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arxiv: 2604.26316 · v1 · submitted 2026-04-29 · 🧮 math.PR

Smallest distances between zeros of Gaussian analytic functions

Renjie Feng , Dong Yao This is my paper

Pith reviewed 2026-05-07 12:54 UTC · model grok-4.3

classification 🧮 math.PR
keywords Gaussian analytic functionszerospoint processesPoisson limitminimal distancesRiemann surfacesGaussian entire functionsrescaling
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The pith

After rescaling by local intensity, the smallest distances between zeros of Gaussian analytic functions converge to a Poisson point process with a universal rate.

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

The paper studies the minimal separations among zeros of random holomorphic functions on compact Riemann surfaces, where the functions are Gaussian with a covariance kernel tied to the surface geometry. It establishes that rescaling these distances according to the local zero density causes the collection of smallest gaps to converge to a Poisson point process whose intensity is the same everywhere. This convergence also implies that the positions of these minimal distances are distributed uniformly according to the surface's area measure. As a direct consequence, the density of the k-th smallest rescaled distance takes the explicit form proportional to x to the power 4k-1 times e to the minus x to the fourth, for any k at least 1. The same limiting behavior holds for Gaussian entire functions on the plane.

Core claim

The point process formed by the rescaled smallest distances between zeros converges in distribution to a Poisson point process with universal intensity; the locations of these minimal distances become uniformly distributed with respect to the volume form on the surface; and therefore the limiting density of the k-th ordered rescaled distance is proportional to x^{4k-1} e^{-x^4}.

What carries the argument

The rescaled point process of minimal inter-zero distances, which is shown to converge to a homogeneous Poisson point process whose rate is independent of the underlying surface or covariance details.

If this is right

  • The probability that the smallest rescaled distance exceeds a value t decays as exp(-c t^4) for a universal constant c.
  • The k-th smallest rescaled distance has density proportional to x^{4k-1} exp(-x^4) for every positive integer k.
  • The locations realizing these minimal distances are asymptotically uniform with respect to the surface volume measure.
  • The same Poisson limit and explicit densities apply to the zeros of Gaussian entire functions on the complex plane.

Where Pith is reading between the lines

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

  • Numerical sampling of zeros on a torus or sphere could directly verify the predicted gap densities without needing the full surface geometry.
  • The result may extend to other random holomorphic sections whose zero intensity is locally constant after suitable normalization.
  • The universal x^4 repulsion in the gap law could be compared against minimal distances in other point processes with quadratic repulsion, such as certain determinantal processes.

Load-bearing premise

The Gaussian analytic functions possess a covariance kernel that produces a simple zero point process whose local intensity matches the volume form, and the rescaling is performed using exactly this local intensity.

What would settle it

Generate many realizations of zeros for a concrete Gaussian analytic function on the sphere or torus, compute the ordered list of minimal distances after rescaling by the local intensity, and check whether their empirical distribution matches the predicted density proportional to x^3 e^{-x^4} for the first gap.

read the original abstract

In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a Poisson point process with a universal rate. Furthermore, the locations where these smallest distances occur tend to follow a uniform measure with respect to the volume form. As a consequence, the limiting density of the $k$-th rescaled smallest distance is proportional to $x^{4k-1}e^{-x^4}$ for any $k\geq 1$. Analogous results hold for the classical Gaussian Entire Functions.

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

0 major / 2 minor

Summary. The paper examines the smallest distances between zeros of Gaussian analytic functions on compact Riemann surfaces. The main result states that after appropriate rescaling, the point process of these smallest distances converges to a Poisson point process with a universal rate. The locations of these smallest distances are asymptotically distributed according to the uniform measure with respect to the volume form. Consequently, the limiting density of the k-th rescaled smallest distance is proportional to x^{4k-1} e^{-x^4} for k ≥ 1. Analogous results are obtained for Gaussian entire functions.

Significance. If the claims are established rigorously, this work provides a universal characterization of the nearest-neighbor distance statistics for the zero sets of GAFs, which are important examples of determinantal point processes. The explicit limiting density offers a concrete prediction that aligns with the expected behavior from the local pair correlation function g(r) ∼ r², leading to the x^4 exponent in the exponential. This extends local statistics to global minimal distances on compact surfaces and includes the entire function case. The use of the volume form for uniformity is a natural and clean feature.

minor comments (2)
  1. The abstract could include a short indication of the proof strategy or key tools used, such as the determinantal property or decorrelation estimates, to help readers assess the result at a glance.
  2. Some notation for the rescaling procedure and the definition of the point process of smallest distances might benefit from additional clarification in the introduction for readers not familiar with point process convergence on manifolds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The assessment accurately reflects the main results on the convergence of rescaled smallest distances to a Poisson point process and the limiting density for the k-th smallest distance. As no specific major comments were provided in the report, we have no points requiring detailed rebuttal or changes to the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation follows from local determinantal repulsion

full rationale

The central result follows from the known local structure of the zero point process of GAFs: the covariance kernel implies a determinantal process whose pair correlation satisfies g(r) ~ r^2 as r -> 0. Rescaling distances by the local intensity (volume form) then yields an intensity measure ~ x^3 dx, from which the void probabilities and order statistics produce the Poisson limit with density proportional to x^{4k-1} e^{-x^4} and spatial uniformity. This reduction uses only the standard microscopic asymptotics of the kernel and decorrelation properties on compact surfaces; it does not reduce the claimed limit to a fitted parameter, self-definition, or load-bearing self-citation. The result is therefore self-contained against external benchmarks for determinantal processes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of Gaussian analytic functions as centered Gaussian holomorphic processes whose zero set is a simple point process with intensity given by the volume form; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Zeros of a Gaussian analytic function form a simple point process whose intensity measure is the volume form on the Riemann surface.
    This is the usual setup for GAFs that allows local rescaling by density.

pith-pipeline@v0.9.0 · 5387 in / 1407 out tokens · 121078 ms · 2026-05-07T12:54:59.460341+00:00 · methodology

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Reference graph

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