Quenched invariance principle for random walks in balanced random environment

TL;DR

This paper proves an invariance principle and transience/recurrence results for random walks in balanced random environments on integer lattices, extending known results to less restrictive conditions using ergodic and percolation methods.

Contribution

It establishes a quenched invariance principle and transience/recurrence criteria for random walks in balanced environments under weaker assumptions than uniform ellipticity.

Findings

Invariance principle holds for $d\\geq 2$ in ergodic environments.

Random walks are transient for $d\geq 3$ and recurrent for $d=2$.

Results extend to i.i.d. environments with mere ellipticity using percolation techniques.

Abstract

We consider random walks in a balanced random environment in $\mathbb{Z}^d$, $d\geq 2$. We first prove an invariance principle (for $d\ge2$) and the transience of the random walks when $d\ge 3$ (recurrence when $d=2$) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.