Determinantal point processes and extrapolation

In joint work with Yanqi Qiu and Alexander Shamov we prove that the zero set of a Gaussian Analytic Function is almost surely a uniqueness set in the Bergman space on the unit disc — equivalently, that any square-integrable holomorphic function is uniquely determined by its restriction to our set. By the Peres-Virag theorem, our random set is a determinantal point process governed by the Bergman kernel. The key lemma is that conditioning preserves the determinantal property. In subsequent joint work with Yanqi Qiu, we show that the Patterson-Sullivan construction recovers the value of any Hardy function at any point of the disc from its restriction to a random configuration of the determinant point process with the Bergman kernel.