Integrability of exit times and ballisticity for random walks in Dirichlet environment

TL;DR

This paper investigates the conditions under which exit times are integrable for random walks in Dirichlet environments, providing explicit criteria and refining existing ballisticity conditions on general directed graphs.

Contribution

It offers an explicit equivalent condition for the integrability of exit times in Dirichlet environments and refines the ballisticity criterion for such random walks.

Findings

Explicit condition for exit time integrability

Refined ballisticity criterion

Applicability to general directed graphs

Abstract

We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.