On Pólya's random walk constants

Robert E. Gaunt; Saralees Nadarajah; Tibor K. Pogány

doi:10.1090/proc/16854

On Pólya's random walk constants

Robert E. Gaunt, Saralees Nadarajah, Tibor K. Pogány

Probability

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Abstract

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

Original language	English
Journal	Proceedings of the American Mathematical Society
DOIs	https://doi.org/10.1090/proc/16854
Publication status	Published - 12 Apr 2024

Keywords

Random walk
return probability
P´olya’s random walk constants
Lauricella function
Watson’s triple integrals
Laplace transform

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10.1090/proc/16854

polya fvAccepted author manuscript, 308 KBLicence: CC BY

Cite this

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title = "On P{\'o}lya's random walk constants",

abstract = "A celebrated result in probability theory is that a simple symmetric random walk on the d-dimensional lattice Zd is recurrent for d=1,2 and transient for d ≥ 3. In this note, we derive a closed-form expression, in terms of the Lauricella function of type C, for the return probability for all d ≥ 3. Previously, a closed-form formula had only been available for d=3.",

keywords = "Random walk, return probability, P´olya{\textquoteright}s random walk constants, Lauricella function, Watson{\textquoteright}s triple integrals, Laplace transform",

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AU - Gaunt, Robert E.

AU - Nadarajah, Saralees

AU - Pogány, Tibor K.

N1 - 4 pages

PY - 2024/4/12

Y1 - 2024/4/12

N2 - A celebrated result in probability theory is that a simple symmetric random walk on the d-dimensional lattice Zd is recurrent for d=1,2 and transient for d ≥ 3. In this note, we derive a closed-form expression, in terms of the Lauricella function of type C, for the return probability for all d ≥ 3. Previously, a closed-form formula had only been available for d=3.

AB - A celebrated result in probability theory is that a simple symmetric random walk on the d-dimensional lattice Zd is recurrent for d=1,2 and transient for d ≥ 3. In this note, we derive a closed-form expression, in terms of the Lauricella function of type C, for the return probability for all d ≥ 3. Previously, a closed-form formula had only been available for d=3.

KW - Random walk

KW - return probability

KW - P´olya’s random walk constants

KW - Lauricella function

KW - Watson’s triple integrals

KW - Laplace transform

U2 - 10.1090/proc/16854

DO - 10.1090/proc/16854

M3 - Article

SN - 0002-9939

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

ER -

On Pólya's random walk constants

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Keywords

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