Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

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 > p_c(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, Lyons, and Schramm (1997). 2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

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