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Mathematische Annalen

, Volume 371, Issue 1–2, pp 127–188 | Cite as

J-Hermitian determinantal point processes: balanced rigidity and balanced Palm equivalence

  • Alexander I. Bufetov
  • Yanqi Qiu
Article

Abstract

We study Palm measures of determinantal point processes with J-Hermitian correlation kernels. A point process \(\mathbb {P}\) on the punctured real line \(\mathbb {R}^* = \mathbb {R}_{+} \sqcup \mathbb {R}_{-}\) is said to be balanced rigid if for any precompact subset \(B\subset \mathbb {R}^*\), the difference between the numbers of particles of a configuration inside \(B\cap \mathbb {R}_+\) and \(B\cap \mathbb {R}_-\) is almost surely determined by the configuration outside B. The point process \(\mathbb {P}\) is said to have the balanced Palm equivalence property if any reduced Palm measure conditioned at 2n distinct points, n in \(\mathbb {R}_+\) and n in \(\mathbb {R}_-\), is equivalent to the \(\mathbb {P}\). We formulate general criteria for determinantal point processes with J-Hermitian correlation kernels to be balanced rigid and to have the balanced Palm equivalence property and prove, in particular, that the determinantal point processes with Whittaker kernels of Borodin and Olshanski are balanced rigid and have the balanced Palm equivalence property.

Mathematics Subject Classification

Primary 60G55 Secondary 30H20 30B20 

Notes

Acknowledgements

The authors were supported by A*MIDEX project (no. ANR-11-IDEX-0001-02), financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency (ANR). A. B. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant agreement no 647133 (ICHAOS), by the Grant MD 5991.2016.1 of the President of the Russian Federation and by the Russian Academic Excellence Project ‘5-100’. Y. Q. 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 as well as by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund, he is also support in part by NSF of China (Grant No. 11688101).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Aix-Marseille Université, Centrale Marseille, CNRS, I2M, UMR7373MarseilleFrance
  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.CNRS, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  6. 6.Institute of MathematicsAMSS, Chinese Academy of Sciences and Hua Loo-Keng Key Laboratory of MathematicsBeijingChina

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