Universality in Random Normal Matrices

Analysis Seminar
Wednesday, January 22, 2014 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005
Dr. Roman Riser – ETH, Zurich
Doron Lubinsky
In the beginning, the basics about random matrix models and some facts about normal random matrices in relation with conformal map- pings will be explained. In the main part we will show that for Gaussian random normal matrices the eigenvalues will fill an elliptically shaped do- main with constant density when the dimension n of the matrices tends to infinity. We will sketch a proof of universality, which is based on orthogonal polynomials and an identity which plays a similar role as the Christoffel- Darboux formula in Hermitian random matrices. Especially we are interested in the density at the boundary where we scale the coordinates with n^(-1/2). We will also consider the off-diagonal part of the kernel and calculate the correlation function. The result will be illustrated by some graphics.