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Critical Percolation on any Nonamenable Group Has no Infinite Clusters
Itai Benjamini, Russell Lyons, Yuval Peres and Oded Schramm
The Annals of Probability
Vol. 27, No. 3 (Jul., 1999), pp. 1347-1356
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2652805
Page Count: 10
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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.
The Annals of Probability © 1999 Institute of Mathematical Statistics