Collisions of random walks

Barlow, Martin T.; Peres, Yuval; Sousi, Perla

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 922-946.

Un graphe récurrent $G$ a la propriété de collisions infinies si deux marches aléatoires indépendantes dans $G$, issues du même état, se rencontrent infiniment souvent presque sûrement. Nous donnons un critère simple à l’aide de fonctions de Green qui implique cette propriété, et nous l’utilisons pour prouver que la propriété de collisions infinies a lieu dans les cas suivants: un arbre de Galton-Watson critique avec variance finie conditionné à survivre, l’amas de percolation critique conditionné à être infini dans ${\mathbb{Z}}^d$ avec $d\ge 19$ et l’arbre couvrant uniforme dans ${\mathbb{Z}}^2$. Pour le graphe en forme de peigne aléatoire avec queues polynomiales et les arbres à symétrie sphérique, nous déterminons précisément la région critique dans l’espace des phases pour les collisions infinies.

A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in ${\mathbb{Z}}^d$ with $d\ge 19$ and the uniform spanning tree in ${\mathbb{Z}}^2$ all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.

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