Fluctuations in random complex zeroes: Asymptotic normality revisited

TL;DR

This paper investigates the statistical behavior of zeros of a specific class of random entire functions, establishing conditions for their fluctuations to be normally distributed and providing examples of atypical behaviors.

Contribution

It extends previous work by computing the variance of linear statistics of these zeros and identifying near-optimal conditions for asymptotic normality.

Findings

Variance of linear statistics computed

Conditions for asymptotic normality established

Examples of abnormal fluctuations provided

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. Extending the previous results of Sodin and Tsirelson, we compute the variance of linear statistics of random complex zeroes, and find close to optimal conditions on a test-function that yield asymptotic normality of fluctuations of the corresponding linear statistics. We also provide examples of test-functions with abnormal fluctuations of linear statistics.