Fluctuation in the zero set of the parabolic gausian analytic function

Abstract

In this project we study the fluctuation of the zero set process of the parabolic Gaussian analytic function, denoted $\mathbb{S}^{2}$ -GAF and where $\mathbb{S}^{2}$ is the Riemann sphere. There exist several ways to measure such fluctuations. One of them is to compute the variance of certain variables counting the number of points of the process inside a given region. Some asymptotics of such variables will lead us to conclude that the $\mathbb{S}^{2}-$ GAF process is more rigid than the Poisson process on $\mathbb{S}^{2}$ having, in mean, the same number of points as the $\mathbb{S}^{2}$ -GAF process. Also, we will see that the $\mathbb{S}^{2}$ -GAF process tends, as the intensity goes to infinity, to the planar GAF. Another point of view to study the fluctuations of the $\mathbb{S}^{2}$ -GAF is the so-called large deviations, i.e., to measure how certain linear statistics deviate from its average by a fraction of its same average. The latter allows us to estimate the hole probability, i.e., the probability that the point process does not meet a given disk.