Extrapolation for point processes

Consider a Gaussian Analytic Function on the disk, that is, a random series whose coefficients are independent complex Gaussians. In joint work with Yanqi Qiu and Alexander Shamov, we show that the zero set of a Gaussian Analytic Function 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 having these zeros. The key role in our argument is played by the determinantal structure of the zeros given by the PeresVir´ag Theorem, and we prove, in general, that the family of reproducing kernels along a realization of a determinantal point process generates the whole ambient Hilbert space, thus settling a conjecture of Lyons and Peres. In a sequel paper, joint with Yanqi Qiu, we study how to recover a holomorphic function from its values on our random set. The talk is based on the preprints arXiv:1806.02306, arXiv:1612.06751, arXiv:1605.01400