Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter $\delta\in\mathbb R$. For recurrence/transience the critical threshold is $|\delta|=1$, for ballisticity it is $|\delta|=2$ and for diffusivity $|\delta|=4$. In this paper we establish a phase transition at $|\delta|=3$. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff $|\delta|>3$. This result follows from an explicit description of the tail behaviour of the return time as a function of $\delta$, which is achieved by diffusion approximation of related branching processes by squared Bessel processes.
"Excursions of excited random walks on integers." Electron. J. Probab. 19 1 - 25, 2014. https://doi.org/10.1214/EJP.v19-2940