Sub-ballistic random walk in Dirichlet environment

TL;DR

This paper studies sub-ballistic random walks in Dirichlet environments on multidimensional integer lattices, establishing invariant measures, characterizing transience, and determining displacement growth rates.

Contribution

It introduces a time-change method to analyze the walk, proves the existence of an invariant measure, and characterizes transience in Dirichlet environments, solving a longstanding problem.

Findings

Invariant measure for accelerated walk established

Directional transience characterized in Dirichlet environment

Displacement magnitude follows a polynomial order

Abstract

We consider random walks in Dirichlet environment (RWDE) on $\Z ^d$, for $ d \geq 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.