On a general many-dimensional excited random walk

• DOI: 10.1214/11-aop678
• Publication Date: 2012-10-08T13:35:23Z
• Authors: Mikhail Menshikov; Serguei Popov; Alejandro F. Ramírez; Marina Vachkovskaia
• Keywords: Central limit theorem

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