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Description

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.