  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.
