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.
翻译:我们在Weyl-Heisenberg组(两端静态)下研究复杂平面上的高斯随机功能,其随机功能不易变。该理论以变换高斯整个功能为模型,但允许非分析性的例子,在这种情况下,通风数字可以是正数,也可以是负数。我们计算出这种功能的零数组的第一强度,既当被视为平面上的点,也可以按其相向风速计算。在后一种情况下,收费显示在某种平均平衡中,不取决于特定的常态结构(通用筛选)。我们调查相应的波动,并显示在许多情况下,这些波动被大尺度(超均度)抑制。这意味着普遍筛选是大尺度上的经验性可观测的。我们还得出这种排量差异的无症状表达方式。作为主要应用,我们获得一般窗口复杂白噪声短时四更短变零的统计,并证明以下的不确定原则:每个单位区域的预期数字是最小的,在所有窗口的功能中,通过普遍化的图像,包括整个图像的功能。