Averaged large deviations for random walk in a random environment

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 Z(d) with d >= 1, 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 A, and that the true velocity of the particle is an element (resp. in the boundary) of A when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in A.