Computable criteria for ballisticity of random walks in elliptic random environment

TL;DR

This paper introduces computable criteria to determine when random walks in elliptic random environments exhibit ballistic behavior, especially in non-uniform cases, by analyzing exit times and trapping geometries.

Contribution

It provides the first explicit, computable conditions for ballisticity in non-uniform elliptic environments, with a geometric classification of trapping mechanisms.

Findings

Introduces a computable condition in 2D for ballisticity.

Extends the condition to higher dimensions based on exit times.

Proves the sharpness of the general condition.

Abstract

We consider random walks in i.i.d. elliptic random environments which are not uniformly elliptic. We introduce a computable condition in dimension $d=2$ and a general condition valid for dimensions $d\ge 2$ expressed in terms of the exit time from a box, which ensure that local trapping would not inhibit a ballistic behavior of the random walk. An important technical innovation related to our computable condition, is the introduction of a geometrical point of view to classify the way in which the random walk can become trapped, either in an edge, a wedge or a square. Furthermore, we prove that the general condition we introduce is sharp.