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The main result of this paper, Theorem 1.5, establishes a conjecture of Lyonsand Peres: for a determinantal point process governed by a reproducing kernel,the system of kernels sampled at the particles of a random configuration iscomplete in the range of the kernel. A key step in the proof, Lemma 1.11,states that conditioning on the configuration in a subset preserves thedeterminantal property, and the main Lemma 1.12 is a new local property forkernels of conditional point processes. In Theorem 1.7 we prove the trivialityof the tail sigma-algebra for determinantal point processes governed byself-adjoint kernels.

Kernels of conditional determinantal measures and the proof of the Lyons-Peres Conjecture