Random walks on discrete point processes

Berger, Noam; Rosenthal, Ron

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 727-755.

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.

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.

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