On Pólya's random walk constants

2023/11/19 by Robert E. Gaunt, Gaunt, Robert E., Saralees Nadarajah +3
Mathematics · Computer Science · #Stochastic processes and statistical mechanics #Bayesian Methods and Mixture Models #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2311.11326

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 F_C, for the return probability for all d≥3. Previously, a closed-form formula had only been available for d=3.

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