Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random walks on discrete point processes

Noam Berger and Ron Rosenthal

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We consider a model for random walks on random environments (RWRE) with a random subset of ${\mathbb{Z} }^{d}$ as the vertices, and uniform transition probabilities on $2d$ points (the closest in each of the coordinate directions). We prove that the velocity of such random walks is almost surely zero, give partial characterization of transience and recurrence in the different dimensions and prove a Central Limit Theorem (CLT) for such random walks, under a condition on the distance between coordinate nearest neighbors.


Nous considérons un modèle de marches aléatoires en milieu aléatoire ayant pour sommets un sous-ensemble aléatoire de ${\mathbb{Z} }^{d}$ et une probabilité de transition uniforme sur $2d$ points (les plus proches voisins dans chacune des directions des coordonnées). Nous prouvons que la vitesse de ce type de marches est presque sûrement zéro, donnons une caractérisation partielle de transience et récurrence dans les différentes dimensions et prouvons un théorème central limite (CLT) pour de telles marches sous une condition concernant la distance entre plus proches voisins.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 727-755.

First available in Project Euclid: 10 April 2015

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Discrete point processes Random walk in random environment


Berger, Noam; Rosenthal, Ron. Random walks on discrete point processes. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 727--755. doi:10.1214/13-AIHP593.

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