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arXiv · 2605.16715

Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments

Sergey Kirgizov·Khaydar Nurligareev·Michael Wallner
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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.

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Sergey Kirgizov, Khaydar Nurligareev, Michael Wallner. 2026-05-15. Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments. https://arxiv.org/abs/2605.16715

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