Local repulsion between zeros and critical points of the Gaussian Entire Function
Antti Haimi, Lukas Odelius, José Luis Romero
TL;DR
This work analyzes the zeros of the Gaussian entire function $G$ and the zeros of its covariant derivative $F$, classifying critical points by index and proving strong local repulsion on small disks. The authors develop approximate Kac–Rice formulas, exploit twisted Bargmann–Fock stationarity, and perform delicate asymptotic expansions of $\mathrm{Jac}\,F$ under conditioning to obtain precise near-diagonal second factorial moments. They establish sharp rates for zeros–zeros, zeros–critical points, and critical points within the same index, including $\mathbb{E}[\mathcal{N}^{z}_\rho(\mathcal{N}^{z}_\rho-1)]\asymp\rho^6$, $\mathbb{E}[\mathcal{N}^{\mathrm{c}}_\rho(\mathcal{N}^{\mathrm{c}}_\rho-1)]\asymp\rho^4$ with $\lim_{\rho\to0^+}\mathbb{E}[\mathcal{N}^{\mathrm{c}}_\rho(\mathcal{N}^{\mathrm{c}}_\rho-1)]/(\mathbb{E}\mathcal{N}^{\mathrm{c}}_\rho)^2=6/25$, and notably $\mathbb{E}[\mathcal{N}^{\mathrm{z}}_\rho\mathcal{N}^{\mathrm{c},-}_\rho]\asymp\rho^{20}$ indicating extremely strong repulsion between zeros and negative-index critical points. The results further connect to time-frequency landmarks of noise and spectrogram analysis via the weighted amplitude $S(z)=e^{-\tfrac{1}{2}|z|^2}G(z)$ and its relation to spectrogram-based noise localization. The methods combine Gaussian regression, intricate conditioning arguments, and diagonal expansions, with applications to signal processing and complex-geometry models of random holomorphic sections.
Abstract
We study the zeros and critical points of different indices of the standard Gaussian entire function on the complex plane (whose zero set is stationary). We provide asymptotics for the second order correlations of all the corresponding number statistics on small observation disks, showing various rates of local repulsion. The results have consequences for signal processing, as they show extremely strong repulsion between the local maxima and zeros of spectrograms of noise computed with respect to Gaussian windows.