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 $${K_\Lambda}(z,w) = \sum\limits_{n \in \Lambda} {(n + 1){z^n}{{\bar w}^n}} ,\,\,\,\,\,z,w \in {\mathbb{D}},$$ with Λ an infinite subset of non-negative integers. Our constructions are given in both deterministic and probabilistic methods. In the deterministic method, our proofs involve the classical Bloch functions.