Quenched large deviations for multidimensional random walk in random environment: a variational formula

TL;DR

This paper establishes a quenched large deviation principle for multidimensional random walks in random environments using a novel minimax approach, providing a variational formula for the rate function under minimal assumptions.

Contribution

It introduces a new technique based on a minimax theorem to derive the quenched large deviation principle, avoiding reliance on the subadditive ergodic theorem.

Findings

Proves a quenched large deviation principle for multidimensional RWRE.

Derives a variational formula for the quenched rate function.

Employs a minimax theorem technique instead of traditional methods.

Abstract

We take the point of view of the particle in a multidimensional nearest neighbor random walk in random environment (RWRE). We prove a quenched large deviation principle and derive a variational formula for the quenched rate function. Most of the previous results in this area rely on the subadditive ergodic theorem. We employ a different technique which is based on a minimax theorem. Large deviation principles for RWRE have been proven for i.i.d. nestling environments subject to a moment condition and for ergodic uniformly elliptic environments. We assume only that the environment is ergodic and the transition probabilities satisfy a moment condition.