Random walk on the randomly-oriented Manhattan lattice

Sean Ledger,Benedek Valkó,Bálint Tóth

doi:10.1214/18-ecp144

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.

Highlights

In the randomly-oriented Manhattan lattice, every line in Zd is assigned a uniform random direction
We study a continuous-time nearest neighbor random walk on Zd in the random environment ω(i, x)
In this note we show how the method transfers to the Manhattan lattice to prove superdiffusivity of the random walk in d = 2 and d = 3

Summary

Introduce the quantitytt

Note that because S is self-adjoint, the term (Aψ, (λ − S)−1Aψ)π is nonnegative. Dropping it from the expression inside the supremum in (1.7) gives the following upper bound:. Since S is the generator of the environment process as seen from a symmetric simple random walk, (φ, (λ − S)−1φ)π can be computed directly, which leads to the upper bounds on EG(λ).

For p

If d then for all p