Chaos and order for determinantal point processes

Determinantal point processes arise in combinatorics, representation theory and mathematical physics, specifically random matrix theory.
Our processes are chaotic: the Central Limit Theorem holds (Costin-Lebowitz, Soshnikov), and so does its functional analogue (Dymov and the speaker), the Kolmogorov 0-1 law also holds (Lyons, Osada-Osada, Qiu, Shamov and the speaker).
At the same time, particles of our random configurations interact at infinite distance, as is shown by the Ghosh-Peres rigidity property or the explicit form of conditional measures for our processes found by the speaker.
As shown by Qiu, Shamov and the speaker, almost every realization of a determinantal point process is a uniqueness set for the governing reproducing kernel Hilbert space, for instance, the Bergman space. While partial results are given by the Patterson-Sullivan construction, the general question of simultaneous extrapolation of Bergman functions from their restrictions to our random configuration remains largely open.