Rigidity of determinantal point processes on the unit disc with sub-Bergman kernels

TL;DR

This paper constructs and analyzes number rigid determinantal point processes on the unit disc using sub-Bergman kernels, introducing both deterministic and probabilistic methods with connections to Bloch functions.

Contribution

It provides new constructions of rigid determinantal point processes with sub-Bergman kernels on the unit disc, employing both deterministic and probabilistic approaches.

Findings

Constructed rigid determinantal point processes with sub-Bergman kernels.

Established deterministic and probabilistic methods for these constructions.

Connected the analysis to classical Bloch functions.

Abstract

We give natural constructions of number rigid determinantal point processes on the unit disc $\mathbb{D}$ with sub-Bergman kernels of the form \[ K_\Lambda(z, w) = \sum_{n\in \Lambda}(n+1) z^n \bar{w}^n, \quad z, w \in \mathbb{D}, \] with $\Lambda$ an infinite subset of the set of non-negative integers. Our constructions are given both in a deterministic method and a probabilisitc method. In the deterministic method, our proofs involve the classical Bloch functions.