Spectral analysis of Sinai's walk for small eigenvalues
- Bovier, Anton
- Faggionato, Alessandra
2010 Mathematics Subject Classification
- 60K37 82B41 82B44
- disordered systems, random dynamics, trap models, ageing, spectral properties
Sinai's walk can be thought of as a random walk on $ZZ$ with random potential $V$, with $V$ weakly converging under diffusive rescaling to a two-sided Brownian motion. We consider here the generator $LL _N$ of Sinai's walk on $[-N,N]cap ZZ$ with Dirichlet conditions on $-N,N$. By means of potential theory, for each $h>0$ we show the relation between the spectral properties of $LL_N$ for eigenvalues of order $oleft(expleft(-h sqrtNright)right)$ and the distribution of the $h$-extrema of the rescaled potential $V_N(x)equiv V(Nx)/sqrtN$ 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.