On the connection between transient and ballistic behaviours for RWRE

TL;DR

This paper investigates the relationship between transient and ballistic behaviors in random walks within random environments, establishing new conditions under which ballisticity and decay properties are equivalent, especially in higher dimensions.

Contribution

It introduces a new ballisticity condition based on polynomial decay that guarantees condition (T), the weakest known ballisticity assumption, and provides an alternative proof of equivalence in one dimension.

Findings

Ballisticity condition implies condition (T) under polynomial decay

Standard polynomial decay degree must exceed d for transience

Alternative proof of equivalence in one dimension

Abstract

We study the strong form of the ballistic conjecture for random walks in random environments (RWRE). This conjecture asserts that any RWRE which is directionally transient for a nonempty open set of directions satisfies condition $(T)$ (annealed exponential decay for the unlikely exit probability). Specifically, we introduce a ballisticity condition which is fulfilled as soon as a polynomial condition of degree greater than $d-1$ holds. Under that hypothesis we prove condition $(T)$, which turns this condition into the weakest-known ballisticity assumption. We recall that standard arguments to prove that a ballisticity condition implies directional transience require at least polynomial decay greater than degree $d$. Furthermore, in the one dimensional case we provide an alternative proof which proves the equivalence between transient behaviour and annealed arbitrary decay for the unlikely exit probability, we expect that this new argument might be used in higher dimensions.