From random matrices to random analytic functions

Abstract

We consider two families of random matrix-valued analytic functions: (1) G₁−zG₂ and (2) G₀+zG₁+z²G₂+⋯, where G_i are n×n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces (“Bargmann–Fock spaces”) of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virág (n=1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).