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

TL;DR

This paper extends the zero-one law for directional transience from nearest-neighbor to bounded jump random walks in random environments on -dimensional integer lattices, covering both i.i.d. and Dirichlet environments.

Contribution

It generalizes existing results on directional transience and zero-one laws to include bounded jumps in two-dimensional and Dirichlet environments.

Findings

Extended zero-one law to bounded jumps in D environments.

Characterized directional transience for bounded jumps in Dirichlet environments.

Connected the results to prior nearest-neighbor cases.

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.