Electronic Journal of Probability

Excursions of excited random walks on integers

Elena Kosygina and Martin Zerner

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]

branching process cookie walk diffusion approximation excited random walk excursion squared Bessel process return time strong transience

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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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