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

The main result of this paper, Theorem 1.5, 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 step in the proof, Lemma...

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Bibliographic Details
Published inarXiv.org
Main Authors Bufetov, Alexander I, Qiu, Yanqi, Shamov, Alexander
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.12.2018
Subjects
Configurations
Kernels
Mathematics - Dynamical Systems
Mathematics - Functional Analysis
Mathematics - Mathematical Physics
Mathematics - Probability
Physics - Mathematical Physics
Set theory
Theorems
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Summary:The main result of this paper, Theorem 1.5, 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 step in the proof, Lemma 1.11, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.12 is a new local property for kernels of conditional point processes. In Theorem 1.7 we prove the triviality of the tail sigma-algebra for determinantal point processes governed by self-adjoint kernels.
ISSN:2331-8422
DOI:10.48550/arxiv.1612.06751