Smallest distances between zeros of Gaussian analytic functions

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.

• The paper demonstrates that the empirical process of pairwise geodesic distances converges to a Poisson point process with a quartic intensity after n^(3/4) normalization.
• It derives a universal density law for the k-th smallest distance, revealing strong zero repulsion and independence from the underlying geometry.
• The study extends the analysis to classical Gaussian Entire Functions and contrasts these results with IID point ensembles, highlighting the unique n^(-3/4) minimal gap scale.

Smallest Distances Between Zeros of Gaussian Analytic Functions

Overview

The paper "Smallest distances between zeros of Gaussian analytic functions" (2604.26316) investigates the local statistics of zeros from Gaussian analytic functions (GAF) on compact Riemann surfaces endowed with positive holomorphic line bundles. It characterizes the scaling limits and statistical structure of the smallest pairwise geodesic distances between zeros, identifying the limiting process and rates, as well as the universal density governing these minimal spacings. This work extends classical results on the zero statistics of random holomorphic sections and relates the fine structure of zero configurations to universal probabilistic phenomena.

Main Results

Scaling Limit and Poisson Structure

The central result establishes that, after suitable normalization, the empirical process of the smallest geodesic distances between zeros on a compact Riemann surface converges in distribution to a Poisson point process. Specifically, if $Z_n$ denotes the zero set of a random holomorphic section of $L^n$ (with $|Z_n| = n$ by Riemann-Roch), the rescaled distances $n^{3/4} d_g(z_i, z_j)$ (for a suitable choice of marked point) form a point process $\mathcal{S}_n$. As $n \to \infty$, $\mathcal{S}_n$ converges to a Poisson process on $[0,\infty) \times M$ (where $M$ is the Riemann surface) with expected value

$$\mathbb{E}[\mathcal{S}([0,a]\times U)] = \frac{1}{8} a^4 \int_U \frac{\omega}{\pi},$$

for any Borel set $L^n$0. This result identifies the universal Poissonian nature of rare small-spacing events, in contrast to the typical $L^n$1-scale "average" zero separation dictated by two-point correlation functions.

Universal Smallest Distance Law

The limiting density governing the $L^n$2-th smallest rescaled distance (as $L^n$3) is derived to be

$L^n$4

after an explicit normalization. This density is independent of the underlying geometry (i.e., is universal across compact Riemann surfaces endowed with positively curved line bundles), and the locations of the zeros participating in minimal pairs asymptotically equidistribute with respect to the normalized volume form.

Extension to Classical GEF and Contrast to IID Points

Analogous statements are rigorously derived for the Gaussian Entire Function (GEF) in the infinite plane, where an analogous scaling ($L^n$5 for disk of radius $L^n$6) yields convergence of minimal pairwise distances among zeros in the disk to the Poisson point process with the same quartic intensity in the rescaled variable.

The study provides a sharp contrast to the case of $L^n$7 independent uniformly distributed points on the sphere: for such ensembles, the smallest distances are of order $L^n$8 and governed by entirely different extremal statistics due to the absence of local repulsion. This identifies the $L^n$9 scaling as a signature of zero repulsion—a property reflected in the vanishing short range of two-point correlation functions for GAF zeros.

Technical Approach

The derivation leverages the established machinery for describing Gaussian holomorphic sections on Kähler manifolds:

• Use of Hermitian line bundles with positive curvature, enabling probabilistic definition of GAF sections (with the covariance kernel given by the Bergman kernel).
• Detailed short- and long-range asymptotics for high-$|Z_n| = n$0 Bergman kernels (Tian-Yau-Zelditch expansion) and correlation functions for zeros (Bleher-Shiffman-Zelditch theory).
• Application of Kac-Rice formulas for multi-point zero densities, with precise error control enabling local expansions in scaling windows of size $|Z_n| = n$1 around diagonal configurations.
• Poisson convergence is established via computation of factorial moments of the small-distance pair process, using a separation-of-clusters argument; the negligible overlap between minimal pairs ensures asymptotic independence, and the effective volume estimate yields the quartic Poisson intensity.
• Rigorous control of error terms leverages exponential decay estimates for off-diagonal Bergman kernels, as well as sharp partitioning techniques employing divided differences for zero/cluster enumeration.

Broader Context and Implications

The analysis achieves a precise characterization of the rare-event extremal regime ("microscopic" spacings much smaller than the mean nearest-neighbor distance) for zeros of GAFs, complementing the well-understood universality at the "mesoscopic" scale. Universal Poissonian limit laws for minimal distances are established, providing a template for further studies in complex and almost-complex settings.

The dimensional dependence of zero repulsion and its impact on the statistics of smallest gaps is also clarified: for Riemann surfaces ($|Z_n| = n$2), zeros exhibit strict short-range avoidance, while for higher-dimensional Kähler manifolds, the behavior is modified (neutral or attractive for $|Z_n| = n$3). This opens avenues for classification of extremal statistics of zeros in broader geometric and arithmetic ensembles.

Practically, these findings have implications for high-degree polynomial root statistics, random matrix theory (through analogies with eigenvalue spacings), and quantum chaotic systems, where GAFs and zeros model wavefunction structure and nodal set fine-scale geometry.

Future Directions

Potential research directions include:

• Extension of these results to more general random holomorphic fields, including quasi-random and arithmetic models, or those associated with degenerating metrics or non-positive curvature.
• Exploration of rigidity phenomena for minimal spacings—namely, the interplay between global constraints (e.g., total number of zeros, real structures) and local repulsion.
• Investigation of analogous minimal spacing laws for zero sets of Gaussian sections on higher-dimensional Kähler or symplectic manifolds, where repulsion statistics are known to differ substantially.
• Links with stochastic geometry (e.g., study of minimal coverings and random Delaunay triangulations induced by zeros) and applications to energy minimization on complex manifolds.

Conclusion

This work rigorously determines the universal limiting law for the smallest pairwise distances between zeros of Gaussian analytic functions on compact Riemann surfaces and for the classical GEF, characterizing the scaling, limiting point process, and density functions. The convergence to a Poisson process with quartic intensity and the identification of $|Z_n| = n$4 as the minimal gap scale affirm the strongly repulsive nature of random zero configurations in the complex analytic setting, distinguishing them fundamentally from systems of independent points. The methods and insights presented provide new tools for understanding the fine structure of zero sets in random complex geometry and pave the way for future investigation into higher-dimensional and non-Euclidean settings.