Invariant Percolation and Harmonic Dirichlet Functions

Abstract

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of 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 ${\ell }^2$ Betti numbers.

to appear in Geometric And Functional Analysis (GAFA)

Cited by in corpus (3)

• On limits of finite graphs
• Weak convergence of finite graphs, integrated density of states and a Cheeger type inequality
• Examples of Groups that are Measure Equivalent to the Free Group