GAMP/OpAlg seminar by Alexander Bufetov (Aix-Marseille Université)

GAMP/OpAlg Seminar

Speaker: Alexander Bufetov (Aix-Marseille Université)

Title: Determinantal Point Processes and Extrapolation

Abstract: Point processes whose correlation functions are given by determinants  arise in the study of very different objects such as random matrices (Dyson), fermions (Macchi),  random series (Peres-Virag, Krishnapur), spanning trees (Burton-Pemantle, Benjamini-Lyons-Peres-Schramm), Young diagrams (Baik-Deift-Johansson, Borodin-Okounkov-Olshanski),  representations of infinite-dimensional groups (Borodin-Olshanski). The talk will start with an elementary introduction to determinantal point processes, proceeding to recent developments and open problems. 

For example, 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 Analyitc 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, 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 set. The second part of the talk is based on the preprints arXiv:1806.02306 and arXiv:1612.06751.