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

TL;DR

This paper proves that random walks in i.i.d. Dirichlet environments on Z^d are directionally transient when parameters are nonsymmetric, using properties of Dirichlet environments and time reversal.

Contribution

It establishes directional transience for Dirichlet environment-based random walks and provides a probabilistic proof of the environment's stability under time reversal.

Findings

Transience occurs in a direction with positive probability for nonsymmetric parameters.

In dimension 2, directional transience is almost sure due to a 0-1 law.

Provides an alternative probabilistic proof of Dirichlet environment stability.

Abstract

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.