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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 -algebra for determinantal point processes governed by self-adjoint kernels.
Cite this article
Alexander I. Bufetov, Yanqi Qiu, Alexander Shamov, Kernels of conditional determinantal measures and the Lyons–Peres completeness conjecture. J. Eur. Math. Soc. 23 (2021), no. 5, pp. 1477–1519