Transience and anchored isoperimetric dimension of supercritical percolation clusters

We establish several equivalent characterisations of the anchored isoperimetric di-mension 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 (Duke Math. J. 2020) 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.

Automated summary

In this paper, we establish several equivalent characterizations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. Based on these characterizations and a theorem by Duminil-Copin, Goswami, Raoufi, Severo, and Yadin (Duke Math. J. 2020), we deduce that if G is a transient transitive graph, the infinite clusters of Bernoulli percolation on G are transient for p sufficiently close to 1. It remains an open question to extend this result to the critical probability. Along the way, we also establish two new cluster repulsion inequalities that are of independent interest.