Communications in Mathematical Physics

, Volume 351, Issue 1, pp 1–44 | Cite as

Determinantal Point Processes Associated with Hilbert Spaces of Holomorphic Functions

Article

Abstract

We study determinantal point processes on \({\mathbb{C}}\) induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc \({\mathbb{D}}\) induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures of the same order are equivalent. The Radon–Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures of different orders are singular. In the second case, we show that all reduced Palm measures, of all orders, are equivalent. The Radon–Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Centrale Marseille, CNRS, I2M, UMR7373Aix-Marseille UniversitéMarseille Cedex 13France
  2. 2.Steklov Institute of MathematicsMoscowRussia
  3. 3.Institute for Information Transmission ProblemsMoscowRussia
  4. 4.National Research University Higher School of EconomicsMoscowRussia
  5. 5.The Chebyshev LaboratorySaint-Petersburg State UniversitySaint-PetersburgRussia
  6. 6.CNRS, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France

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