Random entire functions from random polynomials with real zeros

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.