Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

Details Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

2005-10-27

arxiv.org/abs/math/0510588v1

Mathematics

Probability

25 pages, 1 figure

Scientific paper

10.1007/s10955-006-9159-y

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-->infinity. For the Planar Gaussian analytic function, sum_n a_n z^n/sqrt(n!), we show that this probability is asymptotic to exp(-0.5 m^2 log(m)). For the Hyperbolic Gaussian analytic functions, sum_n sqrt({-rho choose n}) a_n z^n, rho>0, we show that this probability decays like exp(-cm^2). In the planar case, we also consider the problem posed by Mikhail Sodin on moderate and very large deviations in a disk of radius r as r --> infinity. We partly 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.

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