Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous

2018/12/29 by Barbara Dembin, Dembin, Barbara
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1901.00367

openalex publication_date 2018/12/29 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

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 parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C_p [7]. We are interested in the regularity properties in p of the anchored isoperimetric profile of the infinite cluster C_p. For d≥2, we prove that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals [p₀ , p₁ ] ⊂ (p_c (d), 1).

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