Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Giant vacant component left by a random walk in a random d-regular graph

Jiří Černý, Augusto Teixeira, and David Windisch

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Abstract

We study the trajectory of a simple random walk on a d-regular graph with d ≥ 3 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u ∈ (0, ∞) such that, with high probability as n grows, if u < u, then the largest component of the vacant set has a volume of order n, and if u > u, then it has a volume of order logn. The critical value u coincides with the critical intensity of a random interlacement process on a d-regular tree. We also show that the random interlacements model describes the structure of the vacant set in local neighbourhoods.

Résumé

Nous étudions la trajectoire d’une marche aléatoire simple sur un graphe d-régulier avec d ≥ 3 dont la structure ressemble localement à un arbre, quand le nombre de sommets n du graphe croît. Des exemples de tels graphes comprennent des graphes aléatoires d-réguliers et des ‘expanseur de grande maille’. Pour ces graphes, nous étudions les propriétés de percolation de l’ensemble des sommets non visités par la marche jusqu’au moment un, où u > 0 est un paramètre positif fixé. Nous montrons que cet ensemble vacant subit une transition de phase en u dans le sens suivant : il existe un seuil u ∈ (0, ∞) explicitement calculable tel que, avec une forte probabilité quand n croît, si u < u, la plus grande composante de l’ensemble vacant a un volume d’ordre n, et si u > u, elle a un volume d’ordre logn. La valeur critique u coïncide avec l’intensité critique des entrelacs aléatoires sur un arbre d-régulier. Nous montrons aussi que les entrelacs aléatoires décrivent bien la structure de l’ensemble vacant dans des voisinages locaux.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 4 (2011), 929-968.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1317906496

Digital Object Identifier
doi:10.1214/10-AIHP407

Mathematical Reviews number (MathSciNet)
MR2884219

Zentralblatt MATH identifier
1267.05237

Subjects
Primary: 60G50: Sums of independent random variables; random walks 05C80: Random graphs [See also 60B20] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Random walk Vacant set Regular graph Expanders Random interlacement Phase transition

Citation

Černý, Jiří; Teixeira, Augusto; Windisch, David. Giant vacant component left by a random walk in a random d -regular graph. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 929--968. doi:10.1214/10-AIHP407. https://projecteuclid.org/euclid.aihp/1317906496


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