Title: Truncations of Random Unitary Matrices
Speaker: Kathryn Lockwood (PhD Student, Department of Mathematics, Applied Mathematics, & Statistics, Case Western Reserve University)
Abstract: If U is a random n x n Haar distributed unitary matrix, it is well known that the eigenvalues of U are approximately uniformly distributed on the unit circle. Truncating the bottom (n-m) rows and the last (n-m) columns of U produces a m x m matrix. When m is on the order of the square root of n the eigenvalues of the truncated matrix are approximately uniformly distributed on the unit disc. When m is equal to n the truncated matrix is the full unitary matrix U. In between these extreme cases the distribution of the eigenvalues can be approximated by a parametrized family of probability measures on the disc which interpolate between uniform measure on the disc and uniform measure on the circle. A Coulomb gas model can be used to represent the eigenvalues in these in between cases as locations of unit charges in the plane. This model produces a large deviation principle for the eigenvalues of the truncated matrix and determines the large-n limiting measure.