Zeros of Gaussian Weyl-Heisenberg functions and hyperuniformity of charge
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We study Gaussian random functions on the plane whose stochastics are invariant under the Weyl-Heisenberg group (twisted stationarity). We calculate the first intensity of their zero sets, both when considered as points on the plane, or as charges according to their phase winding. In the latter case, charges are always in a certain average equilibrium. We investigate the corresponding fluctuations, and show that in many cases they are suppressed at large scales (hyperuniformity). We also derive an asymptotic expression for the charge variance. Applications include poly-entire functions such as covariant derivatives of Gaussian entire functions, and zero sets of the short-time Fourier transform with general windows. We prove the following uncertainty principle: the expected number of zeros per unit area of the short-time Fourier transform of complex white noise is minimized, among all window functions, exactly by generalized Gaussians.
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