Sub-ballistic random walk in Dirichlet environment

Abstract

We consider random walks in Dirichlet environment (RWDE) on ${\mathbb{Z}}^d$, for $d\ge 3$, in the sub-ballistic case. We associate to any parameter $({\alpha }_1,...,{\alpha }_{2d})$ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's 0-1 law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement.