Fluctuations in random complex zeroes: Asymptotic normality revisited

The Gaussian Entire Function F(Z) = Σ_k=0 ^∞ζk Z^k/√K!(ζ₀, ζ₁,. are Gaussian i.i.d. complex random coefficients) is distinguished by the distribution invariance of its zero set with respect to the isometries of the complex plane. We find close to optimal conditions on a function h that yield asymptotic normality of linear statistics of zeroes n(R,h) = Σa:F(a)=0 h(a/R) when, R → ∞, and construct examples of function h with abnormal fluctuations of linear statistics. We show that the fluctuations of n(R,h) are asymptotically normal when h is either a C ₀^α-function with α>1, or a C ₀^α-function with α ≤ 1 such that the variance of n(R,h) is at least R^-2α+ε with some ε>0. These results complement our recent results from "Clustering of correlation functions for random complex zeroes", where, using different methods, we prove that the fluctuations of n(R,h) are asymptotically normal when h is bounded, and the variance of n(R,h) grows at least as a positive power of R.