Nonnormal Random Matrices and Convergence To a Ring
Nonnormal Random Matrices and Convergence To a Ring

Ofer Zeitouni, University of Minnesota, Weizmann Institute
Sherrerd Hall 101
We study the empirical measure $L_{A_n}$ of the eigenvalues of nonnormal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring, even if the support of $D_n$ consists of several disjoint intervals. In particular, we provide a complete proof of FeinbergZee single ring theorem.