Effective Polynomial Ballisticity Condition for Random Walk in Random Environment

Abstract

The conditions $(T)_\gamma,$ $\gamma \in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. Among others, these conditions entail a ballistic behaviour as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_\gamma$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type. In addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than $(T)_\gamma,$ another advantage of the condition $(\mathcal{P})$ is that it is effective in the sense that it can be checked on finite boxes. In particular, this extends the conjectured equivalence of the conditions $(T)_\gamma,$ $\gamma \in (0,1),$ to all relevant dimensions.