From random matrices to random analytic functions

TL;DR

This paper studies two families of random matrix-valued analytic functions, revealing that their singularities form determinantal point processes on geometric surfaces, unifying several classical results in random matrix theory.

Contribution

It introduces a unified framework linking random matrix-valued analytic functions with determinantal point processes on surfaces, extending classical results.

Findings

Singularities form determinantal point processes on the sphere and hyperbolic plane.

Kernels are reproducing kernels of Hilbert spaces of analytic functions.

Unifies classical results of Peres, Virag, and Ginibre.

Abstract

We consider two families of random matrix-valued analytic functions: (1) G_1-zG_2 and (2) G_0 + zG_1 +z^2G_2+ ..., where G_i are n x n independent random matrices with independent standard complex Gaussian entries. The set of z where these matrix-valued analytic functions become singular, are shown to be determinantal point processes on the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain natural Hilbert spaces of analytic functions on the corresponding surfaces. This gives a unified framework in which to view a result of Peres and Virag (n=1 in the second setting) and a well known theorem of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).