Exit laws from large balls of (an)isotropic random walks in random environment

TL;DR

This paper analyzes the exit laws of large balls for isotropic and anisotropic random walks in small-perturbation random environments, establishing their closeness to symmetric walks and proving transience.

Contribution

It extends previous results by including anisotropic walks and offers a perturbative approach to understanding exit laws and transience in random environments.

Findings

Exit laws are close to those of symmetric random walks.

Bounds on total variation distances are established.

Transience of the random walks is proved.

Abstract

We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581-645]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author's thesis [Long-time behavior of random walks in random environment (2013) Z\"{u}rich Univ.]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.