Un graphe récurrent a la propriété de collisions infinies si deux marches aléatoires indépendantes dans , 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 avec et l’arbre couvrant uniforme dans . 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 has the infinite collision property if two independent random walks on , 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 with and the uniform spanning tree in 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.
@article{AIHPB_2012__48_4_922_0, author = {Barlow, Martin T. and Peres, Yuval and Sousi, Perla}, title = {Collisions of random walks}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, volume = {48}, year = {2012}, pages = {922-946}, doi = {10.1214/12-AIHP481}, mrnumber = {3052399}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_4_922_0} }
Barlow, Martin T.; Peres, Yuval; Sousi, Perla. Collisions of random walks. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 922-946. doi : 10.1214/12-AIHP481. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_4_922_0/
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