Equality and difference of quenched and averaged large deviation rate functions for random walks in random environments without ballisticity

Consider a multidimensional random walk in a uniformly elliptic random environment. Varadhan (\cite{V03}) showed that the rescaled location of the random walks satisfies both an almost sure (quenched) as well as an averaged (annealed) large deviation principle (LDP) and obtained a variational formula for the averaged rate function. Rosenbluth (\cite{R06}) obtained a variational formula for the quenched rate function, which is rather implicit and hard to analyze. When the spatial dimension $d\geq 4$ and the random walk in a random environment satisfies a a certain ballisticity condition, Yilmaz (\cite{Y11}) showed that these two rate functions agree on a neighbourhood of the limiting velocity. In the present context we drop any ballisticity or transience condition and show that, in $d\geq 4$ the quenched and the annealed rate functions agree on an open subset of the boundary of the unit ball when the underlying disorder parameter is sufficiently low. This result also implies an explicit formula for the quenched rate function simplifying the earlier representation significantly. Finally, we prove the existence of a non-trivial critical disorder parameter, such that equality of these two rate functions prevails below and on this threshold, and fails beyond it.