On Pólya’s random walk constants

Gaunt, Robert; Nadarajah, Saralees; Pogány, Tibor

doi:10.1090/proc/16854

On Pólya’s random walk constants
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by Robert E. Gaunt, Saralees Nadarajah and Tibor K. Pogány;

Proc. Amer. Math. Soc. 152 (2024), 3593-3597

DOI: https://doi.org/10.1090/proc/16854

Published electronically: June 14, 2024

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Abstract:

A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb {Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq 3$. Previously, a closed-form formula had only been available for $d=3$.

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