On new examples of ballistic random walks in random environment

Abstract

In this article we show that random walks in random environment on $\mathbb{Z}^d$, $d \ge3$, with transition probabilities which are $\varepsilon$-perturbations of the simple random walk and such that the expectation of the local drift has size bigger than $\varepsilon^\rho $, with $\rho< \frac{5}{2}$, when $d=3$, $\rho< 3$, when $d \ge4$, fulfill the condition (T$^\prime$) introduced by Sznitman [Prob. Theory Related Fields (2002) 122 509-544], when $\varepsilon$ is small. As a result these walks satisfy a law of large numbers with nondegenerate limiting velocity, a central limit theorem and several large deviation controls. In particular, this provides examples of ballistic random walks in random environment which do not satisfy Kalikow's condition in the terminology of Sznitman and Zerner [Ann. Probab. (1999) 27 1851-1869]. An important tool in this work is the effective criterion of Sznitman.