Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Transience of the vacant set for near-critical random interlacements in high dimensions

Alexander Drewitz and Dirk Erhard

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Abstract

The model of random interlacements is a one-parameter family $\mathcal{I}^{u}$, $u\ge0$, of random subsets of ${\mathbb{Z} }^{d}$, which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its volume. Its complement, the so-called vacant set $\mathcal{V}^{u}$, has been shown to undergo a non-trivial percolation phase-transition in $u$; i.e., there exists $u_{*}(d)\in(0,\infty)$ such that for $u\in[0,u_{*}(d))$ the vacant set $\mathcal{V}^{u}$ contains a unique infinite connected component $\mathcal{V}_{\infty}^{u}$, while for $u>u_{*}(d)$ it consists of finite connected components. It is known (Probab. Theory Related Fields 150 (2011) 575–611, Ann. Probab. 39 (2011) 70–103) that $u_{*}(d)\sim\log d$, and in this article we show the existence of $u(d)>0$ with $\frac{u(d)}{u_{*}(d)}\to1$ as $d\to\infty$ such that $\mathcal{V}_{\infty}^{u}$ is transient for all $u\in[0,u(d))$.

Résumé

Le modèle des entrelacs aléatoires est une famille $\mathcal{I}^{u}$, $u\geq0$, de sous-ensembles aléatoires de ${\mathbb{Z} }^{d}$. Cette famille décrit localement la trace d’une marche aléatoire sur le tore de dimension $d$ qui évolue jusqu’au temps $u$ fois le volume du tore. Il est connu que l’ensemble vacant $\mathcal{V}^{u}$ fait l’objet d’une transition de phase non-triviale en $u$, c’est-à-dire qu’il existe $u_{*}(d)\in(0,\infty)$ tel que pour $u\in[0,u_{*}(d))$, l’ensemble vacant $\mathcal{V}^{u}$ a une unique composante infinie connexe $\mathcal{V}^{u}_{\infty}$ tandis que pour $u>u_{*}(d)$, toutes les composantes connexes de $\mathcal{V}^{u}$ sont finies. Il est connu (Probab. Theory Related Fields 150 (2011) 575–611, Ann. Probab. 39 (2011) 70–103) que $u_{*}(d)\sim\log d$; dans cette article nous montrons l’existence de $u(d)>0$, avec $\frac{u(d)}{u_{*}(d)}\to1$ quand $d\to\infty$, tel que $\mathcal{V}_{\infty}^{u}$ est transiente pour tout $u\in[0,u(d))$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 84-101.

Dates
Received: 10 December 2013
Revised: 14 May 2014
Accepted: 29 June 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089261

Digital Object Identifier
doi:10.1214/14-AIHP631

Mathematical Reviews number (MathSciNet)
MR3449295

Zentralblatt MATH identifier
1333.60202

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes 82B43: Percolation [See also 60K35]

Keywords
Random interlacements Percolation Transience Electrical networks

Citation

Drewitz, Alexander; Erhard, Dirk. Transience of the vacant set for near-critical random interlacements in high dimensions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 84--101. doi:10.1214/14-AIHP631. https://projecteuclid.org/euclid.aihp/1452089261


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