Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher

Abstract

Let $d\ge 2$. We consider an i.i.d. supercritical bond percolation on ${\mathbb{Z}}^d$, every edge is open with a probability $p>p_c(d)$, where $p_c(d)$ denotes the critical point. We condition on the event that $0$ belongs to the infinite cluster ${\mathcal{C}}_{\infty }$ and we consider connected subgraphs of ${\mathcal{C}}_{\infty }$ having at most $n^d$ vertices and containing $0$. Among these subgraphs, we are interested in the ones that minimize the open edge boundary size to volume ratio. These minimizers properly rescaled converge towards a translate of a deterministic shape and their open edge boundary size to volume ratio properly rescaled converges towards a deterministic constant.