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

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 $\mathrm{\mid }\delta \mathrm{\mid }<1$ and transient when $\mathrm{\mid }\delta \mathrm{\mid }>1$. In the case $\mathrm{\mid }\delta \mathrm{\mid }=1$ we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when $\mathrm{\mid }\delta \mathrm{\mid }=1$, an interesting new behavior that was not present in previously studied models.