Non-conformality of large deviations of moving average of the random walk in strongly mixing environment

TL;DR

This paper demonstrates that in strongly mixing environments, the quenched and annealed large deviations of a random walk always differ at some point, regardless of disorder level, contrasting with their conformality at low disorder.

Contribution

It reveals that quenched and annealed large deviations do not conform in strongly mixing environments, unlike in low disorder cases, highlighting a fundamental difference.

Findings

Quenched and annealed large deviations always disagree at some interior point.

Conformality of large deviations holds only at low disorder levels.

Disagreement occurs regardless of the disorder strength in strongly mixing environments.

Abstract

The quenched and annealed large deviations of the random walk in random environment are shown to conform on any compact set whenever the level of disorder is sufficiently low. In this work, we show that these two large deviations always disagree at some interior point of the natural domain of the random walk in strongly mixing environment, regardless of the level of disorder.