Transience and anchored isoperimetric dimension of supercritical percolation clusters

Tom Hutchcroft

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arXiv · 2207.05226

Transience and anchored isoperimetric dimension of supercritical percolation clusters

Tom Hutchcroft

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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.

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Tom Hutchcroft. 2022-07-11. Transience and anchored isoperimetric dimension of supercritical percolation clusters. https://arxiv.org/abs/2207.05226

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