Spectral analysis of Sinai's walk for small eigenvalues

Details Spectral analysis of Sinai's walk for small eigenvalues Spectral analysis of Sinai's walk for small eigenvalues

2005-09-16

arxiv.org/abs/math/0509385v2

Annals of Probability 2008, Vol. 36, No. 1, 198-254

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000178 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000178

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 $V_N(x)\equiv V(Nx)/\sqrt{N}$ 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.

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