Point processes and extrapolation of holomorphic functions

Consider a Gaussian Analytic Function on the disk. In joint work with Yanqi Qiu and Alexander Shamov, we show that its zero set is a uniqueness set for the Bergman space on the disk: in other words, almost surely, there does not exist a nonzero square-integrable holomorphic function with these zeros. The distribution of our random subset is invariant under the group of isometries of the Lobachevsky plane; the action of every hyperbolic or parabolic isometry is mixing. It follows, in particular, that our set is not sampling in the sense of Seip. Ia sequel paper, joint with Yanqi Qiu, we use the Patterson-Sullivan construction to recover the values of a holomorphis function from its restriction onto our random set.