Slowdown estimates for ballistic random walk in random environment

TL;DR

This paper investigates the probability of slowdown in high-dimensional ballistic random walks within random environments, providing bounds close to theoretical limits and developing a local quenched CLT under specific conditions.

Contribution

It introduces near-optimal bounds for slowdown probabilities and establishes an almost local quenched CLT for ballistic random walks in high dimensions.

Findings

Upper bound for slowdown probability close to the naive trap lower bound

Development of an almost local quenched CLT under ballisticity conditions

Results applicable for dimensions ≥ 4 with weak ballisticity assumptions

Abstract

For a random walk in an elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the a ballisticity condition slightly weaker than condition (T'), We consider the probability of linear slowdown. We show an upper bound for this probability which is very close to the lower bound obtained by the "naive trap" analysis. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the same ballisticity condition.