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- W3103233742 abstract "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 $o(exp(-hsqrt{N}))$ 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." @default.
- W3103233742 created "2020-11-23" @default.
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- W3103233742 date "2008-01-01" @default.
- W3103233742 modified "2023-09-25" @default.
- W3103233742 title "Spectral analysis of Sinai’s walk for small eigenvalues" @default.
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- W3103233742 doi "https://doi.org/10.1214/009117907000000178" @default.
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