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
topic.
