Recurrence/Transience criteria for excited random walks with finite-drift cookie stacks

TL;DR

This paper establishes recurrence and transience criteria for excited random walks on integers with infinite cookie stacks, showing a phase transition at total drift magnitude 1 and discovering new transient behavior at the critical point.

Contribution

It extends recurrence/transience criteria to finite-drift environments and reveals novel transient behavior at the critical drift magnitude.

Findings

ERW is recurrent when |δ|<1

ERW is transient when |δ|>1

ERW can be transient at |δ|=1 under certain conditions

Abstract

We consider excited random walk (ERW) on $\mathbb{Z}$ in environments with identical stacks of infinitely many cookies at each site, subject to the constraint that the total drift per site $\delta = \sum (2p_j - 1)$ is finite. Building on the methods of Kozma, Orenshtein, and Shinkar (arXiv:1311.7439), we show that ERW in finite-drift environments is recurrent when $|\delta|<1$ and transient when $|\delta|>1$. In the case $|\delta|=1$ we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when $|\delta|=1$, an interesting new behavior that was not present in previously studied models.