A non-backtracking Polya's theorem

Mark Kempton

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arXiv · 1610.04672

A non-backtracking Polya's theorem

Mark Kempton

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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.

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Mark Kempton. 2016-10-15. A non-backtracking Polya's theorem. https://arxiv.org/abs/1610.04672

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