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

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.