The Jancovici - Lebowitz - Manificat law for large fluctuations of random complex zeroes

TL;DR

This paper investigates large fluctuations in the zero sets of a specific class of random entire functions, revealing they follow a known asymptotic law related to Coulomb charge fluctuations, highlighting deep connections between random zeros and physical systems.

Contribution

It demonstrates that the large fluctuations of these random complex zeroes adhere to the Jancovici-Lebowitz-Manificat law, linking random zero distributions to Coulomb system charge fluctuation laws.

Findings

Large zero fluctuations follow the Jancovici-Lebowitz-Manificat asymptotic law.

Zero set distribution is invariant under complex plane isometries.

Results connect random entire functions with Coulomb charge fluctuation phenomena.

Abstract

By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant with respect to isometries of the complex plane. We study large fluctuations of random complex zeroes and show that they obey the asymptotic law that was discovered some time ago by Jancovici, Lebowitz and Manificat for charge fluctuations of a Coulomb system of particles.