Universality in Random Normal Matrices

Analysis Seminar
Wednesday, January 22, 2014 - 14:00
1 hour (actually 50 minutes)
Skiles 005
ETH, Zurich
Abstract: In the beginning, the basics about random matrix models andsome facts about normal random matrices in relation with conformal map-pings will be explained. In the main part we will show that for Gaussianrandom 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 orthogonalpolynomials and an identity which plays a similar role as the Christoff el-Darboux formula in Hermitian random matrices.Especially we are interested in the density at the boundary where we scalethe coordinates with n^(-1/2). We will also consider the off -diagonal part of thekernel and calculate the correlation function. The result will be illustratedby some graphics.