Barlow, Martin T.; Peres, Yuval; Sousi, Perla
Collisions of random walks
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4 , p. 922-946
MR 3052399 | 1 citation dans Numdam
doi : 10.1214/12-AIHP481
URL stable : http://www.numdam.org/item?id=AIHPB_2012__48_4_922_0

Classification:  60J10,  60J35,  60J80,  05C81
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 d avec d19 et l’arbre couvrant uniforme dans 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 d with d19 and the uniform spanning tree in 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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