  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.
