Random walk on the randomly-oriented Manhattan lattice

TL;DR

This paper investigates a random walk on a randomly-oriented Manhattan lattice, demonstrating superdiffusive behavior in low dimensions and diffusive behavior in higher dimensions, revealing how lattice orientation affects walk dynamics.

Contribution

It introduces a new model of directed lattice random walk with random orientations and analyzes its diffusive properties across dimensions.

Findings

Superdiffusive behavior in 2 and 3 dimensions.

Diffusive behavior in 4 or more dimensions.

Orientation randomness significantly impacts walk diffusion.

Abstract

In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z}^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $\mathbb{Z}^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.