Random walks in Dirichlet environment: an overview

TL;DR

This paper reviews the properties and recent advances in understanding Random Walks in Dirichlet Environment, highlighting their invariance properties, regimes, and providing new large deviation results in one dimension.

Contribution

It offers a comprehensive overview of RWDE, emphasizing their invariance properties and presenting new large deviation rate function calculations for 1D cases.

Findings

Invariance under time reversal simplifies analysis.

Identification of transient and ballistic regimes.

New large deviation rate function for 1D RWDE.

Abstract

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights $(\alpha_i)_{i=1, \ldots, 2d}$, one for each direction of $\Bbb{Z}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.