Percolation Perturbations in Potential Theory and Random Walks

TL;DR

This paper investigates the properties of percolation clusters on Cayley graphs, revealing differences between amenable and nonamenable groups in terms of harmonic functions, transience, and isoperimetric properties.

Contribution

It introduces the novel result that nonamenable Cayley graph clusters admit invariant random subgraphs with positive isoperimetric constants, and explores harmonic functions on these clusters.

Findings

Nonamenable clusters are transient for random walk.

Clusters have positive speed and admit bounded harmonic functions.

Amenable clusters almost surely admit no nonconstant harmonic Dirichlet functions.

Abstract

We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph almost surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, almost surely the infinite clusters of $p$-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when $p$ is sufficiently close to 1. Many conjectures and questions are presented.