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

Černý, Jiří; Teixeira, Augusto; Windisch, David

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 929-968.

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.

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.

Classification : 60G50, 05C80, 82B41
Mots clés : random walk, vacant set, regular graph, expanders, random interlacement, phase transition

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