On a general many-dimensional excited random walk

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

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 (Xn,n=0,1,2,…) taking values on Zd, d≥2, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction ℓ; 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 ℓ so that lim infn→∞Xn⋅ℓn>0. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than n1/2+α distinct sites by time n, where α 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.

Citation

Menshikov, M., Popov, S., Ramírez, A. F., & Vachkovskaia, M. (2012). On a general many-dimensional excited random walk. Annals of Probability, 40(5), 2106-2130. https://doi.org/10.1214/11-aop678

• Journal Article Type: Article
• Publication Date: Sep 1, 2012
• Deposit Date: Oct 18, 2012
• Publicly Available Date: Jul 29, 2014
• Journal: Annals of Probability
• Print ISSN: 0091-1798
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 40
• Issue: 5
• Pages: 2106-2130
• DOI: https://doi.org/10.1214/11-aop678
• Public URL: https://durham-repository.worktribe.com/output/1471855