Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

@inproceedings{Krishnapur2008OvercrowdingEF,
  title={Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions},
  author={Manjunath Krishnapur},
  year={2008}
}
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 m → ∞. For the Planar Gaussian analytic function, ∑ n≥0 anz √ n! , we show that this probability is asymptotic to e− 1 2 m2 . For the Hyperbolic Gaussian analytic functions,