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

Abstract

This paper has two main results, which are connected through the fact that the first is a key ingredient in the second. Both are extensions of results concerning directional transience of nearest-neighbor random walks in random environments to allow for bounded jumps. Zerner and Merkl proved a 0-1 law for directional transience for planar random walks in random environments. We extend the result to non-planar i.i.d. random walks in random environments on $\mathbb{Z}^2$ with bounded jumps. Sabot and Tournier characterized directional transience for a given direction for nearest-neighbor random walks in Dirichlet environments on $\mathbb{Z}^d$, $d\geq1$. We extend this characterization to random walks in Dirichlet environments with bounded jumps.