Recurrence and Polya number of general one-dimensional random walks

TL;DR

This paper derives a simple formula for the Pólya number of a general 1D random walk, showing recurrence depends on equal left and right move probabilities, using creative telescoping for proof.

Contribution

It provides a closed-form expression for the Pólya number of a 1D random walk with stay probability, establishing recurrence criteria based on move probabilities.

Findings

Pólya number formula: P=1-Δ, where Δ=|l-r|

Walk is recurrent iff l=r

Rigorous proof via creative telescoping

Abstract

The recurrence properties of random walks can be characterized by P\'{o}lya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities $l$ and $r$, or remain at the same position with probability $o$ ($l+r+o=1$). We calculate P\'{o}lya number $P$ of this model and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$ ($\Delta=|l-r|$). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability $l$ equals to the right-moving probability $r$.