Transience and anchored isoperimetric dimension of supercritical percolation clusters

TL;DR

This paper characterizes the anchored isoperimetric dimension of supercritical percolation clusters on transitive graphs and shows that for transient graphs, these clusters are transient near the percolation threshold.

Contribution

It provides new equivalent characterizations of the anchored isoperimetric dimension and proves the transience of infinite clusters near the critical probability for certain graphs.

Findings

Equivalent characterizations of anchored isoperimetric dimension

Infinite clusters are transient for p close to 1 on transient graphs

Introduction of two new cluster repulsion inequalities

Abstract

We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that if $G$ is a transient transitive graph then the infinite clusters of Bernoulli percolation on $G$ are transient for $p$ sufficiently close to $1$. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.