An invariance principle for a class of non-ballistic random walks in random environment

TL;DR

This paper proves a quenched invariance principle for certain non-ballistic random walks in i.i.d. environments, showing convergence to Brownian motion with a specific diffusion matrix under diffusive scaling.

Contribution

It extends previous results by establishing an invariance principle under weaker symmetry assumptions in the environment, specifically balance in one coordinate direction.

Findings

Almost sure convergence to Brownian motion under diffusive scaling

Explicit description of the diffusion matrix

Precise estimates on mean sojourn times in large balls

Abstract

We are concerned with random walks on $\mathbb{Z}^d$, $d\geq 3$, in an i.i.d. random environment with transition probabilities $\epsilon$-close to those of simple random walk. We assume that the environment is balanced in one fixed coordinate direction, and invariant under reflection in the coordinate hyperplanes. The invariance condition was used in Baur and Bolthausen (2014) as a weaker replacement of isotropy to study exit distributions. We obtain precise results on mean sojourn times in large balls and prove a quenched invariance principle, showing that for almost all environments, the random walk converges under diffusive rescaling to a Brownian motion with a deterministic (diagonal) diffusion matrix. We also give a concrete description of the diffusion matrix. Our work extends the results of Lawler (1982), where it is assumed that the environment is balanced in all coordinate directions.