Random entire functions from random polynomials with real zeros

TL;DR

This paper establishes a criterion for polynomial convergence to entire functions in the Laguerre-Pólya class and shows that such functions can be approximated by rescaled characteristic polynomials of random Hermitian matrices, extending to $eta$-ensembles.

Contribution

It provides a simple convergence criterion for polynomials to $ ext{LP}$ class functions and links these functions to limits of characteristic polynomials of random matrices, including $eta$-ensembles.

Findings

Random $ ext{LP}$ functions can be obtained as limits of rescaled characteristic polynomials.

Rescaled characteristic polynomials of infinite unitarily invariant Hermitian matrices converge to $ ext{LP}$ functions.

Results extend naturally to $eta$-ensembles and related point processes.

Abstract

We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-P\'{o}lya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this to show that any random $\mathcal{LP}$ function can be obtained as the uniform limit of rescaled characteristic polynomials of principal submatrices of an infinite unitarily invariant random Hermitian matrix. Conversely, the rescaled characteristic polynomials of principal submatrices of any infinite random unitarily invariant Hermitian matrix converge uniformly to a random $\mathcal{LP}$ function. This result also has a natural extension to $\beta$-ensembles. Distinguished cases include random entire functions associated to the $\beta$-Sine, and more generally $\beta$-Hua-Pickrell, $\beta$-Bessel and $\beta$-Airy point processes studied in the literature.