Asymptotic direction of random walks in Dirichlet environment

TL;DR

This paper proves that in i.i.d. Dirichlet environments on Z^d, random walks almost surely have an asymptotic direction aligned with the initial drift unless it is zero, and it also determines specific probability distributions.

Contribution

It generalizes previous results on directional transience and identifies exact probabilities, extending understanding of reinforced random walks in Dirichlet environments.

Findings

Random walks in Dirichlet environments have a.s. asymptotic direction matching initial drift.

The paper determines the exact distribution of certain probabilities related to the walk.

It confirms and extends a conjecture on reinforced random walks.

Abstract

In this paper we generalize the result of directional transience from [SabotTournier10]. This enables us, by means of [Simenhaus07], [ZernerMerkl01] and [Bouchet12] to conclude that, on Z^d (for any dimension d), random walks in i.i.d. Dirichlet environment, or equivalently oriented-edge reinforced random walks, have almost-surely an asymptotic direction equal to the direction of the initial drift, unless this drift is zero. In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of [SaTo10].