Random walks in random hypergeometric environment

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.