Kernels of conditional determinantal measures

Alexander I. Bufetov (1, 2, 3, 4) , Yanqi Qiu (5) , Alexander Shamov (6)

1 I2M - Institut de Mathématiques de Marseille
2 SMI | RAS - Steklov Mathematical Institute [Moscow]
3 Institute for Information Transmission Problems
4 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
5 IMT - Institut de Mathématiques de Toulouse UMR5219
6 Weizmann Institute of Science - Department of Mathematics

For determinantal point processes governed by self-adjoint kernels, we prove in Theorem 1.2 that conditioning on the configuration in a subset preserves the determinantal property. In Theorem 1.3 we show the tail sigma-algebra for our determinantal point processes is trivial, proving a conjecture by Lyons. If our self-adjoint kernel is a projection, then, establishing a conjecture by Lyons and Peres, we show in Theorem 1.5 that reproducing kernels corresponding to particles of almost every configuration generate the range of the projection. Our argument is based on a new local property for conditional kernels of determinantal point processes stated in Lemma 1.7.

Mots clés: Determinantal point processes; conditional measures; tail triviality; operator-valued martingales; Palm measures; measure-valued martingales

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

Alexander I. Bufetov, Yanqi Qiu, Alexander Shamov. Kernels of conditional determinantal measures. 2016. ⟨hal-01483603⟩