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

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 pr...

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Published inJournal of the European Mathematical Society : JEMS Vol. 23; no. 5; pp. 1477 - 1519
Main Authors Bufetov, Alexander I., Qiu, Yanqi, Shamov, Alexander
Format Journal Article
LanguageEnglish
Published European Mathematical Society 01.01.2021
Subjects
Mathematics
Determinantal point processes
tail triviality
conditional measures
Lyons–Peres completeness conjecture
operator-valued martingales
measure-valued martingales
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Summary: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 σ-algebra for determinantal point processes governed by self-adjoint kernels.
ISSN:1435-9855
1435-9863
DOI:10.4171/jems/1038