Abstract: The hyperbolic Gaussian analytic function (GAF) is a random holomorphic function on the unit disc. It satisfies the remarkable property that the distribution of its zero set is invariant under automorphisms of the disc, and it is essentially the only GAF with this property. A hole is the event that there are no zeroes in a given hyperbolic disc. I will discuss the asymptotic decay of the probability of this event, under various regimes. Joint work with A. Nishry, R. Peled and M. Sodin.