Collisions of random walks
Barlow, Martin T. ; Peres, Yuval ; Sousi, Perla
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 922-946 / Harvested from Numdam

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/12-AIHP481
Classification:  60J10,  60J35,  60J80,  05C81
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     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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