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 $K_Λ(z, w) = \sum_{n\in Λ}(n+1) z^n \bar{w}^n, \quad z, w \in \mathbb{D},$ with $Λ$ 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.

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