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September 2012 On a general many-dimensional excited random walk
Mikhail Menshikov, Serguei Popov, Alejandro F. Ramírez, Marina Vachkovskaia
Ann. Probab. 40(5): 2106-2130 (September 2012). DOI: 10.1214/11-AOP678

Abstract

In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86–92] by Benjamini and Wilson. We consider a discrete-time stochastic process $(X_{n},n=0,1,2,\ldots)$ taking values on ${\mathbb{Z}}^{d}$, $d\geq2$, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction $\ell$; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction $\ell$ so that $\liminf_{n\to\infty}\frac{X_{n}\cdot\ell}{n}>0$. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than $n^{{1/2}+\alpha}$ distinct sites by time $n$, where $\alpha$ is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem.

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Mikhail Menshikov. Serguei Popov. Alejandro F. Ramírez. Marina Vachkovskaia. "On a general many-dimensional excited random walk." Ann. Probab. 40 (5) 2106 - 2130, September 2012. https://doi.org/10.1214/11-AOP678

Information

Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1272.60023
MathSciNet: MR3025712
Digital Object Identifier: 10.1214/11-AOP678

Subjects:
Primary: 60J10 , 82B41

Keywords: Ballisticity , cookie random walk , excited random walk , ‎range‎ , transience

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • September 2012
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