We confirm Flandrin’s prediction for the expected average of local maxima of
    spectrograms of complex white noise with Gaussian windows (Gaussian
    spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions),
    a consequence of the conjectured double honeycomb mean model for the network of
    zeros and local maxima, where the area of local maxima centered hexagons is
    three times larger than the area of zero centered hexagons. More precisely, we
    show that Gaussian spectrograms, normalized such that their expected density of
    zeros is 1, have an expected density of 5/3 critical points, among those 1/3
    are local maxima, and 4/3 saddle points, and compute the distributions of
    ordinate values (heights) for spectrogram local extrema. This is done by first
    writing the spectrograms in terms of Gaussian Entire Functions (GEFs). The
    extrema are considered under the translation invariant derivative of the Fock
    space (which in this case coincides with the Chern connection from complex
    differential geometry). We also observe that the critical points of a GEF are
    precisely the zeros of a Gaussian random function in the first higher Landau
    level. We discuss natural extensions of these Gaussian random functions:
    Gaussian Weyl-Heisenberg functions and Gaussian bi-entire functions. The paper
    also contains a bibliographic review of recent results on the theory and
    applications of white noise spectrograms, connections between several
    developments, and is partially intended as a pedestrian introduction to the

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