Slowdown estimates for ballistic random walk in random environment

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'). 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, we show an almost local version of the quenched central limit theorem under the assumption of the same condition.