Limit laws for transient random walks in random environment on $\mathbb{Z}$

Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2469-2508.

Nous considérons les marches aléatoires en milieu aléatoire sur $\mathbb{Z}$ transientes et de vitesse nulle. D’après un résultat classique de Kesten, Kozlov et Spitzer, le temps d’atteinte du niveau $n$ converge en loi, après renormalisation, vers une variable aléatoire stable positive, mais ces auteurs n’obtiennent pas la description de son paramètre. Une preuve différente est présentée, qui permet d’obtenir une caractérisation complète de cette loi stable. Le cas d’environnements de Dirichlet s’avère être particulièrement explicite.

We consider transient random walks in random environment on $\mathbb{Z}$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

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