Slowdown Estimates for Ballistic Random Walk in Random Environment

We consider random walk in elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the bal-listicity condition (T ′). We show that for every ǫ > 0 and n large enough, the annealed probability of linear slowdown is bounded above by exp`− (log n) d−ǫ´. This is almost matching 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 the main result, we show an almost local version of the quenched central limit theorem under the assumption of condition (T ′).