The main result of this paper establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a reproducing kernel, the system of kernels sampled at the particles of a random configuration is complete in the range of the kernel. A key ingredient states that conditioning on the configuration in a subset preserves the determinantal property, whose proof relies on a new local property for kernels of conditional point processes. Along the way, we prove  the conjecture of Lyons that the tail sigma-algebra is trivial for determinantal point processes governed by self-adjoint kernels.

Publication:

-    JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, (2021)

Authors:

-    Alexander Bufetov (Aix-Marseille Universit′e, Centrale Marseille, CNRS, Institut de Math′ematiques de Marseille)

-    Yanqi Qiu (Institute of Mathematics, AMSS, CAS and Hua Loo-Keng Key Laboratory of Mathematics, Beijing, China; CNRS, Institut de Math′ematiques de Toulouse, Universit′e Paul Sabatier)

-    Alexander Shamov (Department of Mathematics, Weizmann Institute of Science)