The Annals of Probability

On a general many-dimensional excited random walk

Mikhail Menshikov, Serguei Popov, Alejandro F. Ramírez, and Marina Vachkovskaia

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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.

Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 2106-2130.

Dates
First available in Project Euclid: 8 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1349703317

Digital Object Identifier
doi:10.1214/11-AOP678

Mathematical Reviews number (MathSciNet)
MR3025712

Zentralblatt MATH identifier
1272.60023

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Excited random walk cookie random walk transience ballisticity range

Citation

Menshikov, Mikhail; Popov, Serguei; Ramírez, Alejandro F.; Vachkovskaia, Marina. On a general many-dimensional excited random walk. Ann. Probab. 40 (2012), no. 5, 2106--2130. doi:10.1214/11-AOP678. https://projecteuclid.org/euclid.aop/1349703317


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