Asymptotic directions in random walks in random environment revisited

TL;DR

This paper revisits and improves upon recent results on the asymptotic behavior of elliptic random walks in random environments, establishing equivalences related to transience and asymptotic directions.

Contribution

It provides an improved proof of the equivalence between transience and the existence of a deterministic asymptotic direction, and discusses open problems in the field.

Findings

Established equivalence between transience and asymptotic direction existence

Improved upon previous proofs of asymptotic behavior

Reviewed open problems in random walk in random environment

Abstract

Recently Simenhaus proved that for any elliptic random walk in random environment, transience in the neighborhood of a given direction is equivalent to the a.s. existence of a deterministic asymptotic direction and to transience in any direction in the open half space defined by this asymptotic direction. Here we prove an improved version of this result and review some open problems.