Averaged large deviations for random walk in a random environment

TL;DR

This paper investigates the properties of the averaged large deviation rate function for a random walk in a random environment, establishing convexity, analyticity, and identifying minimizers under certain conditions.

Contribution

It extends Varadhan's 2003 results by proving convexity and analyticity of the rate function under Sznitman's transience condition and characterizing the true velocity.

Findings

The rate function $I_a$ is strictly convex and analytic on a non-empty open set.

The true velocity lies in the interior or boundary of this set depending on the walk's nestling condition.

The unique minimizer of the variational formula is identified for any velocity in the set.

Abstract

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\mathbb{Z}^d$ with $d\geq1$, and gives a variational formula for the corresponding rate function $I_a$. Under Sznitman's transience condition (T), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity of the particle is an element (resp. in the boundary) of $\mathcal{A}$ when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in $\mathcal{A}$.