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] 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 N^{1/2}) and the distribution of the h-extrema of the rescaled potential V_N/(x) = V(Nx)/N^{1/2} defined on [-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 lo calization theorem.