Zeros of random functions generated with de Branges kernels

TL;DR

This paper investigates the distribution of real zeros in random functions constructed from de Branges space kernels, providing explicit formulas and characterizations that extend known results in Gaussian analytic functions.

Contribution

It introduces an explicit formula for the first intensity function of zeros in de Branges spaces and shows this characterizes the space, extending Calabi rigidity results.

Findings

Explicit formula for the first intensity function in terms of the phase.

The first intensity fully characterizes the de Branges space.

Extension of Calabi rigidity to real zeros of de Branges space functions.

Abstract

We study the point process given by the set of real zeros of random sums of orthonormal bases of reproducing kernels of de Branges spaces. Examples of these kernels are the cardinal sine, Airy and Bessel kernels. We find an explicit formula for the first intensity function in terms of the phase of the Hermite-Biehler function. We prove that the first intensity of the point process completely characterizes the underlying de Branges space. This result is a real version of the so called Calabi rigidity for GAFs proved by M. Sodin.