A zero-one law for random walks in random environments on $\mathbb{Z}^2$ with bounded jumps

Daniel J. Slonim

Published 2021-08-25Version 1

Zerner and Merkl proved a 0-1 law for directional transience for planar random walks in random environments. Later, Zerner provided a simplified proof. We extend the result to non-planar i.i.d. random walks in random environments on $\Z^2$ with bounded jumps. As an application, we complete a characterization of directional transience (for a given direction) for random walks in Dirichlet environments with bounded jumps in all dimensions. Such a characterization was previously known for the nearest-neighbor case of Dirichlet environments.