Valleys and the maximum local time for random walk in random environment

Let $\xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $\xi^*(n) = \max_x \xi(n,x)$. It is known that $\limsup \xi^*(n)/n$ is a positive constant a.s. We prove that $\liminf_n (\log\log\log n)\xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. R\'ev\'esz (1990). The proof is based on an analysis of the {\em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.