Abstract
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, including all transitive, nonamenable, Gromov hyperbolic graphs. In notation, the conjecture states that $p_{c}<p_{2\to 2}$, where for each $q\in [1,\infty ]$ we define $p_{q\to q}$ to be the supremal value of $p$ for which the operator norm $\|T_{p}\|_{q\to q}$ is finite. 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 (a.k.a. chemical) 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.
Highlights
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
We are interested in phase transitions, which occur when qualitative features of G[p] change abruptly as p is varied through some critical value
Summary
If G is a connected, locally finite, quasi-transitive graph that satisfies the triangle condition the estimates (1.1), (1.2), and (1.3) all hold for every vertex v of G. The probability that the cluster of the origin in G[pc] 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. We apply some of the results above together with duality arguments to study the critical behaviour of percolation at the uniqueness threshold pu on nonamenable, quasi-transitive, connected planar maps (e.g. tesselations of the hyperbolic plane), which are always Gromov hyperbolic [13] and have pc < p2→2 by the results of [23]. 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
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