Invariant measure for the process viewed from the particle for 2D random walks in Dirichlet environment

TL;DR

This paper investigates the existence of invariant measures for the process viewed from the particle in 2D random walks within Dirichlet environments, revealing conditions under which such measures exist or do not, based on recurrence and transience properties.

Contribution

It introduces a new identity inspired by the $ extstar$-VRJP and applies it to classify cases of invariant measure existence for 2D RWDE.

Findings

No invariant measure exists in the recurrent case if the walk is symmetric.

An invariant measure exists under certain transience and trapping conditions.

The new identity links RWDE to random Schrödinger operators and generalizes known properties in 1D.

Abstract

In this paper, we consider random walks in Dirichlet random environment (RWDE) on $\mathbb{Z}^2$. We prove that, if the RWDE is recurrent (which is strongly conjectured when the weights are symmetric), then there does not exist any invariant measure for the process viewed from the particle which is absolutely continuous with respect to the static law of the environment. Besides, if the walk is directional transient and under condition $\mathbf{(T')}$, we prove that there exists such an invariant probability measure if the trapping parameter verifies $\kappa > 1$ or after acceleration of the process by a local function of the environment. This gives strong credit to a conjectural classification of cases of existence or non-existence of the invariant measure for two dimensional RWDE. The proof is based on a new identity, stated on general finite graphs, which is inspired by the representation of the $\star$-VRJP, a non-reversible generalization of the Vertex reinforced Jump Process, in terms of random Schr\"odinger operators. In the case of RWDE on 1D graph, the previous identity entails also a discrete analogue of the Matsumoto-Yor property for Brownian motion.