Invariant Percolation and Harmonic Dirichlet Functions

D. Gaboriau

Published:

Geom. Funct. Anal., 15 (2005), no. 5, 1004-1051.

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Also HAL (CNRS) http://hal.ccsd.cnrs.fr/ccsd-00001606 or ArXiV http://arxiv.org/abs/math/0405458

Abstract

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99: ``Percolation perturbations in potential theory and random walks''] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of l² Betti numbers.