Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

Abstract

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as $r\rightarrow \infty$. For the planar Gaussian analytic function, $\sum_{n \geq 0}\frac{a_n z^n}{\sqrt{n!}}$, we show that this probability is asymptotic to ${e^{-\frac{1}{2}m^2\log(m)}}$. For the hyperbolic Gaussian analytic functions, ${\sum_{n\geq 0}{-\rho \choose n}^{1/2} a_n z^n}, \rho > 0$, we show that this probability decays like $e^{-cm^2}$.

In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as $r\rightarrow \infty$. We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.