Regularity of isoperimetric and time constants for a supercritical Bernoulli percolation

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. We are interested in the regularity properties of two distinct objects defined on this infinite cluster: the isoperimetric (or Cheeger) constant, and the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ C_p corresponds to the length of the shortest path in C_p joining the two points. The chemical distance between 0 and nx grows asymptotically like n μ_p (x). We aim to study the regularity properties of the map p → μ_p on the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G_p = p δ_1 + (1 − p) δ_∞ , p > p_c (d). It is already known that the map p → μ_p is continuous. We prove here that p → μ_p satisfies stronger regularity properties, this map is almost Lipschitz continuous up to a logarithmic factor in [p_0 , 1] for any p_0 > p_c (d). We prove an analog result for the Cheeger constant in dimension 2 for all intervals [p_0 , p_1 ] ⊂ (1/2, 1). For d ≥ 3, we prove that the modified Cheeger constant defined by Gold is Lipschitz continuous on all intervals [p_0 , p_1 ] ⊂ (p_c (d), 1).
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https://hal.archives-ouvertes.fr/hal-01721917
Contributor : Barbara Dembin <>
Submitted on : Monday, March 5, 2018 - 10:33:08 AM
Last modification on : Wednesday, April 8, 2020 - 1:40:03 PM
Document(s) archivé(s) le : Wednesday, June 6, 2018 - 12:34:40 PM

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• HAL Id : hal-01721917, version 1
• ARXIV : 1803.03141

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Barbara Dembin. Regularity of isoperimetric and time constants for a supercritical Bernoulli percolation. 2018. ⟨hal-01721917v1⟩

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