A non-backtracking Polya's theorem

TL;DR

This paper extends Pólya's theorem to non-backtracking random walks, showing they are recurrent in 2D and transient in 1D and higher dimensions, with exact enumeration and combinatorial insights.

Contribution

It proves a version of Pólya's theorem for non-backtracking walks and provides exact enumeration and combinatorial links on 2D grids.

Findings

Non-backtracking walk is recurrent in 2D

Non-backtracking walk is transient in 1D and higher dimensions

Exact enumeration links to trinomial coefficients

Abstract

P\'olya's random walk theorem states that a random walk on a $d$-dimensional grid is recurrent for $d=1,2$ and transient for $d\ge3$. We prove a version of P\'olya's random walk theorem for non-backtracking random walks. Namely, we prove that a non-backtracking random walk on a $d$-dimensional grid is recurrent for $d=2$ and transient for $d=1$, $d\ge3$. Along the way, we prove several useful general facts about non-backtracking random walks on graphs. In addition, our proof includes an exact enumeration of the number of closed non-backtracking random walks on an infinite 2-dimensional grid. This enumeration suggests an interesting combinatorial link between non-backtracking random walks on grids, and trinomial coefficients.