On the limiting velocity of random walks in mixing random environment

Guo, Xiaoqin

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 375-402.

Nous considérons des marches aléatoires dans un environnement Gibbsien fortement mélangeant dans ${\mathbb{Z}}^d$, $d\ge 2$. A l’aide d’arguments de renouvellement, nous donnons d’abord une preuve alternative de la loi conditionnelle des grands nombres de Rassoul-Agha (Electron. Commun. Probab. 10 (2005) 36-44) pour des environnements mélangeants. Ensuite, par des méthodes de couplage, nous montrons qu’il existe au plus une vitesse limite non nulle en grande dimension ($d\ge 5$).

We consider random walks in strong-mixing random Gibbsian environments in ${\mathbb{Z}}^d$, $d\ge 2$. Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment (Electron. Commun. Probab. 10 (2005) 36-44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ($d\ge 5$).

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