Type transition of simple random walks on randomly directed regular lattices1

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 the simple random walk, i.e. its being recurrent or transient, and show that there exists a critical value of the decay power, above which the walk is almost surely recurrent and below which is almost surely transient. Supported in part by the Actions Internationales programme of the Université de Rennes 1. 1991 Mathematics Subject Classification: 60J10, 60K20