Fluctuations in the zero set of the hyperbolic Gaussian analytic function

Abstract: The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disc whose distribution is invariant under automorphisms of the disc. We study the variance of the number of points in a disc of increasing radius. Somewhat surprisingly, we find a change of behaviour at a certain value of the `intensity' of the process, which appears to be novel.