Kernels of conditional determinantal measures and the proof of the\n Lyons-Peres Conjecture
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Abstract
The main result of this paper, Theorem 1.5, establishes a conjecture of Lyons\nand Peres: for a determinantal point process governed by a reproducing kernel,\nthe system of kernels sampled at the particles of a random configuration is\ncomplete in the range of the kernel. A key step in the proof, Lemma 1.11,\nstates that conditioning on the configuration in a subset preserves the\ndeterminantal property, and the main Lemma 1.12 is a new local property for\nkernels of conditional point processes. In Theorem 1.7 we prove the triviality\nof the tail sigma-algebra for determinantal point processes governed by\nself-adjoint kernels.\n
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