . It remains to prove that η p,q > 0 for 0 ≤ p < p q→q . Let n, m ≥ 0 and let 0 < p < p q→q . Noting that C p,ℓ C p,r whenever ℓ ≥ r and applying (2.2), we obtain that and hence that C p,n+m q→q ≤ T p q→q m + 1 C p,n q→q .
Applying this bound inductively and using that C p,0 = T p we deduce that C p,kn q→q ≤ T p q→q n + 1 k T p q→q for every k, n ≥ 0, and hence that η p,q ≥ 1 n log n + 1 T p q→q for every n ≥ 1. Taking n = ⌈e T p q→q ⌉ − 1 completes the proof. Note that we also obtain the explicit bound We have bounded e 1/e by 2 for aesthetic reasons.) , and let 0 < p < p q→q . Then we have by Hölder's inequality that It follows that ξ p ≥ η p,q ≥ e −1 T p −1 q→q > 0 for every 0 < p < p q→q (G).

Cluster ballisticity and the extrinsic radius exponent
We now turn to our results concerning cluster ballisticity and the extrinsic radius exponent. We begin by discussing the extrinsic radius exponent. Let G = (V, E) be a connected, locally finite graph and let p ∈ [0, 1]. Let d denote the graph distance on G, and let d int denote the graph distance on the subgraph As discussed in the introduction, mean-field theory predicts that critical percolation on sufficiently high-dimensional quasi-transitive graphs (say, of at least seven-dimensional volume growth) resembles the trace of a critical branching random walk on the graph. Many of the specific predictions suggested by this heuristic are known to hold for any quasi-transitive graph satisfying the triangle condition. For example, Barsky and Aizenman [6] showed that in this case P pc |K v | ≥ n ≍ n −1/2 , and Kozma and Nachmias [29] showed that (Some parts of the proof of Kozma and Nachmias were specific to Z d , see [22,Section 7] for an adaptation to general quasi-transitive graphs satisfying ∇ pc < ∞.) These exponents are the same as those governing a critical Galton-Watson tree of finite variance, which arises as the genealogical tree of a critical branching random walk.
Determining the tail of the extrinsic radius is a more subtle matter, since it varies from graph to graph even in the mean-field regime and therefore cannot be deduced from the triangle condition alone: on a regular tree of degree at least three we have that extrinsic and intrinsic distances coincide and hence that P pc rad(K v ) ≥ n = P pc rad int (K v ) ≥ n ≍ n −1 , while for high-dimensional Euclidean lattices it was proven in the breakthrough work of Kozma and Nachmias [30] that P pc rad(K v ) ≥ n ≍ n −2 . Intuitively, the difference between these two exponents is explained by the fact that simple random walk is ballistic on the tree and diffusive on Z d . Since random walk is ballistic on every quasi-transitive nonamenable graph, it is reasonable to conjecture that P pc rad(K v ) ≥ n ≍ n −1 for every such graph. Our next result verifies this conjecture under the assumption that p c < p 2→2 .
Theorem 3.1. Let G be a connected, locally finite, nonamenable, quasi-transitive graph, and suppose that p c (G) < p 2→2 (G). Then for every v ∈ V we have that Note that the upper bound of (3.2) follows from the corresponding bound for the intrinsic radius, so that it remains only to prove the lower bound. Theorem 3.1 was known in the case that G has a quasi-transitive nonunimodular subgroup, using arguments specific to that setting [22,Theorem 1.6]. We believe the theorem is new in essentially all other cases, including for tessellations of the hyperbolic plane.
They key step in the proof of Theorem 3.1 is the following intrinsic variant on Proposition 2.4, which can be used to derive various ballisticity results for percolation below p 2→2 . For each p ∈ [0, 1] and n ≥ 0 we define the matrix C int Proposition 3.2. Let G be a connected, locally finite graph, and let p < p q→q (G). Then

Proof of Theorem 3.1 given Proposition 3.2.
Since p c < p 2→2 implies that ∇ pc < ∞, we have that P pc (rad int (K v ) ≥ n) ≍ n −1 . Therefore, since the extrinsic radius is bounded by the intrinsic radius, it suffices to prove the lower bound. For every n ≥ 1 and m ≥ n we trivially have that The second term on the right can be bounded by and applying Proposition 3.2 we deduce that Taking m = ⌈Cn⌉, we deduce that there exists a positive constant C ′ such that from which the claim follows immediately.
Proposition 3.2 can also be used to derive various other other results concerning the ballisticity of percolation clusters. Examples of two such statements are given in the following theorem, whose proof is very straightforward and is omitted. Theorem 3.3. Let G be a connected, locally finite, quasi-transitive graph. Then for every p < p 2→2 (G), there exist positive constants c p , C p and λ p such that for every u ∈ V and t ≥ 1, and similarly for every u, v ∈ V and n ≥ λ p d(u, v).
Note that the special case of Theorem 3.3 in which p < p c can also be deduced from the fact that the cluster volume in subcritical percolation has an exponential tail [3] (although the argument given here will give better control on the rate of divergence of the constants as p ↑ p c ).
See [5] for a related result concerning supercritical percolation on Euclidean lattices.
In contrast to these results, in Section 6 we show that for percolation in the hyperbolic plane, the quantity P pu d int (u, v) ≥ n | u ↔ v can have a power law tail at p u .
We now turn to the proof of Proposition 3.2. We write = n} for the intrinsic ball and sphere of radius n around v respectively, and for each p ∈ [0, 1] and m ≥ n ≥ 0 define the matrices for every p ∈ [0, 1] and m, n ≥ 0.
Remark 3.5. It may be instructive for the reader to reflect on why Reimer's inequality does not imply the submultiplicative inequality C int Proof. For each three vertices u, v, w, let G n,m (u, v, w) be the event that w is connected to v with n ≤ d int (u, w) ≤ n + m and that v is the nth vertex on some geodesic from u to w in G[p]. The definitions ensure that We now claim that Indeed, on the event G n,m (u, v, w), the subgraph of G induced by the intrinsic ball of radius n around u together with all the closed edges incident to this ball is a witness for {v ∈ ∂B int (u, n)}, while the geodesic in G[p] from v to w is a witness for {w ∈ B int (v, m)} that is disjoint from the former witness. Thus, we can apply Reimer's inequality to obtain that which is equivalent to the claimed inequality (3.5).
We now prove Proposition 3.2. The proof is quite different to that of Proposition 2.4 since submultiplicativity is not available, and relies instead on a certain averaging trick that will be used again in the proof of Proposition 3.6.

Proof of Proposition 3.2. Applying (3.5) and using the trivial inequality
for every k, n ≥ 1 and r ≤ kn. Averaging this inequality over (k − 1)n ≤ r ≤ kn we obtain that for every k, n ≥ 1. Applying this bound inductively we deduce that

Mean-field lower bounds on norms of ball and sphere operators
In this subsection, we note that methods similar to those used to derive Propositions 3.2 and 2.4 can also be used to prove the following simple mean-field lower bounds. These bounds complement those of [24, Corollary 2.5], which states that if G is a connected, locally finite graph then for every 0 ≤ p < p q→q , where A is the adjacency matrix. Similarly to the way that that corollary is used to prove the main results of [24], one could potentially use Proposition 3.6 to establish that p c < p 2→2 by showing that, for example, S int pc,n 2→2 → 0 as n → ∞.
The proof of Proposition 3.6 is inspired by the proof of [12, Theorem 1.1]. Note that the sphere bounds do not obviously imply the ball bounds when q / ∈ {1, ∞}.
Proof. We will prove (3.8), the proof of (3.7) being similar. Let Λ ⊂ E be finite. We write {u Λ ← → v} for the event that u and v are connected by an open path using only edges of Λ, and for each p ∈ [0, 1] define the matrix T p,Λ (u, v) = P p (u Λ ← → v). Note that, since G has bounded degrees and B int p,n is symmetric, we have by the Riesz-Thorin theorem that for every n ≥ 0 and q ∈ [1, ∞], and similarly that T p,Λ q→q ≤ T p,Λ 1→1 < ∞ for every Λ ⊂ E finite. Applying Reimer's inequality as in the proof of Lemma 3.4, we obtain that which is equivalent to the bound

Norm exponents
In [24, Proposition 2.3], we proved that if G is quasi-transitive and p c (G) < p 2→2 (G) then p c (G) < p q→q (G) for every q ∈ (1, ∞). Our next result establishes a sharp quantitative 2 version of this fact, which will yield very strong quantitative information about the critical two-point function.
Theorem 4.1. Let G be a connected, locally finite, nonamenable, quasi-transitive graph, and suppose that p c (G) < p 2→2 (G). Then The proof of Theorem 4.1 is given in the following two subsections. This theorem is complemented by the lower bound for every 0 < p < p q→q (4.1) which is proven in [ The claim follows by taking q = log |W | and applying Theorem 4.1.
A further corollary of Theorem 4.1 concerns the asymptotic density of slightly supercritical clusters. Let G = (V, E) be a connected, locally finite, quasi-transitive graph. For each 0 < p ≤ 1, we define the annealed upper logarithmic density of clusters in G[p] by where B(v, n) denotes the ball of radius n around v in G. Note that Markov's inequality implies that for every v ∈ V almost surely. We now begin to work towards the proof of Theorem 4.1. We first lower bound the rate that T pq→q q ′ →q ′ blows up as q ′ → q. The resulting estimate can be though of as a mean-field lower bound. Given a graph G, we define The fact that this limit exists follows by Fekete's lemma, since the sequence sup v∈V |B(v, n)| is submultiplicative. Moreover, if G has degrees bounded by M then we have that γ(G) ≤ log(M − 1) < ∞.
with the convention that the right hand side is equal to q ′ /eγ when q = ∞.
The proof of Proposition 4.4 will apply the following elementary lemma.
The case q 2 = ∞ will use the following similarly elementary fact. Proof. We may assume that M q→q > 0, since the claim is trivial otherwise. Note that for each f ∈ V R , the norm f q is decreasing in q for every f ∈ V R , and is continuous in q ∈ [1, ∞] if f is finitely supported (i.e., zero at all but finitely many vertices). For each a < M q→q , there exists a finitely supported f ∈ V R such that M f q / f q ≥ a. For such f we have lim q ′ ↑q f q ′ = f q and lim inf q ′ ↑q M f q ′ ≥ M f q . We deduce that lim inf q ′ ↑q M q ′ →q ′ ≥ a, and the claim follows since a < M q→q was arbitrary.
Proof of Lemma 4.5. We may assume that M q 1 →q 1 < ∞ and sup v∈V {u : M (v, u) = 0} < ∞, since the claim is trivial otherwise. First suppose that q 2 = ∞. Let f ∈ L q 2 (V ) be such that f (v) ≥ 0 for every v ∈ V . By Hölder's inequality, we have that and putting these two bounds together we obtain that Noting that f q 2 /q 1 q 1 q 1 = f q 2 q 2 , taking the supremum over f ∈ L q 2 (V ), and dividing both sides by M q 2 −q 1 q 2 →q 2 yields the desired inequality. The case q 2 = ∞ follows from the case q 2 < ∞ by an application of Lemma 4.6.
We now turn to the proof of Proposition 4.4.
Proof of Proposition 4.4. Applying Lemma 4.5 with q 1 = q ′ , q 2 = q, and M = S pq→q,n , we have that for every n ≥ 0, where the second inequality follows from Proposition 3.6. It follows from this inequality and Proposition 2.4 that and the claim follows immediately.
We next give a related bound on the rate of change of p q→q as a function of q. Recall that the ξ p denotes the exponential decay rate of the two-point function, as defined in Section 2.
Proposition 4.7. Let G be a connected, bounded degree graph, let q ∈ (2, ∞] and q ′ ∈ [2, q), and suppose that ξ pq→q > 0. Then with the convention that the prefactor on the right hand side is equal to q ′ ξ pq→q /(γ + q ′ ξ pq→q ) when q = ∞.
Recall that it follows from Theorem 2.2 that ξ pq→q > 0 whenever p q→q < p 2→2 . We therefore immediately deduce the following corollary. Recall from [16, Theorem 2.38] that if A is an increasing event, then log P p (A)/ log p is a nonincreasing function of p ∈ (0, 1). Thus, if 0 < p 1 ≤ p 2 < 1 then we have that Thus, it follows by definition of ξ p 1 that S p 2 ,n exp log(p 1 /p 2 )ξ p log p 2 n + o(n) S p 1 ,n as n → ∞.
and hence that lim inf On the other hand, applying Lemma 4.5 as in the proof of Proposition 4.4 we obtain that if n log S p 2 ,n q ′ →q ′ ≥ lim inf n→∞ 1 n log S p 1 ,n q→q + log(p 1 /p 2 )ξ p log p 2 − (q − q ′ )γ qq ′ for every 1 ≤ q ′ ≤ q and 0 < p 1 ≤ p 2 < 1. Taking p 1 = p q→q and p 2 = p given by we deduce from (4.3) and (4.4) that lim inf n→∞ 1 n log S p,n q ′ →q ′ ≥ 0, and hence that p q ′ →q ′ ≤ p as claimed.
It remains to prove a complimentary upper bound on T pc q→q and lower bound on p q→q . Proposition 4.9. Let G be a connected, locally finite, quasi-transitive graph, and suppose that p c < p 2→2 . Then there exist positive constants c and C such that Proof. The claimed lower bound on p q→q − p c follows immediately from the claimed upper bound on T pc q→q together with (4.1). Since p c < p 2→2 , we have that ∇ pc < ∞, and hence by the results of [29,41] that there exists a constant C such that for every n ≥ 0. Since G is quasi-transitive we deduce that there exists a constant C ′ such that Let q ∈ (1, 2) and let θ = θ(q) ∈ (0, 1) be such that 1/q = (1 − θ)/1 + θ/2.
for each 0 ≤ N < ∞. Applying (4.5) and sending N → ∞ we obtain that (n + 1), and hence by calculus that there exists a positive constant C ′′ such that lim sup q↓1 θ(q) T pc q→q ≤ C ′′ T pc 2→2 .
Proof of Corollary 4.3. We begin with the upper bound. For each p ∈ (0, p 2→2 ) let q(p) = sup{q ∈ [2, ∞] : p < p q→q }. If p < p 2→2 then for every q ∈ [2, q(p)) and v ∈ V we have by Hölder's inequality that It follows that δ log (p) ≤ q(p) −1 for every 0 < p < p 2→2 , so that the claimed upper bound may be deduced immediately from Theorem 4.1. For the lower bound, we apply (4.2) to deduce that for every v ∈ V and 0 < p 1 ≤ p 2 ≤ 1. Using this together with Proposition 3.6, we easily deduce that if p 1 ≥ p c then and hence that Applying Theorem 2.2 gives that ξ pc > 0, which completes the proof.

Multiple arms
In this section, we prove that if p c < p 2→2 then the probability of various 'multiple arm' events are of the same order as the upper bound given by the estimates for the corresponding 'one arm' events and the BK inequality. Besides their intrinsic interest, these results will also be applied in our study of percolation in the hyperbolic plane at the uniqueness threshold in Section 6. We write ≍ ℓ to denote an equality depending on the choice of ℓ but not on any of the other parameters in question.
for every n 1 , . . . , n ℓ ≥ 1 and for every p c < p ≤ 1, every n ≥ 1, and every The proof combines two techniques from the literature: the 'inverse BK' method used in the proof of [30,Theorem 3], which establishes a similar result for extrinsic radii in Euclidean lattices, and the ghost field technique, which was introduced to percolation by Aizenman and Barsky [1] and, for our purposes, allows us to apply the inverse BK method to study the volume of the clusters rather than the radii. Let G = (V, E) be a connected, locally finite graph, and let G[p] be Bernoulli-p bond percolation on G. A ghost field of intensity h on G is a random subset of V , independent of G[p], such that each vertex v of G is included in G independently at random with inclusion probability 1 − e −h . For each v ∈ V , 0 < p < 1, and h > 0, we define the magnetization (5.5) Note that if G is quasi-transitive then an easy FKG argument implies that there exist a positive for every h > 0 and 0 < p < 1. It is proven in [6] (and follows from eq. (1.3)) that if G is quasi-transitive and satisfies the triangle condition then (5.6) and in fact the lower bound of (5.6) holds for every quasi-transitive graph [1]. (Some aspects of the proof of [6] are specific to the case of Z d , see [22,Section 7] for an overview of the changes needed to handle arbitrary quasi-transitive graphs.) We begin the proof of Theorems 5.1 and 5.2 with the following lemma, which is inspired by and based closely on [30, Lemma 6.1]. Given 0 < p < 1 and h = (h 1 , h 2 , . . . , h ℓ ) ∈ (0, ∞) ℓ , we write P p,h for the joint law of G[p] and ℓ mutually independent ghost fields G 1 , . . . , G ℓ of intensities h 1 , . . . , h ℓ .
Proof of Lemma 5.3. By inducting on ℓ, it suffices to prove that for every 0 < p < 1 and h ∈ (0, ∞) ℓ . For this proof, we will change notation and denote our percolation configuration by ω rather than G[p]. Let ω 0 , ω ∞ be independent copies of Bernoulli-p bond percolation on G, independent of the ghost fields G 1 , . . . , G ℓ , and let P ⊗ denote the joint law of all these random variables. Let e 1 , e 2 , . . . be an enumeration of the edge set of G, and for each m ≥ 0 define Let A = {v 1 ↔ G 1 } • · · · • {v ℓ−1 ↔ G ℓ−1 } and B = {v ℓ ↔ G ℓ }, and observe that and that Thus, to prove (5.7), it suffices to prove that Observe that on the event A 0 • B m \ A 0 • B m−1 we must have that ω ∞ (e m ) = 1. Moreover, the events have the same probability: Indeed, the proof of the BK inequality (see in particular the presentation in [16, Section 2.3]) establishes that there is a measure-preserving bijection between these sets. Thus, we have that in ω m−1 } all occur disjointly. Moreover, on this event, the events {v i ↔ G i in ω m−1 } for i / ∈ {j, ℓ} also occur disjointly from each other and from these events. Thus, we may apply the BK inequality to deduce that which is at most The claim now follows by noting that for every v ∈ V and m ≥ 1 by the Harris-FKG inequality, and hence that Remark 5.4. Lemma 5.3, and the proof of [30, Lemma 6.1] that inspired it, are remarkable as a rare instance where it is the convergence of the bubble diagram rather than the triangle diagram that is indicative of mean-field type behaviour. In particular, if 0 denotes the origin in Z d , it seems one should expect that not just for d > 6, but also for some d slightly smaller than 6, possibly including d = 5.
Next, we compare the probability that the events all occur disjointly to the probability that the events all hold with all clusters distinct.
Proof of Lemma 5.5. Let A be the event that K v 1 , . . . , K v ℓ are disjoint and v i ↔ G i for every 1 ≤ i ≤ n, and let B be the event {v 1 ↔ G 1 } • · · · • {v ℓ ↔ G ℓ }. Clearly A ⊆ B. Suppose that B \ A occurs. Since B occurs, there must exist a collection of edge-disjoint open paths γ 1 , . . . , γ ℓ such that γ i connects v i to a vertex of G i . (Note that we may have that v i ∈ G i , in which case we may take γ i to be a degenerate length zero path.) On the other hand, since A does not occur, there must exist an open path γ connecting two distinct vertices from the set {v 1 , . . . , v ℓ }. Suppose that this path γ starts at the vertex v j 1 , and let γ ′ be the segment of γ between the last time it visits a vertex visited by γ j 1 and the first subsequent time that it visits a vertex visited by one of the paths γ i for i = j 1 . Call these two vertices w 1 and w 2 , and let j 2 = j 1 be such that γ j 2 visits w 2 . (It may be that w 1 = w 2 , in which case γ ′ has length zero.) Observe that, with this choice of j 1 , j 2 , w 1 and w 2 , we have that the events {v j 1 → w 1 }, {v j 2 ↔ w 2 }, {w 1 ↔ w 2 }, {w 1 ↔ G j 1 }, and {w 2 ↔ G j 2 } all occur disjointly. Moreover, the events {v i ↔ G i } for i / ∈ {j 1 , j 2 } also occur disjointly from each other and from these events. Thus, applying the BK inequality and summing over the possible choices of j 1 , j 2 , w 1 , and w 2 and we obtain that concluding the proof.
We are now ready to prove Theorems 5.1 and 5.2.
Proof of Theorem 5.1. It suffices to prove the lower bound, since the upper bound is an immediate consequence of (1.3) and the BK inequality. Fix ℓ ≥ 1, and let C 1 be the constant from (5.5). Let 0 < p < 1 and let h = (h 1 , . . . , h ℓ ) ∈ (0, ∞) ℓ . Let G[p] be Bernoulli-p bond percolation, let G 1 , . . . , G ℓ be independent ghost fields of intensities h 1 , . . . , h ℓ > 0, and write P p,h for the joint law of G[p] and G 1 , . . . , G ℓ . Since G is quasi-transitive and ∇ pc < ∞, we have by the result of [28] that the open triangle condition also holds, so that for every ε > 0 there exists r < ∞ such that if u, v have distance at least r in G then T 3 pc (u, v) ≤ ε. In particular, it follows that there exists K(ℓ) such that if u, v are vertices of G with distance at least K(ℓ) then where M is the maximum degree of G. (Note that if p c < p 2→2 we can take K(ℓ) to be O(ℓ), and if G is transitive with p c < p 2→2 we can take K(ℓ) to be O(log ℓ).) With this choice of K(ℓ), we deduce from Lemmas 5.3 and 5.5 that if v 1 , . . . , v ℓ are such that d(v i , v j ) ≥ K(ℓ) for every 1 ≤ i < j ≤ ℓ, then for every h 1 , . . . , h ℓ > 0. Since G is quasi-transitive, we may apply the lower bound of (5.6) to deduce that there exists a constant c 1 such that On the other hand, the BK inequality implies that and hence by (5.6) that, since ∇ pc < ∞, there exists a constant C 2 such that for every h 1 , . . . , h ℓ > 0. It remains to convert the magnetization estimates (5.8) and (5.9) into the claimed estimate (5.1). Let δ ℓ > 0 be a sufficiently small that where c 1 and C 2 are the constants from (5.8) and (5.9) respectively. Let n 1 , n 2 , . . . , n ℓ ≥ 1 and let h i = δ ℓ /n i for each 1 ≤ i ≤ ℓ. Let D be the event that K v 1 , . . . , K v ℓ are disjoint, let A be the event that v i ↔ G i for every i ≥ 1, and let B be the event that |K v i | ≥ n i for every i ≥ 1. Then we have that For each 1 ≤ i, j ≤ ℓ let h i,j be defined by h i,j = h i if i = j and h j,j = 1/n j , and let h j = (h 1,j , . . . , h ℓ,j ). Then for each 1 ≤ j ≤ n we can write Applying (5.9) we deduce that and hence by (5.8) that It follows that for every 1 ≤ i < j ≤ ℓ and every n 1 , . . . , n ℓ ≥ 1.
Proof of Theorem 5.2. It suffices to prove the lower bound, since the upper bound is an immediate consequence of (1.2) and the BK inequality. Fix ℓ ≥ 1, and let C 1 be the constant from (5.5). Let 0 < p < 1 and let h > 0. Let G[p] be Bernoulli-p bond percolation, let G 1 , . . . , G ℓ be independent ghost fields of intensity h > 0, and write P p,h for the joint law of G[p] and G 1 , . . . , G ℓ . Fix p c < p 0 < p 2→2 . As in the proof of Theorem 5.1, there exists K(ℓ) such that if u, v are vertices of G with distance at least K(ℓ) then we have by Lemmas 5.3 and 5.5 that for every 0 ≤ p < p 0 and h > 0. Taking the limit as h ↓ 0, we obtain that for every 0 ≤ p < p 0 and v 1 , . . . , v ℓ satisfying d(v i , v j ) ≥ K(ℓ) for every 1 ≤ i < j ≤ ℓ. Thus, we may deduce (5.2) from (1.2). We now deduce the lower bounds of (5.3) and (5.4) from the lower bound of (5.2). First, notice that . . . , v ℓ are all in distinct infinite clusters) ℓ n −ℓ for every n ≥ 1 and every v 1 , . . . , v ℓ satisfying d(v i , v j ) ≥ K(ℓ) for every 1 ≤ i < j ≤ ℓ: the first inequality is trivial and the second follows from (5.2). Now consider coupling percolation at p c and p c + 1 n in the standard monotone fashion. If the clusters of v 1 , . . . , v ℓ are all distinct with intrinsic radius at least n in percolation at p c + 1 n , then the conditional probability that this continues to hold in percolation at p c is at least [p c /(p c + 1 n )] nℓ . To see this, take a length-n intrinsic geodesic from v i in each (p c + 1 n )-cluster, and observe that, for the property to no longer hold at p c , at least one edge in at least one of these geodesics must change from open to closed when we move from p c + 1 n to p c . Thus, we deduce that P pc (K v 1 , . . . , K v ℓ are disjoint and rad int (K v i ) ≥ n for every 1 ≤ i ≤ ℓ) ≥ p c p c + 1 n nℓ P pc+ 1 n (K v 1 , . . . , K v ℓ are disjoint and rad int (K v i ) ≥ n for every 1 ≤ i ≤ ℓ), so that the lower bound of (5.3) follows from the lower bound of (5.2). Finally, the lower bound of (5.4) can be deduced from the lower bound of (5.3) using Proposition 3.2 in a very similar manner to the proof of Theorem 3.1.
Let G be a connected, locally finite graph and consider a Bernoulli bond percolation G[p] on G. A vertex v of G is said to be a furcation point if K v is infinite and deleting v would split K v into at least three distinct infinite clusters, and say that v is a trifurcation point if deleting v would split K v into exactly three distinct infinite clusters. These points come up memorably in the Burton-Keane [10] proof of uniqueness of the infinite cluster in amenable transitive graphs, where it is argued that if G[p] has infinitely many infinite clusters, then it must have furcation points; see also [35,Section 7.3]. Using Theorem 5.2 allows one to make this proof quantitative, leading to the following corollary. (Note that the upper bound follows trivially from (1.2) and the BK inequality.) Corollary 5.6. Let G be a connected, locally finite, quasi-transitive graph, and suppose that p c < p 2→2 . Then there exists a vertex v of G such that P p v is a trifurcation point ≍ (p − p c ) 3 as p ↓ p c .
(Note that, under the triangle condition, being a furcation point but not a trifurcation point has probability of order at most (p − p c ) 4 as p ↓ p c by (1.2) and the BK inequality.)

Applications to percolation in the hyperbolic plane
In this section we apply the results of the previous sections to study percolation on quasi-transitive nonamenable simply connected planar maps with locally finite dual. In particular, we apply duality arguments to compute various critical exponents governing the geometry of clusters at the uniqueness threshold p u .
Let us now briefly recall the relevant terminology. A (locally finite) map is a connected, locally finite graph G together with an equivalence class of proper embeddings of G into orientable surfaces (without boundary), where we consider two embeddings to be equivalent if there is an orientation preserving homeomorphism between the two surfaces sending one embedding to the other. A map is said to be simply connected if the surface it is embedded in is (in which case it is homeomorphic to the plane or the sphere). Maps can also be defined combinatorially as graphs equipped with cyclic orderings of the oriented edges emanating from each vertex. Every map M has a dual M † whose vertices are the faces of M and whose faces correspond to the vertices of M , and there is a natural bijection between the edges of M and the edges of M † sending each edge e of M to the unique edge e † of M † that crosses e. See [33] or [4, Section 2.1] for further background. Let us remark again that if M is a simply connected, quasi-transitive, locally finite, nonamenable map then it is Gromov hyperbolic [13] and hence has p c < p 2→2 by the results of [24].
Let M be a map with locally finite dual M † . Observe that if ω is distributed as Bernoullip bond percolation on M , then the configuration ω † ∈ {0, 1} E † defined by ω † (e † ) = 1 − ω(e) is distributed as Bernoulli-(1−p) percolation on M † . This duality is extremely useful in the case that M is simply connected. In particular, Benjamini and Schramm proved that if M is a nonamenable, quasi-transitive, locally finite, simply connected map with locally finite dual, then See also the earlier work of Lalley [31]. We now apply this duality to deduce various results about percolation at p u on M by converting them into results about critical percolation on M † . Given two vertices x and y that are in the same cluster, we write ConRad(x, y) for the minimal r such that x and y are connected by an open path in the union of extrinsic balls B(x, r) ∪ B(y, r). where the suprema are taken over all neighbouring pairs of vertices in M .
The reader may find it interesting to compare these exponents to the corresponding exponents for the free uniform spanning forest [26], which are 1/2 and 1 respectively. Theorem 6.1 is complemented by forthcoming work of the author with Hermon [19], which establishes in particular that, under the hypotheses of Theorem 6.1, for every p ∈ (0, p u ) ∪ (p u , 1), there exists a constant c p > 0 such that P p d int (x, y) ≥ n | x ↔ y ≤ e −cpn (6.2) for every n ≥ 1 and every pair of neighbouring vertices x and y.
Note that the lower bounds of Theorem 6.1 theorem cannot be strengthened to apply to every edge, since the local geometry of the graph might make it impossible for the endpoints of some edges to be in distinct large clusters. The proof shows that the lower bound holds whenever the edge dual to that between x and y lies on a doubly infinite geodesic in M † .
Proof of Theorem 6.1. Let ω be p u percolation on M . Then ω † is p c percolation on M † and consequently has only finite clusters by the results of [7,21] since M † is quasi-transitive and nonamenable. Let e be an edge of M , let f and g be the two faces of M on either side of e, and let K 1 and K 2 be the clusters of f and g in ω † \ {e † } respectively. Observe that if x and y are connected in ω with d int (x, y) > 1 then we must have that ω(e) = 0 and that K 1 = K 2 , since otherwise the path between f and g in ω † \ {e † } would disconnect x from y. Moreover, we claim that there exist positive constants c, C such that c min{|K 1 |, |K 2 |} ≤ d int (x, y) ≤ C min{|K 1 |, |K 2 |} (6.3) and c min{diam(K 1 ), diam(K 2 )} ≤ ConRad(x, y) ≤ C min{diam(K 1 ), diam(K 2 )} (6.4) on this event. Indeed, if η is any open path from x to y in ω \ {e} then η ∪ {e} forms a cycle that surrounds one of the two clusters K 1 or K 2 . The lower bound of (6.3) follows by nonamenability of M together with the fact that M and M † both have bounded degrees, these facts together implying that any simple cycle in M must have length comparable to the number of faces it bounds. For the upper bound of (6.3), we note that the collection of dual edges other than e in the boundary of K i must contain a primal path from x to y on the event that K 1 and K 2 are distinct. Thus, d int (x, y) is at most this number of dual edges, which is at most a constant multiple of |K i | since M † has bounded degrees. The bounds of (6.4) follow by similar arguments. The upper bounds of the theorem follow immediately from the estimates (6.3), (6.4), the bounds (1.3) and of Theorem 3.1 (applied to M † ) together with the BK inequality. On the other hand, the lower bounds are very close to those of Theorems 5.1 and 5.2 and can be deduced from those theorems via a finite energy argument. We will give only a brief sketch of how this is done; an essentially identical argument is given in detail in the proof of [23, Theorem 1.3]. By Theorems 5.1 and 5.2 applied to M † , there exist positive constants c and k such that if f, g are faces of M with distance at least k (in M † ) then P(K f , K g distinct and |K f |, |K g | ≥ n) ≥ c n and P(K f , K g distinct and diam(K f ), diam(K g ) ≥ n) ≥ c n 2 .
Fix two such f and g. If we start with a configuration in which K f and K g are distinct and then force the edges along the geodesic in M † between f and g to be open one at a time, there must be some first time when the two clusters merge when the next edge is added. At the time immediately before this one, the edge about to be added has endpoints in distinct clusters, both of which are at least as large (both in terms of volume and diameter) as K f and K g were in the original configuration. Since there are a constant number of choices of what this edge could be, and since forcing a bounded number of edges to be included in the percolation configuration increases probabilities of events by at most a constant, the claim follows.
Remark 6.2. Outside of the planar case, the behaviour of percolation at p u is very poorly understood. In particular, no good characterisation of whether or not there is uniqueness at p u is known. It is known that quasi-transitive simply connected planar maps have uniqueness at p u , but that graphs defined as infinite products [40] and Cayley graphs of infinite Kazhdan groups [36] have nonuniqueness at p u . It is open whether uniform lattices in H d with d ≥ 3 have nonuniqueness at p u or not. Indeed, this is also open for transitive graphs that are rough-isometric to H 2 but are not planar. In [24,Section 6] it is shown that there is always nonuniqueness at p 2→2 , and one possibility is that there is uniqueness at p u if and only if p u > p 2→2 .