Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Collisions of random walks

Martin T. Barlow, Yuval Peres, and Perla Sousi

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Abstract

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 $d\ge19$ 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.

Résumé

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 $d\geq19$ 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 922-946.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098434

Digital Object Identifier
doi:10.1214/12-AIHP481

Mathematical Reviews number (MathSciNet)
MR3052399

Zentralblatt MATH identifier
1285.60073

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C81: Random walks on graphs

Keywords
Random walks Collisions Transition probability Branching processes

Citation

Barlow, Martin T.; Peres, Yuval; Sousi, Perla. Collisions of random walks. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 922--946. doi:10.1214/12-AIHP481. https://projecteuclid.org/euclid.aihp/1353098434


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