Smallest distances between zeros of Gaussian analytic functions

TL;DR

This paper investigates the distribution of the smallest distances between zeros of Gaussian analytic functions on compact Riemann surfaces, revealing a universal Poisson process limit after rescaling.

Contribution

It establishes the convergence of the smallest distances to a universal Poisson point process and describes their limiting density, extending to classical Gaussian Entire Functions.

Findings

Smallest distances, after rescaling, converge to a Poisson process.

Locations of smallest distances are uniformly distributed with respect to volume form.

Limiting density of the k-th smallest distance follows a specific exponential-polynomial form.

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.