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 ε > 0 and n large enough, the annealed probability of linear slowdown is bounded from above by exp(−(log n)d_−_ε). 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.