Gaussian Weyl-Heisenberg函数的零点与电荷的超均匀性
Zeros of Gaussian Weyl-Heisenberg functions and hyperuniformity of charge
摘要：
本文研究了复平面上的高斯随机函数，其随机性在Weyl-Heisenberg群下是不变的（扭曲平稳性）。该理论是仿照平移不变高斯整函数，但允许非解析的例子，在这种情况下，卷绕数可以是正的或负的。我们计算第一强度的零集的功能，无论是当被认为是在平面上的点，或根据其相绕组的收费。在后一种情况下，收费被证明是在一定的平均均衡独立的特定协方差结构（通用筛选）。我们调查了相应的波动，并表明，在许多情况下，他们被抑制在大尺度（超均匀性）。这意味着普遍筛查在大规模上是经验上可观察的。我们还推导出电荷方差的渐近表达式。作为主要应用，我们得到了一般窗条件下复白噪声短时傅里叶变换零点集的统计量，并证明了如下测不准原理：在所有窗函数中，单位面积上期望的零点个数恰好由广义高斯函数最小化.进一步的应用包括多整函数，例如高斯整函数的共变导数。
英文摘要：
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.
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