A Stationary, Mixing and Perturbative Counterexample to the 0-1-law for Random Walk in Random Environment in Two Dimensions

TL;DR

This paper constructs a two-dimensional stationary, mixing, and perturbative environment where a random walk exhibits non-trivial probability of drifting, challenging the 0-1-law known for i.i.d. environments.

Contribution

It provides the first counterexample in 2D showing the failure of the 0-1-law for RWRE under stationary, mixing, and perturbative conditions.

Findings

Counterexample environment with non-trivial drift probability

Challenges the 0-1-law for stationary, mixing environments

Demonstrates limitations of existing theoretical assumptions

Abstract

We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d.\ environments.