Spectral analysis of Sinai's walk for small eigenvalues

Sinai's walk can be thought of as a random walk on Z with random potential V, with V weakly converging under diffusive rescaling to a two-sided Brownian motion. We consider here the generator L-N of Sinai's walk on [-N, N] boolean AND Z with Dirichlet conditions on -N, N. By means of potential theory, for each h > 0, we show the relation between the spectral properties of L-N for eigenvalues of order o(exp(-h root N)) and the distribution of the h-extrema of the rescaled potential V-N(x) V(Nx)/root N definedon [-1, 1]. Information about the h-extrema of V-N is derived from a result of Neveu and Pitman concerning the statistics of h-extrema of Brownian motion. As first application of our results, we give a proof of a refined version of Sinai's localization theorem.