Alexander Bufetov: Conditional measures of determinantal point processes: the Gibbs property and the completeness of reproducing kernels

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.