Reversed Dirichlet environment and directional transience of random walks in Dirichlet random environment

Christophe Sabot (1) , Laurent Tournier (1)

1 ICJ - Institut Camille Jordan

We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Sa09]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used in that article.

Mots clés: Random walk; random environment; Dirichlet distribution; time reversal; transience

Domaines: Probabilités [math.PR]

Christophe Sabot, Laurent Tournier. Reversed Dirichlet environment and directional transience of random walks in Dirichlet random environment. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2011, 47 (1), pp.1-8. ⟨10.1214/09-AIHP344⟩. ⟨hal-00387166v2⟩