The mean square displacement of random walk on the Manhattan lattice

TL;DR

This paper derives an explicit formula for the mean square displacement of a random walk on the Manhattan lattice across all dimensions and steps, providing a comprehensive mathematical characterization.

Contribution

It presents the first explicit formula for the mean square displacement of random walks on the Manhattan lattice for any dimension and number of steps.

Findings

Explicit formula for mean square displacement in all dimensions

Applicable for all step counts n

Enhances understanding of random walk behavior on Manhattan lattice

Abstract

We give an explicit formula for the mean square displacement of the random walk on the $d$-dimensional Manhattan lattice after $n$ steps, for all $n$ and all dimensions $d \geq 2$.