Type Transition of Simple Random Walks on Randomly Directed Regular Lattices

Simple random walks on a partially directed version of Z are considered. More precisely, vertical edges between neighbouring vertices of Z 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 it is almost surely recurrent and below which it is almost surely transient.