Large Deviations for First Hitting time of Random Walk in Random Environment (lower level)

Let $T_{-n}$, $n \in\mathbb{N}$, be a hitting time of level -n for a random walk in random environment (RWRE). The exact asymptotics $P(T_{-n} = k)$ are proved. Here $k =k(n)$, $n$-k is even for every n and the ratio $k/n$ belongs to a compact set.