  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.
