Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments

TL;DR

This paper provides a combinatorial interpretation of moments of random walks in dimensions 2 and 4, linking them to lattice paths and Dyck paths, and derives formulas for related path statistics.

Contribution

It introduces a new combinatorial interpretation for the moments of random walks in dimensions 2 and 4 using lattice paths and Dyck path bijections.

Findings

Interpretation of moments as counts of lattice paths in dimension m-1

Bijection between Dyck paths with peaks and specific words

Closed formulas for lattice path statistics

Abstract

We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$, the $n$th moment can be interpreted as the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known. The goal of this paper is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths with a prescribed number of peaks and words of a certain type. In addition, this bijection allows us to derive closed formulas for the number of lattice paths provided with certain statistics.