Type transition of simple random walks on randomly directed regular lattices

TL;DR

This paper investigates how the recurrence or transience of simple random walks on a randomly directed lattice depends on the decay rate of horizontal edge orientations, identifying a critical decay power that determines the walk's behavior.

Contribution

It introduces a model of random walks on a partially directed lattice with a decay-dependent orientation perturbation and establishes a phase transition in recurrence versus transience.

Findings

Existence of a critical decay power for recurrence/transience transition

Walks are almost surely recurrent above the critical power

Walks are almost surely transient below the critical power

Abstract

Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ 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.