Determinantal point processes and the reconstruction of holomorphic functions

In joint work with Yanqi Qiu and Alexander Shamov we prove that the sequence of reproducing kernels sampled along a random trajectory of a determinant point process is complete in the ambient Hilbert space. From this result and the Peres-Virag Theorem it follows, in particular, 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.
In 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. This extrapolation result is then extended to real and complex hyperbolic spaces of higher dimension. Recovering continuous functions by the Patterson-Sullivan construction is also shown to be possible in more general Gromov hyperbolic spaces.