Open Access
January 2008 Spectral analysis of Sinai’s walk for small eigenvalues
Anton Bovier, Alessandra Faggionato
Ann. Probab. 36(1): 198-254 (January 2008). DOI: 10.1214/009117907000000178


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 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.


Download Citation

Anton Bovier. Alessandra Faggionato. "Spectral analysis of Sinai’s walk for small eigenvalues." Ann. Probab. 36 (1) 198 - 254, January 2008.


Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1154.60078
MathSciNet: MR2370603
Digital Object Identifier: 10.1214/009117907000000178

Primary: 60K37 , 82B41 , 82B44

Keywords: Ageing , Disordered systems , random dynamics , spectral properties , trap models

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • January 2008
Back to Top