Slowdown estimates for ballistic random walk in random environment

Abstract

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to $4$, satisfying a condition slightly weaker than the ballisticity condition ($T^{\prime }$). We show that for every $\epsilon >0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $\exp (-(\log n{)}^{d-\epsilon })$. This bound almost matches the known lower bound of $\exp (-C(\log n{)}^d)$, and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability of slowdown. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the assumption of the same condition.