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

Date: 
Friday 10th April 2015 - 11:30 to 12:30
Venue: 
INI Seminar Room 1
Abstract: 
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.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons