Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz, Balázs Ráth, and Artëm Sapozhnikov

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Abstract

Random interlacements at level $u$ is a one parameter family of connected random subsets of $\mathbb{Z}^{d}$, $d\geq3$ (Ann. Math. 171 (2010) 2039–2087). Its complement, the vacant set at level $u$, exhibits a non-trivial percolation phase transition in $u$ (Comm. Pure Appl. Math. 62 (2009) 831–858; Ann. Math. 171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab. 19 (2009) 454–466).

In this paper we study local percolative properties of the vacant set of random interlacements at level $u$ for all dimensions $d\geq3$ and small intensity parameter $u>0$. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level $u$. In particular, this implies that finite connected components of the vacant set at level $u$ are unlikely to be large. These results are new for $d\in\{3,4\}$. The case of $d\geq5$ was treated in (Probab. Theory Related Fields 150 (2011) 529–574) by a method that crucially relies on a certain “sausage decomposition” of the trace of a high-dimensional bi-infinite random walk. Our approach is independent from that of (Probab. Theory Related Fields 150 (2011) 529–574). It only exploits basic properties of random walks, such as Green function estimates and Markov property, and, as a result, applies also to the more challenging low-dimensional cases. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

Résumé

Un entrelac aléatoire au niveau $u$ est une famille à un paramètre de sous-ensembles connexes aléatoires de $\mathbb{Z}^{d}$, $d\geq3$, introduit dans (Ann. Math. 171 (2010) 2039–2087). Son complémentaire, l’ensemble vacant au niveau $u$, possède une transition de percolation non triviale en $u$, comme il a été montré dans (Comm. Pure Appl. Math. 62 (2009) 831–858) et (Ann. Math. 171 (2010) 2039–2087). La composante connexe infinie, lorsqu’elle existe, est presque sûrement unique, voir (Ann. Appl. Probab. 19 (2009) 454–466).

Dans ce papier, nous étudions les propriétés percolatives locales de l’ensemble vacant au niveau $u$ en toutes dimensions $d\geq3$ et pour un petit paramètre d’intensité $u$. Nous donnons une borne exponentielle tendue sur la probabilité qu’un grand (hyper)cube contienne deux composantes macroscopiques distinctes de l’ensemble vacant au niveau $u$. Nos résultats impliquent qu’il est peu probable que les composantes connexes finies de l’ensemble vacant au niveau $u$ soient grandes. Ces résultats ont été prouvés dans (Probab. Theory Related Fields 150 (2011) 529–574) pour $d\geq5$. Notre approche est différente (de celle de (Probab. Theory Related Fields 150 (2011) 529–574)) et est valide pour $d\geq3$.

L’un des ingrédients principaux de la preuve est une certaine propriété d’indépendence conditionelle des entrelacs aléatoires, qui est intéressante en elle-même.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1165-1197.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555496

Digital Object Identifier
doi:10.1214/13-AIHP540

Mathematical Reviews number (MathSciNet)
MR3269990

Zentralblatt MATH identifier
1319.60180

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

Keywords
Random interlacement Random walk Large finite cluster Supercriticality Conditional independence

Citation

Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm. Local percolative properties of the vacant set of random interlacements with small intensity. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1165--1197. doi:10.1214/13-AIHP540. https://projecteuclid.org/euclid.aihp/1413555496


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