Determinantal point processes and completeness of reproducing kernels

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 neither sampling nor interpolating in the sense of Seip. Nonetheless, in a sequel paper, joint with Yanqi Qiu, we give an explicit procedure to recover a Bergman function from its values on our set.
By the Peres and Virag Theorem, zeros of a Gaussian Analytic Function on the disk are a determinantal point process governed by the Bergman kernel, and we prove, for general determinantal point processes, that reproducing kernels sampled along a trajectory form a complete system in the ambient Hilbert space. The key step in our argument is that the determinantal property is preserved under conditioning.