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Journal of the European Mathematical Society

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Volume 23, Issue 5, 2021, pp. 1477–1519
DOI: 10.4171/JEMS/1038

Published online: 2021-01-19

Kernels of conditional determinantal measures and the Lyons–Peres completeness conjecture

Alexander I. Bufetov[1], Yanqi Qiu[2] and Alexander Shamov[3]

(1) Aix-Marseille Université, France, Steklov Mathematical Institute of RAS, Moscow and Saint-Petersburg State University, R
(2) Chinese Academy of Sciences, Beijing, China and Université Paul Sabatier, Toulouse, France
(3) Weizmann Institute of Science, Rehovot, Israel

The main result of this paper, Theorem 1.4, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a self-adjoint reproducing kernel, the system of kernels sampled at the points of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.9, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.10 is a new local property for kernels of conditional point processes. In Theorem 1.6 we prove the triviality of the tail $\sigma$-algebra for determinantal point processes governed by self-adjoint kernels.

Keywords: Determinantal point processes, Lyons–Peres completeness conjecture, conditional measures, tail triviality, measure-valued martingales, operator-valued martingales

Bufetov Alexander I., Qiu Yanqi, Shamov Alexander: Kernels of conditional determinantal measures and the Lyons–Peres completeness conjecture. J. Eur. Math. Soc. 23 (2021), 1477-1519. doi: 10.4171/JEMS/1038