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Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher (1810.11239v1)

Published 26 Oct 2018 in math.PR

Abstract: Let $d\geq 2$. 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 $nd$ 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.

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