Zeros of Gaussian Weyl–Heisenberg Functions and Hyperuniformity of Charge

• Published in: Journal of statistical physics Vol. 187; no. 3; p. 22
• Main Authors: Haimi, Antti, Koliander, Günther, Romero, José Luis
• Format: Journal Article
• Language: English
• Published: New York Springer US 2022; Springer; Springer Nature B.V
• Subjects: Entire functions; Fourier transforms; Invariants; Mathematical analysis; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Screening; Statistical Physics and Dynamical Systems; Theoretical; Uncertainty principles; White noise; Winding; Window functions; Gaussian Weyl–Heisenberg function; Twisted convolution; 60G55; Charge; 42A61; Hyperuniformity; 94A12; Zero set; 60G15; Short-time Fourier transform
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-022-02917-3

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.