Spectral analysis of Sinai's walk for small eigenvalues

Sinai’s walk can be thought of as a random walk on $\mathbb{Z}$ 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 $[-N,N]\cap \mathbb{Z}$ 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 $V_N(x)\equiv V(Nx)\mathrm{/}\sqrt{N}$ 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.