Excursions of excited random walks on integers

Elena Kosygina (Baruch College and the CUNY Graduate Center)
Martin P. W. Zerner (University of Tuebingen)

Abstract


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.

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Pages: 1-25

Publication Date: February 28, 2014

DOI: 10.1214/EJP.v19-2940

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