# Slowdown estimates for ballistic random walk in random environment

### Noam Berger

Hebrew University, Jerusalem, Israel

## 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 ($T'$). 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.

## Cite this article

Noam Berger, Slowdown estimates for ballistic random walk in random environment. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 127–174

DOI 10.4171/JEMS/298