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


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.


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Itai Benjamini. Russell Lyons. Yuval Peres. Oded Schramm. "Critical Percolation on Any Nonamenable Group has no Infinite Clusters." Ann. Probab. 27 (3) 1347 - 1356, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0961.60015
MathSciNet: MR1733151
Digital Object Identifier: 10.1214/aop/1022677450

Primary: 60B99
Secondary: 60D05 , 82B43

Keywords: amenability , Cayley graphs , percolation

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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