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Rare events for the Sine_beta process

Banff International Research Station for Mathematical Innovation and Discovery

The Gaussian Unitary/Orthogonal Ensembles (GUE/GOE) are some of the most studied Hermitian random matrix models. They may be generalized to a one parameter family of point processes indexed by beta called the beta-Hermite ensembles. When appropriately rescaled, the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine_beta process. One expects the Sine_beta process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We find the asymptotic probability of two rare events. The first is a large deviation problem for the density of points in a large interval. The second is the asymptotic probability of overcrowding in a fixed interval.

Mathematics; Probability theory and stochastic processes; Statistical mechanics, structure of matter

video/mp4

Author affiliation: University of Arizona

2017-02-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59450

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/