Diane Holcomb

Abstract: The Gaussian Unitary and Orthogonal Ensembles (GUE, GOE) are some 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. On large intervals 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 solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.