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arxiv: 2605.17296 · v1 · pith:H6GSNRQGnew · submitted 2026-05-17 · 🧮 math.PR · math-ph· math.CA· math.MP

Zero correlations and averaged fields of orthonormal Gaussian functions

Lu\'is Daniel Abreu , Tomoyuki Shirai This is my paper

Pith reviewed 2026-05-19 23:12 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CAmath.MP
keywords Gaussian entire functionszero point processespair correlationsLandau raising operatoraveraged fieldsfunctional central limit theoremtime-frequency analysis
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The pith

Iterated orthonormal Gaussian entire functions produce zero processes with index-dependent short-range correlations and averaged fields converging almost surely to 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a sequence of pointwise orthonormal Gaussian random functions by applying the Landau raising operator repeatedly to the standard Gaussian entire function. It proves that the normalized pair correlations between zeros of functions indexed n and n+k display repulsion when k equals 1, attraction when k equals 2, and independence at short range for all k at least 3, as the points approach each other. This correlation structure is complemented by an almost-sure convergence result: the average of the squared moduli of the first N normalized functions converges to the constant 1 uniformly on any compact subset of the complex plane. The paper further derives a functional central limit theorem showing that the properly scaled fluctuations around this limit converge in distribution to a Gaussian process obtained by integrating real white noise over the unit disk centered at each point z.

Core claim

The normalized pair correlations g_{n,n+k}(z,w) of the zero point processes Z_{f_n} and Z_{f_{n+k}} exhibit repulsion for k=1, attraction for k=2, and no short-range second-order correlation for k >= 3 as w approaches z. The averaged fields (1/N) sum from n=0 to N-1 of |f_n(z, bar z) exp(-|z|^2/2)|^2 converge almost surely to 1 in C(K) for every compact K subset C, and the corresponding scaled fluctuations converge to the Gaussian process G(z) = 1/sqrt(pi) int_C 1_{B(z,1)}(u) dW_R(u) where W_R is real white noise.

What carries the argument

The family of zero point processes {Z_{f_n}} generated by the orthonormal Gaussian entire functions f_n obtained through iterated application of the Landau raising operator to the base function f_0 = sum_{k=0}^infty zeta_k z^k / sqrt(k!) with zeta_k i.i.d. complex standard normals, while enforcing the orthonormality condition E[e^{-|z|^2} f_n(z, bar z) bar f_{n'}(z, bar z)] = delta_{n n'}.

If this is right

  • The zero correlations of the family reproduce the classical interlacing pattern of zeros of orthogonal polynomials.
  • The results confirm the conjectures of Flandrin and of Bayram and Baraniuk concerning white-noise spectrograms.
  • The correlation pattern supplies a mathematical rationale for the observed efficiency of the ConceFT high-resolution time-frequency algorithm.
  • The almost-sure convergence of averaged fields yields a stable deterministic limit for representations used in signal processing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same raising-operator construction could be repeated with other base entire functions or different raising operators to generate further families with tunable zero statistics.
  • The limiting Gaussian process G could be analyzed for its covariance kernel or sample-path regularity in the context of other random fields on the plane.
  • The observed correlation pattern may link the present model to determinantal point processes or exactly solvable systems in random matrix theory.
  • Finite-n simulations of the pair correlations could be compared against the asymptotic formulas to quantify the rate of convergence.

Load-bearing premise

The functions f_n are obtained by iterating the Landau raising operator on the base Gaussian entire function while exactly preserving the given pointwise orthonormality condition in expectation.

What would settle it

Direct numerical evaluation of the limit of g_{n,n+1}(z,w) as w approaches z, which must be strictly less than 1 to confirm repulsion, or Monte Carlo sampling of the averaged field on a compact set to check whether it stabilizes at exactly 1 for large N.

Figures

Figures reproduced from arXiv: 2605.17296 by Lu\'is Daniel Abreu, Tomoyuki Shirai.

Figure 1
Figure 1. Figure 1: The zeros of f (30) k (z, z) for k = 0 (blue), k = 1 (red), k = 2 (green), and k = 3 (pink), where f (N) k (z, z) is the truncated polynomial of fk(z, z) up to degree N. The numbers of zeros are 30 (k = 0), 59 (k = 1), 94 (k = 2), 103 (k = 3). From left to right, (0, 1), (0, 2) and (0, 3) correlations are illustrated. should approach 1. This has been conjectured in [35, Section 10.2], using similar conside… view at source ↗
Figure 2
Figure 2. Figure 2: The white noise spectrograms with the Gaussian (left) and the first Hermite window (right). Dark blue areas correspond to low energy (around zeros) and orange to high energy (around local maxima). One can observe that the locations with local maxima of the first picture have low energy on the second, while the local maxima of the second picture tend to occupy low energy regions of the first. Thus, the shor… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a sequence of Gaussian entire functions f_n by iterated application of the Landau raising operator to the standard GEF f_0 = sum ζ_k z^k / sqrt(k!), preserving the pointwise orthonormality E[e^{-|z|^2} f_n(z, conj(z)) conj(f_{n'}(z, conj(z)))] = δ_{nn'}. It establishes that the normalized pair correlations g_{n,n+k}(z,w) of the zero processes Z_{f_n} and Z_{f_{n+k}} exhibit repulsion for k=1, attraction for k=2, and g_{n,n+k} → 1 for k ≥ 3 as w → z. It further proves that the averaged fields (1/N) ∑_{n=0}^{N-1} |f_n(z, conj(z)) e^{-|z|^2/2}|^2 converge almost surely to 1 in C(K) for compact K ⊂ ℂ, and that the centered and scaled fluctuations converge to the Gaussian process G(z) = (1/sqrt(π)) ∫ 1_{B(z,1)}(u) dW_R(u) with real white noise W_R.

Significance. If the results hold, the work supplies a rigorous probabilistic foundation for observed zero-repulsion and averaging phenomena in time-frequency analysis. The explicit raising-operator construction, the short-range correlation limits reminiscent of orthogonal-polynomial interlacing, and the functional CLT to a white-noise integral over the unit disk directly confirm conjectures of Flandrin, Bayram-Baraniuk and others, while providing a rationale for the numerical efficiency of ConceFT-type algorithms. The use of explicit covariance kernels and standard ergodic averaging arguments constitutes a clear strength.

major comments (2)
  1. [§4, Theorem 4.1] §4, Theorem 4.1: the almost-sure C(K) convergence of the averaged fields is stated to follow from ergodic averaging of the discrete sequence indexed by n; the argument would benefit from an explicit uniform integrability or moment bound on the variance of the partial averages that is uniform over z ∈ K, since the pointwise orthonormality alone does not automatically control the supremum norm.
  2. [§3.3, Eq. (3.12)] §3.3, Eq. (3.12): the short-range limit g_{n,n+2}(z,w) → c < 1 (attraction) is obtained from the joint Gaussian structure via the Kac-Rice formula; the explicit determinant computation of the 4×4 covariance matrix for the pair (f_n, f_{n+2}) at (z,w) should be displayed to confirm that the constant c is indeed independent of n and z.
minor comments (2)
  1. [Introduction] The definition of the normalized pair correlation g_{n,n+k} is introduced only after the statement of the main theorem; moving the definition to the introduction or §2 would improve readability.
  2. [§5] Notation for the real white noise W_R and the indicator 1_{B(z,1)} is used without a preliminary reminder that the underlying measure is Lebesgue on ℂ; a short sentence in §5 would remove any ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments, which help clarify the presentation. We address each major comment below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the almost-sure C(K) convergence of the averaged fields is stated to follow from ergodic averaging of the discrete sequence indexed by n; the argument would benefit from an explicit uniform integrability or moment bound on the variance of the partial averages that is uniform over z ∈ K, since the pointwise orthonormality alone does not automatically control the supremum norm.

    Authors: We agree that an explicit uniform bound strengthens the argument for almost-sure convergence in C(K). While the ergodic theorem applies pointwise from the stationarity and orthonormality, controlling the supremum requires a uniform variance estimate. In the revision we add Lemma 4.2, which uses the explicit covariance kernel of the iterated functions (derived from the raising operator) to show that sup_{z∈K} Var[(1/N)∑_{n=0}^{N-1} |f_n(z) e^{-|z|^2/2}|^2] ≤ C_K/N for any compact K. This bound is uniform in z because the kernel depends only on |z-w| and is bounded on K. Combined with a standard Borel-Cantelli argument on a dense countable subset and continuity of the sample paths, this yields the claimed a.s. convergence in C(K). revision: yes

  2. Referee: [§3.3, Eq. (3.12)] §3.3, Eq. (3.12): the short-range limit g_{n,n+2}(z,w) → c < 1 (attraction) is obtained from the joint Gaussian structure via the Kac-Rice formula; the explicit determinant computation of the 4×4 covariance matrix for the pair (f_n, f_{n+2}) at (z,w) should be displayed to confirm that the constant c is indeed independent of n and z.

    Authors: We agree that displaying the matrix makes the independence explicit. In the revised §3.3 we insert the full 4×4 covariance matrix of the (real and imaginary parts of the) pair (f_n(z), f_{n+2}(z), f_n(w), f_{n+2}(w)) evaluated at nearby points. The entries follow directly from the pointwise orthonormality preserved by the raising operator and the translation-invariant form of the covariance kernel. After normalization, the determinant appearing in the Kac-Rice formula for the pair correlation is independent of both n and the base point z; the resulting short-range limit c<1 is therefore universal. The calculation occupies one additional displayed equation block. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs the sequence f_n explicitly by iterated application of the Landau raising operator to the base GEF f_0 while enforcing the given pointwise orthonormality relation E[e^{-|z|^2} f_n bar f_{n'}] = delta_{nn'}. All subsequent claims—the short-range pair-correlation limits (repulsion for k=1, attraction for k=2, g→1 for k≥3), the a.s. C(K) convergence of the averaged fields to 1, and the functional CLT to the white-noise integral over the unit disk—are obtained by direct computation of the resulting covariance kernels followed by standard ergodic averaging and Kac-Rice-type formulas. No step reduces a target quantity to a fitted parameter or to a self-citation chain; the derivation remains independent of the stated limits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of the Gaussian entire function together with the iterative construction via the Landau raising operator and the stated orthonormality relation; no free parameters, ad-hoc axioms, or new invented entities are introduced.

axioms (2)
  • domain assumption The base function f_0 is the Gaussian entire function with i.i.d. complex Gaussian coefficients zeta_k ~ N_C(0,1).
    This is the canonical starting point for the family of random analytic functions under study.
  • domain assumption Iteration of the Landau raising operator preserves the pointwise orthonormality condition E[e^{-|z|^2} f_n bar f_{n'}] = delta_{nn'}.
    This property is invoked to define the sequence f_n and to derive the correlation and averaging results.

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