The Annals of Probability

Critical Percolation on Any Nonamenable Group has no Infinite Clusters

Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm

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We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is a “mass-transport” method, which is a technique of averaging in nonamenable settings.

Article information

Ann. Probab., Volume 27, Number 3 (1999), 1347-1356.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

Primary: 60B99: None of the above, but in this section
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B43

Percolation Cayley graphs amenability


Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Critical Percolation on Any Nonamenable Group has no Infinite Clusters. Ann. Probab. 27 (1999), no. 3, 1347--1356. doi:10.1214/aop/1022677450.

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