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

DOI10.1007/S00222-020-01011-3arXiv1904.10448WikidataQ115608941 ScholiaQ115608941MaRDI QIDQ6317645

Publication date: 23 April 2019

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 [mathbf{P}_p(n leq |K| < infty) leq e^{-c_p n}] for every

n g e q 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.

Mathematics Subject Classification ID

Random graphs (graph-theoretic aspects) (05C80) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Percolation (82B43)

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