On Pólya’s random walk constants

Abstract

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