Isoperimetry in supercritical bond percolation in dimensions three and\n higher

2016/02/17 by Julian Gold, Gold, Julian
Mathematics · #52B60 #60K35 #82B43 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1602.05598

openalex publication_date 2016/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the isoperimetric subgraphs of the infinite cluster\n\C_\∞ for supercritical bond percolation on \ℤ^d with\nd\≥ 3. Specifically, we consider the subgraphs of \C_\∞ \∩\n[-n,n]^d which have minimal open edge boundary to volume ratio. We prove a\nshape theorem for these subgraphs, obtaining that when suitably rescaled, these\nsubgraphs converge almost surely to a translate of a deterministic shape. This\ndeterministic shape is itself an isoperimetric set for a norm we construct. As\na corollary, we obtain sharp asymptotics on a natural modification of the\nCheeger constant for \C_\∞ \∩ [-n,n]^d. This settles a\nconjecture of Benjamini for the version of the Cheeger constant defined here.\n

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