Anchored isoperimetry in percolation

Abstract

We show that, for any infinite transitive graph $G$, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of $G$. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, ${\mathbb{Z}}^d$, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.

arXiv comment11 pages, 2 figures