The Jancovici–Lebowitz–Manificat Law for Large Fluctuations of Random Complex Zeroes

Abstract

Consider a Gaussian Entire Function $$f(z) = \sum_{k=0}^\infty \zeta_k \frac{z^k}{\sqrt{k!}}\, ,$$ where $${\zeta_0, \zeta_1, \dots }$$ are Gaussian i.i.d. complex random variables. The zero set of this function is distribution invariant with respect to the isometries of the complex plane. Let n(R) be the number of zeroes of f in the disk of radius R. It is easy to see that $${\mathbb{E}n(R) = R^2}$$ , and it is known that the variance of n(R) grows linearly with R (Forrester and Honner). We prove that, for every α > 1/2, the tail probability $${\mathbb{P} \{ |n(R) - R^2 | > R^\alpha \}}$$ behaves as exp $${\left[-R^{\varphi (\alpha)}\right]}$$ with some explicit piecewise linear function $${\varphi(\alpha)}$$ . For some special values of the parameter α, this law was found earlier by Sodin and Tsirelson, and by Krishnapur. In the context of charge fluctuations of a one-component Coulomb system of particles of one sign embedded into a uniform background of another sign, a similar law was discovered some time ago by Jancovici, Lebowitz and Manificat.