Zero correlations and averaged fields of orthonormal Gaussian functions
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arXiv
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| Natura: | Preprint |
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2026
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| author | Abreu, Luís Daniel Shirai, Tomoyuki |
| author_facet | Abreu, Luís Daniel Shirai, Tomoyuki |
| contents | 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 \[ f_{0}(z):=\sum_{k=0}^{\infty} ζ_{k} \frac{z^{k}}{\sqrt{k!}}, \quad ζ_{k} \sim N_{\mathbb{C}}(0,1)\text{ i.i.d.} \] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \[ \mathbb{E}\left[ e^{-\left\vert z\right\vert^{2}}f_{n}(z,\overline{z})\overline{f_{n^{\prime }}(z,\overline{z})}\right] ={δ}_{nn'}. \] 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 \[\mathcal{G}(z) = \frac{1}{\sqrtπ} \int_{\mathbb{C}} \mathbf{1}_{B(z,1)}(u) dW_{\mathbb{R}}(u), \] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17296 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Zero correlations and averaged fields of orthonormal Gaussian functions Abreu, Luís Daniel Shirai, Tomoyuki Probability Mathematical Physics Classical Analysis and ODEs Primary 60G55, Secondary 60G15, 30B20, 30H20, 42C40 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 \[ f_{0}(z):=\sum_{k=0}^{\infty} ζ_{k} \frac{z^{k}}{\sqrt{k!}}, \quad ζ_{k} \sim N_{\mathbb{C}}(0,1)\text{ i.i.d.} \] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \[ \mathbb{E}\left[ e^{-\left\vert z\right\vert^{2}}f_{n}(z,\overline{z})\overline{f_{n^{\prime }}(z,\overline{z})}\right] ={δ}_{nn'}. \] 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 \[\mathcal{G}(z) = \frac{1}{\sqrtπ} \int_{\mathbb{C}} \mathbf{1}_{B(z,1)}(u) dW_{\mathbb{R}}(u), \] 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. |
| title | Zero correlations and averaged fields of orthonormal Gaussian functions |
| topic | Probability Mathematical Physics Classical Analysis and ODEs Primary 60G55, Secondary 60G15, 30B20, 30H20, 42C40 |
| url | https://arxiv.org/abs/2605.17296 |