Spectral analysis of Sinai’s walk for small eigenvalues

Abstract

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 $\mathbb {L}_{N}$ 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 $\mathbb {L}_{N}$ for eigenvalues of order $o(\exp(-h\sqrt{N}))$ and the distribution of the h-extrema of the rescaled potential $V_{N}(x)\equiv V(Nx)/\sqrt{N}$ 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 localization theorem.