Phase transition for the vacant set left by random walk on the giant component of a random graph

Wassmer, Tobias

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 756-780.

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).

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).

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