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Rare events for the Sine_beta process

Mathematical Physics & Probability

Speaker: Diane Holcomb, University of Arizona
Location: 1147 MSB
Start time: Wed, Nov 4 2015, 4:10PM

The Gaussian Unitary Ensembles (GUE) is one of the most studied Hermitian random matrix models. When appropriately rescaled the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine process. One expects the Sine 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. Our proofs work for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.