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 deviations from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectrogram 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. In the same vein, our second order asymptotics for spectrogram maxima show that certain heuristic proxy models used in signal processing are inaccurate at large scales.

Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal's charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete bandwidth-based Ansatz that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover's (Wasserstein) distance to measure spike discrepancy.

The Gaussian beta-ensemble is a real n-point configuration {xj}n1 picked randomly with respect to the Boltzmann factor e-beta 2Hn, where Hn=Sigma i/=j log |xi1-xj|+n1i=1 2 x2 1 i . It is well known that the point process {xj}n1 tends to follow the semicircle law / sigma(x) = 1 (4-x2)+ in certain average senses. 2 pi A Fekete configuration (minimizer of Hn) is spread out in a much more uniform way in the interval [-2, 2] with respect to the regularization sigma n (x) = max{sigma(x),n-1 3 } of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to sigma n(x), in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences beta n of inverse temperatures, which guarantee almost sure equidistribution as n -> infinity. We find that a necessary and sufficient condition is that beta n grows at least logarithmically in n: beta n >= logn. We call this growth rate the perfect freezing regime. Along the way, we give several further results on the distribution of particles when beta n >= log n, for example on minimal spacing and discrepancies, and that with high probability a random sample solves certain sampling and interpolation problems for weighted polynomials. (In this context, Fekete sets correspond to beta equivalent to infinity.) The condition beta n >= logn was introduced in earlier works due to some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Interestingly, although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.