Invariant Percolation and Harmonic Dirichlet Functions

TL;DR

This paper investigates the relationship between harmonic Dirichlet functions on graphs and their infinite clusters in percolation models, extending previous results to more general invariant percolations and establishing the nonuniqueness phase.

Contribution

It extends the understanding of harmonic Dirichlet functions in percolation to broader models, including the Random-Cluster model, and proves the existence of the nonuniqueness phase using $ ext{l}^2$ Betti numbers.

Findings

Established the existence of the nonuniqueness phase for Bernoulli percolation.

Extended results to the Random-Cluster model.

Connected harmonic Dirichlet functions to percolation phases using $ ext{l}^2$ Betti numbers.

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.