Zeros of the Spectrogram of White Noise with Hermite Windows

Abstract

We study the zeros of the spectrogram of complex white noise analyzed with Hermite windows of order n, extending the classical Gaussian-window correspondence with Gaussian Analytic Functions to the polyanalytic setting. Using the polyanalytic Bargmann transform, these zeros split into two families according to the sign of the local Jacobian, giving rise to a natural notion of charge, and we show that these zeros are almost surely simple, isolated, and form a translation and rotation-invariant point process. We derive closed-form expressions for the one-point intensity, decomposed by charge, and also for the unsigned, signed, and hybrid two-point intensities. Besides, we prove that unlike the total repulsion observed with a Gaussian window, the Hermite window of order n generates a pair correlation function that converges at the origin to a strictly positive constant depending on n. Finally, we propose a numerical method for detecting spectrogram zeros via the winding number around candidate local minima.