Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Isoperimetry in supercritical bond percolation in dimensions three and higher

Julian Gold

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Abstract

We study the isoperimetric subgraphs of the infinite cluster $\mathbf{C}_{\infty}$ for supercritical bond percolation on $\mathbb{Z}^{d}$ with $d\geq3$. Specifically, we consider subgraphs of $\mathbf{C}_{\infty}\cap[-n,n]^{d}$ having minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs: when suitably rescaled, they converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\mathbf{C}_{\infty}\cap[-n,n]^{d}$, settling a conjecture of Benjamini for the version of the Cheeger constant defined here.

Résumé

Nous étudions les sous-graphes isopérimétriques du cluster infini $\mathbf{C}_{\infty}$ pour la percolation par arêtes surcritique sur $\mathbb{Z}^{d}$ avec $d\geq3$. Plus précisément, nous considérons les sous-graphes de $\mathbf{C}_{\infty}\cap[-n,n]^{d}$ qui ont une frontière ouverte minimale par rapport au volume. Nous prouvons un théorème de forme pour ces sous-graphes: convenablement normalisés, ils convergent presque surement vers une translation d’une forme limite déterministe. Cette forme déterministe est elle aussi un ensemble isopérimétrique pour une norme que nous définissons. Comme corollaire, nous obtenons une estimée précise sur une modification naturelle de la constante de Cheeger pour $\mathbf{C}_{\infty}\cap[-n,n]^{d}$, résolvant ainsi une conjecture de Benjamini pour cette version de la constante de Cheeger.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2092-2158.

Dates
Received: 29 August 2017
Accepted: 20 September 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849794

Digital Object Identifier
doi:10.1214/17-AIHP866

Mathematical Reviews number (MathSciNet)
MR3865668

Zentralblatt MATH identifier
06996560

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 52B60: Isoperimetric problems for polytopes

Keywords
Percolation Limit shapes Isoperimetry Cheeger constant

Citation

Gold, Julian. Isoperimetry in supercritical bond percolation in dimensions three and higher. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2092--2158. doi:10.1214/17-AIHP866. https://projecteuclid.org/euclid.aihp/1539849794


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