Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Percolation and isoperimetry on roughly transitive graphs

Elisabetta Candellero and Augusto Teixeira

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Abstract

In this paper we study percolation on a roughly transitive graph $G$ with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that $p_{c}<1$, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When $G$ is transitive, the fact that $p_{c}<1$ was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov’s theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.

Résumé

Dans cet article, nous étudions la percolation sur un graphe grossièrement transitif $G$ à croissance polynomiale et de dimension isopérimétrique plus grande que 1. Pour ces graphes, nous prouvons que $p_{c}<1$ ou, en d’autres termes, nous prouvons qu’il existe une phase de percolation. Les résultats principaux de l’article sont valables à la fois pour les processus de percolation dépendants ou indépendants, car ils s’appuient sur des arguments de renormalisation assez robustes. Quand $G$ est transitif, le fait que $p_{c}<1$ était déjà connu. Mais même dans ce cas notre preuve donne des résultats nouveaux et est entièrement probabiliste, évitant l’utilisation du théorème de Gromov sur les groupes à croissance polynomiale. Nous concluons l’article par quelques exemples de percolation dépendante pour lesquels nos résultats s’appliquent.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 1819-1847.

Dates
Received: 17 February 2016
Revised: 18 July 2017
Accepted: 27 July 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849785

Digital Object Identifier
doi:10.1214/17-AIHP857

Mathematical Reviews number (MathSciNet)
MR3865659

Zentralblatt MATH identifier
06996551

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Keywords
Percolation Isoperimetric inequalities Roughly transitive graphs Dependent percolation Decoupling inequalities

Citation

Candellero, Elisabetta; Teixeira, Augusto. Percolation and isoperimetry on roughly transitive graphs. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 1819--1847. doi:10.1214/17-AIHP857. https://projecteuclid.org/euclid.aihp/1539849785


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