Spectral analysis of Sinai's walk for small eigenvalues

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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 Lₙ of Sinai's walk on [-N,N]∩ 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 Lₙ for eigenvalues of order o((-h√N)) and the distribution of the h-extrema of the rescaled potential Vₙ(x)≡ V(Nx)/√N defined on [-1,1]. Information about the h-extrema of Vₙ 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.