Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

TL;DR

This paper compares averaged and quenched large deviation principles for random walks in dynamic random environments, linking their entropy formulations and exploring implications for directed polymer models.

Contribution

It establishes a detailed relationship between averaged and quenched rate functions and their entropies, providing new insights into large deviations in dynamic environments.

Findings

The averaged rate function is a specific relative entropy.

The quenched rate function is a Donsker-Varadhan type entropy.

Connections are made between these entropies and variational principles.

Abstract

We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.