Consider a Gaussian Analytic Function on the disk.
In joint work with Yanqi Qiu and Alexander Shamov,
we show that, almost surely, there does not exist a nonzero
square-integrable holomorphic function with these zeros.
By the Peres-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,
the conjecture of Russell Lyons and Yuval Peres 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.
The problem, posed by Russell Lyons, of describing these conditional
measures explicitly remains open even
for the sine-process, and I will report on partial progress: the
analogue of the Gibbs property for one-dimensional determinantal
processes governed by integrable kernels.
The talk is based on the preprint arXiv:1605.01400
as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and
Alexander Shamov.