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On Pólya's random walk constants

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

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 $d\geq3$. Previously, a closed-form formula had only been available for $d=3$.

On Pólya's random walk constants

Abstract

A celebrated result in probability theory is that a simple symmetric random walk on the -dimensional lattice is recurrent for and transient for . In this note, we derive a closed-form expression, in terms of the Lauricella function , for the return probability for all . Previously, a closed-form formula had only been available for .
Paper Structure (2 sections, 3 theorems, 14 equations)

This paper contains 2 sections, 3 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Closed-form expression for the return probability

Key Result

Theorem 2.1

For any positive integer $d\geq3$,

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof