The limiting characteristic polynomial of classical random matrix ensembles

TL;DR

This paper proves that the characteristic polynomial of classical random matrix ensembles converges to a universal function at microscopic scales, revealing a common limiting behavior across different ensembles.

Contribution

It introduces a general limit theorem for the convergence of random entire functions with regularly spaced zeros, applied to classical ensembles.

Findings

Characteristic polynomial converges to a universal function

Results apply to classical compact groups and GUE

Establishes a universal microscopic scale behavior

Abstract

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.