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Level statistics for 1-dimensional Schr"odinger operator and beta-ensemble

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Periodic and Ergodic Spectral Problems

A part of this talk is based on joint work with Prof. Kotani. We consider the following two classes of 1-dimensional random Schr”odinger operators : (1) operators with decaying random potential, and (2) operators whose coupling constants decay as the system size becomes large. Our problem is to identify the limit $xi_{infty}$ of the point process consisting of rescaled eigenvalues. The result is : (1) for slow decay, $xi_{infty}$ is the clock process ; for critical decay $xi_{infty}$ is the $Sine_{beta}$ process, (2) for slow decay, $xi_{infty}$ is the deterministic clock process ; for critical decay $xi_{infty}$ is the $Sch_{tau}$ process. As a byproduct of (1), we have a proof of coincidence of the scaling limits of circular and Gaussian beta ensembles.

This talk is part of the Isaac Newton Institute Seminar Series series.

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