Abstract
Kernels of conditional determinantal measures and the Lyons–Peres completeness conjecture
Highlights
Lemma 1.9 establishes that, for any determinantal point process PK induced by a selfadjoint locally trace class kernel K, the conditional measures PK (· | X, C) are themselves determinantal and governed by explicitly given self-adjoint kernels
We prove that for determinantal point processes governed by self-adjoint kernels, conditioning on the configuration in any Borel subset preserves the determinantal property
As an application of the local property for the conditional kernels, in Theorem 1.6 we establish the triviality of the tail σ -algebra for determinantal point processes governed by self-adjoint kernels
Summary
Mathematics Subject Classification (2020): Primary 60G55, 60G57; Secondary 60D05, 60G46, 37A25. Denote A2(D) the Bergman space of holomorphic functions on D square-integrable with respect to the Lebesgue measure Leb. A subset X ⊂ D is called a uniqueness set for A2(D) if a function h ∈ A2(D) satisfying h X = 0 must be the zero function. A subset X ⊂ D is called a uniqueness set for A2(D) if a function h ∈ A2(D) satisfying h X = 0 must be the zero function In this particular case, our main result is Theorem 1.1. Recall that a discrete subset Z ⊂ D is called an A2(D)-sampling set if there exist C1, C2 > 0 such that for any g ∈ A2(D) we have. The subset Z(fD) is almost surely neither A2(D)-sampling nor A2(D)interpolating. Lemma 1.3, proved in Section 8.3 below with the use of ergodicity, under the measure PKD , of the action of one-parameter groups of isometries of the Lobachevsky plane (this ergodicity is due to Hough–Krishnapur–Peres–Virag [19, Proposition 2.3.7]), implies Proposition 1.2
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