Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher

TL;DR

This paper proves that for high-dimensional random walks in random environments satisfying certain conditions, the averaged and quenched large deviation rate functions are equal on a significant set, clarifying their relationship in these settings.

Contribution

It establishes the equality of averaged and quenched large deviation rate functions in dimensions four and higher under Sznitman's transience condition (T).

Findings

Rate functions are finite and equal on a large set.

Equality holds for all nonzero velocities where the rate functions vanish.

Results apply specifically to dimensions four and above with condition (T).

Abstract

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and Sznitman's transience condition (T) is satisfied, we prove that these rate functions are finite and equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish.