Hyperuniformity and non-hyperuniformity of zeros of Gaussian Weyl-Heisenberg Functions

We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set). We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so. Our work contributes to recent developments in statistical signal processing, where the time-frequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram (``silent points''). We show that empirical spectrogram zero counts enjoy moderate deviation from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectogram maxima (``loud points") fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners.

Estimation of binary time-frequency masks from ambient noise

We investigate the retrieval of a binary time-frequency mask from a few observations of filtered white ambient noise. Confirming household wisdom in acoustic modeling, we show that this is possible by inspecting the average spectrogram of ambient noise. Specifically, we show that the lower quantile of the average of $\mathcal{O}(\log(|\Omega|/\varepsilon))$ masked spectrograms is enough to identify a rather general mask $\Omega$ with confidence at least $\varepsilon$, up to shape details concentrated near the boundary of $\Omega$. As an application, the expected measure of the estimation error is dominated by the perimeter of the time-frequency mask. The estimator requires no knowledge of the noise variance, and only a very qualitative profile of the filtering window, but no exact knowledge of it.