Annealed and quenched fluctuations for ballistic random walks in random environment on Z

Nathanaël Enriquez (1, 2) , Christophe Sabot (3) , Laurent Tournier (4) , Olivier Zindy (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MODAL'X - Modélisation aléatoire de Paris X
3 ICJ - Institut Camille Jordan
4 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We consider transient random walks in random environment on $\Z$ in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following [7], where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. Furthermore, our proof enables us to give a description of the quenched distribution of hitting times. The case of Dirichlet environment turns out to be remarkably explicit.

Mots clés: Random walk in random environment; Stable laws; Fluctuation theory of random walks; Beta distributions; Quenched distribution

Domaines: Probabilités [math.PR]

Nathanaël Enriquez, Christophe Sabot, Laurent Tournier, Olivier Zindy. Annealed and quenched fluctuations for ballistic random walks in random environment on Z. 2010. ⟨hal-00543882v1⟩