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 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.

25 pages, 1 figure

References in corpus (3)

• Determinantal Processes and Independence
• Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
• Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

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• Large charge fluctuations in the hierarchical Coulomb gas