Random walks in random hypergeometric environment

TL;DR

This paper studies one-dependent random walks in a broad class of random hypergeometric environments on multi-dimensional integer lattices, establishing transience, return time properties, and conditions for invariant measures.

Contribution

It generalizes previous results on Dirichlet environments to hypergeometric environments, characterizing invariant measures based solely on edge weights.

Findings

Walks are almost surely transient in the considered environments.

Return times have finite positive moments.

Existence of invariant measures depends only on edge weights, not on pairwise edge weights.

Abstract

We consider one-dependent random walks on $\mathbb{Z}^d$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function $\kappa$ of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that $\kappa$ coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges.