Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Cycle structure of percolation on high-dimensional tori

Remco van der Hofstad and Artëm Sapozhnikov

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Abstract

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdős–Rényi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal.

In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus $\{-\lfloor r/2\rfloor,\ldots,\lceil r/2\rceil-1\}^{d}$. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width $r/4$ centered around each of their vertices, have length of order $r^{d/3}$, as on the critical Erdős–Rényi random graph. On the Erdős–Rényi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt (2010).

Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on $\mathbb{Z}^{d}$ with constraints on their lengths. We believe these estimates are interesting in their own right.

Résumé

Plusieurs propriétés du comportement des grandes composantes connexes de la percolation critique sur le tore en dimensions grandes ont été récemment établies, telles la taille et le diamétre. L’ordre de grandeur de ces quantités est égal à celle de la percolation sur le graphe complet ou sur le graphe aléatoire de Erdős–Rényi. Ce résultat suggère la question de savoir si les limites d’échelles des plus grandes composantes connexes, telles qu’identifiées par Aldous (1997), sont aussi égales.

Dans ce travail, nous étudions la structure des cycles des plus grandes composantes connexes pour la percolation critique en grande dimension sur le tore $\{-\lfloor r/2\rfloor,\ldots,\lceil r/2\rceil-1\}^{d}$. Alors que les amas de percolation ont plusieurs cycles courts, nous montrons que les cycles longs, c’est-à-dire ceux qui passent à travers la frontière de chacun des cubes de largeur $r/4$ centrés aux sommets du cycle, ont une longueur de l’ordre $r^{d/3}$, comme dans le cas du graphe aléatoire critique d’Erdős–Rényi. Sur ce dernier, les cycles jouent un rôle essentiel dans la limite d’échelle des grands amas critiques tels qu’identifiés par Addario-Berry, Broutin and Goldschmidt (2010).

Les preuves sont basées de manière cruciale sur de nouvelles estimations de la probabilités d’existence de chemins ouverts dans la percolation critique de type Bernouilli sur $\mathbb{Z}^{d}$ avec des contraintes sur leurs longueurs. Ces estimations sont potentiellement intéressantes en soi.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 999-1027.

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1403277006

Digital Object Identifier
doi:10.1214/13-AIHP565

Mathematical Reviews number (MathSciNet)
MR3224297

Zentralblatt MATH identifier
1297.05222

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
Random graph Phase transition Critical behavior Percolation Torus Cycle structure

Citation

van der Hofstad, Remco; Sapozhnikov, Artëm. Cycle structure of percolation on high-dimensional tori. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 999--1027. doi:10.1214/13-AIHP565. https://projecteuclid.org/euclid.aihp/1403277006


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