Type transition of simple random walks on randomly directed regular lattices

Abstract

Simple random walks on a partially directed version of Z² are considered. More precisely, vertical edges between neighbouring vertices of Z² can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function; the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.