Zero correlations and averaged fields of orthonormal Gaussian functions

TL;DR

This paper studies the zeros of Gaussian random functions generated from the Gaussian Entire Function, revealing their correlation patterns, convergence properties, and implications for signal processing algorithms like ConceFT.

Contribution

It introduces new results on the correlation structure, convergence, and fluctuations of zeros of Gaussian functions, linking them to signal processing applications and confirming existing conjectures.

Findings

Pair correlations show a pattern similar to orthogonal polynomial zeros

Averaged fields converge almost surely to 1 on compact sets

Scaled fluctuations converge to a Gaussian process involving white noise

Abstract

We consider the family of point processes $\{\mathcal{Z}_{f_{n}}\}_{n=0}^{\infty}$ of zeros of Gaussian random functions $\{f_{n}(z,\overline{z})\}_{n=0}^{\infty} $, arising from the Gaussian Entire Function \[ f_{0}(z):=\sum_{k=0}^{\infty} \zeta_{k} \frac{z^{k}}{\sqrt{k!}}, \quad \zeta_{k} \sim N_{\mathbb{C}}(0,1)\text{ i.i.d.} \] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \[ \mathbb{E}\left[ e^{-\left\vert z\right\vert^{2}}f_{n}(z,\overline{z})\overline{f_{n^{\prime }}(z,\overline{z})}\right] ={\delta }_{nn'}. \] We first show that the normalized pair correlations $g_{n,n+k}(z,w)$ of the pairs $(\mathcal{Z}_{f_{n}},\mathcal{Z}_{f_{n+k}})$ exhibit \emph{a pattern reminiscent of the classical interlacing of zeros of orthogonal polynomials}: when $w\rightarrow z$, $g_{n,n+k}$ displays repulsion for $k=1$, attraction for $k=2$, and no short-range second-order correlation for $k \ge 3$. We complement this with the convergence of real-valued averaged fields on compacts $K \subset \mathbb{C}$, \[ \lim_{N \to \infty} \frac{1}{N}\sum_{n=0}^{N-1}\left\vert f_{n}(z,\overline{z})e^{-\frac{\left\vert z\right\vert^{2}}{2}} \right\vert^{2} \rightarrow 1 \quad \text{ almost surely in $C(K)$}, \] and a functional central limit theorem for the corresponding scaled fluctuations, which converge to the Gaussian process \[\mathcal{G}(z) = \frac{1}{\sqrt{\pi}} \int_{\mathbb{C}} \mathbf{1}_{B(z,1)}(u) dW_{\mathbb{R}}(u), \] where $W_{\mathbb{R}}$ denotes real white noise on $\mathbb{C}$ and $B(z,1)$ is the unit disk centered at $z$. The results are motivated by problems in signal processing. Due to an identification with white noise spectrograms, they confirm conjectures of Flandrin and Bayram-Baraniuk and provide a rationale for the efficiency of high resolution time-frequency algorithms, namely \emph{ConceFT}, by Daubechies, Wang and Wu.