  We consider the family $\{f_L\}_{L>0}$ of Gaussian analytic functions in the
unit disk, distinguished by the invariance of their zero set with respect to
hyperbolic isometries. Let $n_L\left(r\right)$ be the number of zeros of $f_L$
in a disk of radius $r$. We study the asymptotic probability of the rare event
where there is an overcrowding of the zeros as $r\uparrow1$, i.e. for every
$L>0$, we are looking for the asymptotics of the probability
$\mathbb{P}\left[n_L(r)\geq V(r)\right]$ with $V\left(r\right)$ large compared
to the $\mathbb{E}\left[n_L\left(r\right)\right]$. Peres and Vir\'ag showed
that for $L=1$ (and only then) the zero set forms a determinantal point
process, making many explicit computations possible. Curiously, contrary to the
much better understood planar model, it appears that for $L<1$ the exponential
order of decay of the probability of overcrowding when $V$ is close to
$\mathbb{E}\left[n_L\left(r\right)\right]$ is much less than the probability of
a deficit of zeros.
