Phase transition for the vacant set left by random walk on the giant component of a random graph
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 756-780.

We study the simple random walk on the giant component of a supercritical Erdős–Rényi random graph on n vertices, in particular the so-called vacant set at level u, the complement of the trajectory of the random walk run up to a time proportional to u and n. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter u : For u<u the vacant set has with high probability a unique giant component of order n and all other components small, of order at most log 7 n, whereas for u>u it has with high probability all components small. Moreover, we show that u coincides with the critical parameter of random interlacements on a Poisson–Galton–Watson tree, which was identified in (Electron. Commun. Probab. 15 (2010) 562–571).

Nous étudions la marche aléatoire sur la composante principale d’un graphe aléatoire d’Erdős–Rényi avec n sommets, en particulier l’ensemble vacant au niveau u, le complément de la trajectoire de la marche aléatoire jusqu’à un moment proportionnel à u et n. Nous prouvons que la structure de composant montre une transition de phase à un valeur critique u : Pour u<u l’ensemble vacant se compose, avec une forte probabilité quand n croît, d’une seule composante principale avec volume d’ordre n et des composantes petites d’ordre au plus log 7 n, alors que pour u>u tous les composants sont petits. En outre nous montrons que u coïncide avec le paramètre critique des entrelacs aléatoires sur un arbre de Poisson–Galton–Watson identifié en (Electron. Commun. Probab. 15 (2010) 562–571).

DOI: 10.1214/13-AIHP596
Classification: 05C81,  05C08,  60J10,  60K35
Keywords: random walk, vacant set, Erdős–Rényi random graph, giant component, phase transition, random interlacements
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Wassmer, Tobias. Phase transition for the vacant set left by random walk on the giant component of a random graph. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 756-780. doi : 10.1214/13-AIHP596. http://www.numdam.org/articles/10.1214/13-AIHP596/

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