Fluctuations in random complex zeroes: Asymptotic normality revisited

Understand

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.