Critical Percolation on any Nonamenable Group has no Infinite Clusters

By Itai Benjamini, Russell Lyons, Yuval Peres, Oded Schramm. 2011

80 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1583c3c687da2aa90a4fde1014d56689a3b8ade7

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Hyperbolic and Parabolic Unimodular Random Maps (2016, background): “Benjamini, Lyons, Peres and Schramm [13] also relate analytic properties of G to the WUSF and FUSF of G. Theorem 5.8 ([13]; Theorem 1, equivalence of items (12) (11)).”