Excited random walks with non-nearest neighbor steps

TL;DR

This paper studies a class of self-interacting random walks with non-nearest neighbor steps, establishing conditions for recurrence and transience based on the expected step size, extending known results to more general jump distributions.

Contribution

It provides a new criterion for recurrence and transience of excited random walks with arbitrary jump distributions, generalizing previous nearest-neighbor results.

Findings

Walks are recurrent if expected step size ≤ 1.

Walks are transient if expected step size > 1.

Results apply to a broad class of non-nearest neighbor jumps.

Abstract

Let $W$ be an integer valued random variable satisfying $E[W] =: \delta \geq 0$ and $P(W<0)>0$, and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $x\in \mathbb{Z}$ the size of the next step is an independent random variable with the same distribution as $W$. We show that this self-interacting random walk is recurrent if $\delta\leq 1$ and transient if $\delta>1$. This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.