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We consider excited random walk (ERW) on $\mathbb{Z}$ in environments withidentical stacks of infinitely many cookies at each site, subject to theconstraint 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), weshow that ERW in finite-drift environments is recurrent when $|\delta|<1$ andtransient when $|\delta|>1$. In the case $|\delta|=1$ we prove that ERW isrecurrent under mild assumptions on the environment. In addition, we show thatERW may be transient when $|\delta|=1$, an interesting new behavior that wasnot present in previously studied models.

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