Gap probabilities for the cardinal sine

TL;DR

This paper investigates the probability of zero-free intervals in random sums of cardinal sine functions, revealing exponential decay in the likelihood of no zeros as interval length increases.

Contribution

It introduces a model for zero distributions of random analytic functions based on cardinal sine sums and analyzes their zero-free probability decay.

Findings

Zero-free probability decays exponentially with interval length.

The model uses Gaussian coefficients in the cardinal sine sum.

Provides insights into zero distribution behavior in Paley-Wiener spaces.

Abstract

We study the zero set of random analytic functions generated by a sum of the cardinal sine functions that form an orthogonal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.