On a general manydimensional excited random walk
Abstract
In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 8692] by Benjamini and Wilson. We consider a discretetime stochastic process $(X_n,n=0,1,2,...)$ taking values on ${\mathbb{Z}}^d$, $d\geq2$, described as follows: when the particle visits a site for the first time, it has a uniformlypositive 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2010
 arXiv:
 arXiv:1001.1741
 Bibcode:
 2010arXiv1001.1741M
 Keywords:

 Mathematics  Probability
 EPrint:
 Published in at http://dx.doi.org/10.1214/11AOP678 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)