  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.
