## Electronic Journal of Probability

### Excursions of excited random walks on integers

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 25, 25 pp.

Dates
Accepted: 28 February 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065667

Digital Object Identifier
doi:10.1214/EJP.v19-2940

Mathematical Reviews number (MathSciNet)
MR3174837

Zentralblatt MATH identifier
1291.60093

Rights

#### Citation

Kosygina, Elena; Zerner, Martin. Excursions of excited random walks on integers. Electron. J. Probab. 19 (2014), paper no. 25, 25 pp. doi:10.1214/EJP.v19-2940. https://projecteuclid.org/euclid.ejp/1465065667

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