Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size di
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Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution Jonathan Hermon1 · Tom Hutchcroft2
Received: 10 May 2019 / Accepted: 13 October 2020 © The Author(s) 2020
Abstract Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > pc (G) then there exists a positive constant c p such that P p (n ≤ |K | < ∞) ≤ e−c p n for every n ≥ 1, where K is the cluster of the origin. We deduce the following two corollaries: 1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999). 2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated twopoint function are analytic functions of p throughout the supercritical phase.
B Tom Hutchcroft
[email protected] Jonathan Hermon [email protected]
1
Department of Mathematics, University of British Columbia, Vancouver, Canada
2
Statslab, DPMMS, University of Cambridge, Cambridge, UK
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J. Hermon, T. Hutchcroft
1 Introduction Let G = (V, E) be a connected, locally finite graph. In Bernoulli bond percolation, we choose to either delete or retain each edge of G independently at random with retention probability p ∈ [0, 1] to obtain a random subgraph ω of G with law P p = P Gp . The connected components of ω are referred to as clusters, and we denote the cluster of v in ω by K v = K v (ω). Percolation theorists are particularly interested in the geometry of the open clusters, and how this geometry changes as the parameter p is varied. It is natural to break this study up into several cases according to the relationship between p and the critical probability pc = pc (G) = inf p ∈ [0, 1] : ω has an infinite cluster P p -a.s. , which always satisfies 0 < pc < 1 when G is transitive and has superlinear growth [21] (this result is easier and older if G has polynomial growth [47, Corollary 7.19] or exponential growth [34,46]). Among the different regimes this leads one to consider, the subcritical phase 0 < p < pc is by far the easiest to understand. Indeed, the basic features of subcritical percolation have been well understood since the breakthrough 1986 works of Menshikov [49] and Aizenman and Barsky [1] which, together with the work of Aizenman and Newman [4], establish in particular that if G is a connected, locally finite, transitive graph then
ζ ( p) := − lim sup n→∞
1 log P p (n ≤ |E(K v )| < ∞) > 0 n
(1.1)
for every 0 ≤ p < pc , where we write E(K v ) for the set of edges that have at least one endpoint in the cluster K v . See also [22,23,39] for alternative proofs, and [16] for more refined results. Note that ζ ( p) = ζ ( p, G) is well-define
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