Zeros of Gaussian Weyl-Heisenberg functions and hyperuniformity of charge

TL;DR

This paper investigates the zero sets and charge distributions of Weyl-Heisenberg invariant Gaussian functions, revealing universal screening, hyperuniformity, and optimality of Gaussian windows in the short-time Fourier transform context.

Contribution

It introduces the concept of twisted stationarity for Gaussian functions, calculates zero set intensities, and demonstrates universal screening and hyperuniformity phenomena, with applications to Fourier analysis and uncertainty principles.

Findings

Charges exhibit universal screening independent of covariance.

Zero set fluctuations are suppressed at large scales, indicating hyperuniformity.

Generalized Gaussians minimize the expected zero density among all window functions.

Abstract

We study Gaussian random functions on the complex plane whose stochastics are invariant under the Weyl-Heisenberg group (twisted stationarity). The theory is modeled on translation invariant Gaussian entire functions, but allows for non-analytic examples, in which case winding numbers can be either positive or negative. We calculate the first intensity of zero sets of such functions, both when considered as points on the plane, or as charges according to their phase winding. In the latter case, charges are shown to be in a certain average equilibrium independently of the particular covariance structure (universal screening). We investigate the corresponding fluctuations, and show that in many cases they are suppressed at large scales (hyperuniformity). This means that universal screening is empirically observable at large scales. We also derive an asymptotic expression for the charge variance. As a main application, we obtain statistics for the zero sets of the short-time Fourier transform of complex white noise with general windows, and also prove the following uncertainty principle: the expected number of zeros per unit area is minimized, among all window functions, exactly by generalized Gaussians. Further applications include poly-entire functions such as covariant derivatives of Gaussian entire functions.