Local repulsion between zeros and critical points of the Gaussian Entire Function

We study the zeros and critical points of different indices of the standard Gaussian entire function on the complex plane (whose zero set is stationary). We provide asymptotics for the second order correlations of all the corresponding number statistics on small observation disks, showing various rates of local repulsion. The results have consequences for signal processing, as they show extremely strong repulsion between the local maxima and zeros of spectrograms of noise computed with respect to Gaussian windows.