  We study the zeroes of a family of random holomorphic functions on the unit
disc, distinguished by their invariance with respect to the hyperbolic
geometry. Our main finding is a transition in the limiting behaviour of the
number of zeroes in a large hyperbolic disc.
  We find a normal distribution if the covariance decays faster than a certain
critical value. In contrast, in the regime of 'long-range dependence' when the
covariance decays slowly, the limiting distribution is skewed. For a closely
related model we emphasise a link with Gaussian multiplicative chaos.
