Slowdown estimates for ballistic random walk in random environment

We consider a random walk in random environment in dimension greater than or equal to three satisfying any of the standard ballisticity conditions (either $T$, or $T^{\prime }$ or $T_{\gamma }$). We consider the event $A(n)$ that at time $n$ the distance of the walker from he origin is less than half of its expected value. We show that for every $\alpha <d$, $$P(A(n))<e^{-C(\log n{)}^{\alpha }}.$$ This is almost matching the known lower bound $$P(A(n))>e^{-C(\log n{)}^d}.$$

The lower bound is conjectured (Sznitman, 2001) to be the right value of this probability. In the talk we show the main steps of the proof, and in particular a new quenched CLT.

The talk will not assume knowledge of RWRE.