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

2018/10/26 by Dembin, Barbara
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1810.11239

Abstract

Let d≥ 2. We consider an i.i.d. supercritical bond percolation on 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 C_∞ and we consider connected subgraphs of C_∞ 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.

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