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Probability Theory and Related Fields

, Volume 172, Issue 1–2, pp 31–69 | Cite as

Equivalence of Palm measures for determinantal point processes governed by Bergman kernels

  • Alexander I. Bufetov
  • Shilei Fan
  • Yanqi Qiu
Article
  • 132 Downloads

Abstract

For a determinantal point process induced by the reproducing kernel of the weighted Bergman space \(A^2(U, \omega )\) over a domain \(U \subset \mathbb {C}^d\), we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain U contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the \(H^\infty (U)\)-module structure of \(A^2(U, \omega )\). A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of U.

Keywords

Bergman kernel Determinantal point process Conditional measure Deletion and insertion tolerance Palm equivalence Monotone coupling 

Mathematics Subject Classification

Primary 60G55 Secondary 32A36 

Notes

Acknowledgements

We are deeply grateful to Alexei Klimenko for useful discussions and very helpful comments. The research of A. Bufetov and S. Fan on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under Grant agreement No 647133 (ICHAOS). A. Bufetov has also been funded by the Grant MD 5991.2016.1 of the President of the Russian Federation, by the Russian Academic Excellence Project ‘5-100’ and by the Chaire Gabriel Lamé at the Chebyshev Laboratory of the SPbSU, a joint initiative of the French Embassy in the Russian Federation and the Saint-Petersburg State University. Y. Qiu is supported by the Grant IDEX UNITI-ANR-11-IDEX-0002-02, financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency. Part of this work was carried out at the Institut Henri Poincaré and at the Centre international de rencontres mathématiques in the framework of the CIRM “recherche en petits groupes” programme. We are deeply grateful to these institutions for their warm hospitality.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Alexander I. Bufetov
    • 1
    • 2
    • 3
    • 4
    • 5
  • Shilei Fan
    • 1
    • 6
  • Yanqi Qiu
    • 7
    • 8
  1. 1.Centrale Marseille, CNRS, Institut de Mathématiques de Marseille, UMR 7373Aix-Marseille UniversitéMarseilleFrance
  2. 2.Steklov Mathematical Institute of RASMoscowRussia
  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.School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanChina
  7. 7.CNRS, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  8. 8.Institute of Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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