ON A GENERAL MANY-DIMENSIONAL EXCITED RANDOM WALK

• 成果类型: Article
• 署名作者: Menshikov, Mikhail; Popov, Serguei; Ramirez, Alejandro F.; Vachkovskaia, Marina
• 署名单位: Durham University; Universidade Estadual de Campinas; Pontificia Universidad Catolica de Chile
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP678
• 发表日期: 2012
• 页码: 2106-2130
• 关键词: integers; transience; recurrence

摘要

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, ...) taking values on Z(d), 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 inf(n ->infinity) X-n.l/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 a 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.