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


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 



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).


  1. 1.
    Bufetov, A.I., Dabrowski, Y., Qiu, Y.: Linear rigidity of stationary stochastic processes. Ergod. Theory Dynam. Sys. (2017).
  2. 2.
    Borodin, A., Olshanski, G.: Distributions on partitions, point processes, and the hypergeometric kernel. Commun. Math. Phys. 211(2), 335–358 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, A., Olshanski, G.: Random partitions and the gamma kernel. Adv. Math. 194(1), 141–202 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boas Jr., R.P.: Lipschitz behaviour and integrability of characteristic functions. Ann. Math. Stat. 38, 32–36 (1967)CrossRefzbMATHGoogle Scholar
  5. 5.
    Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13(3), 481–515 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borodin, A.: Point processes and the infinite symmetric group, part II: higher correlation functions. arXiv:math/9804087
  7. 7.
    Borodin, A., Rains, E.M.: Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121(3–4), 291–317 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bufetov, A.I.: Quasi-symmetries of determinantal point processes. Ann. Probab. (2014). arXiv:1409.2068 (to appear)
  9. 9.
    Bufetov, A.I.: Rigidity of determinantal point processes with the airy, the Bessel and the Gamma kernel. Bull. Math. Sci. 6(1), 163–172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bufetov, A.I.: On multiplicative functionals of determinantal processes. Uspekhi Mat. Nauk 67(1(403)), 177–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bufetov, A.I.: Infinite determinantal measures. Electron. Res. Announc. Math. Sci. 20, 12–30 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bufetov, A.I.: Action of the group of diffeomorphisms on determinantal measures. Russ. Math. Surv. 70(5), 953–954 (2015)CrossRefzbMATHGoogle Scholar
  13. 13.
    Bufetov, A.I., Qiu, Y.: Determinantal point processes associated with Hilbert spaces of holomorphic functions. Commun. Math. Phys. 351(1), 1–44 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. In: General theory and structure, 2nd edn. Probability and its applications, vol II. Springer, New York (2008). ISBN: 978-0-387-21337-8Google Scholar
  15. 15.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. Based, in Part, on Notes Left by Harry Bateman. McGraw-Hill Book Company, Inc, New York (1953)Google Scholar
  16. 16.
    Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields 163(3–4), 643–665 (2015)Google Scholar
  17. 17.
    Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. Duke Math. J. 166(10), 1789–1858 (2017)Google Scholar
  18. 18.
    Grothendieck, A.: La théorie de Fredholm. Bull. Soc. Math. Fr. 84, 319–384 (1956)CrossRefzbMATHGoogle Scholar
  19. 19.
    Grümm, H.R.: Two theorems about \({\fancyscript {C}}_{p}\). Rep. Math. Phys. 4, 211–215 (1973)Google Scholar
  20. 20.
    Ben Hough, J., Krishnapur, M., Peres, Y., Virág, B., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, vol. 51. University Lecture Series. American Mathematical Society, Providence (2009)Google Scholar
  21. 21.
    Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes. Electron. J. Probab. 18(74), 24 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. In: Proceedings of the Conference on Yang–Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, vol. 4, pp. 1003–1037 (1990)Google Scholar
  23. 23.
    Kadison, R.V.: Strong continuity of operator functions. Pac. J. Math. 26, 121–129 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kallenberg, O.: Random measures. Akademie-Verlag, Berlin (1986)zbMATHGoogle Scholar
  25. 25.
    Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. I. Arch. Ration. Mech. Anal. 59(3), 219–239 (1975)MathSciNetGoogle Scholar
  26. 26.
    Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lytvynov, E.: Determinantal point processes with \(J\)-Hermitian correlation kernels. Ann. Probab. 41(4), 2513–2543 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83–122 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Olshanski, G.: Point processes and the infinite symmetric group, part V: analysis of the matrix Whittaker kernel. arXiv:math/9810014
  30. 30.
    Olshanski, G.: The quasi-invariance property for the Gamma kernel determinantal measure. Adv. Math. 226(3), 2305–2350 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Osada, H., Shirai, T.: Absolute continuity and singularity of Palm measures of the Ginibre point process. Probab. Theory Relat. Fields 165(3–4), 725–770 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Simon, B.: Notes on infinite determinants of Hilbert space operators. Adv. Math. 24(3), 244–273 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Simon, B.: Trace Ideals and Their Applications, 2nd edn, vol. 120. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005). ISBN: 0-8218-3581-5Google Scholar
  34. 34.
    Soshnikov, A.: Determinantal random point fields. Uspekhi Mat. Nauk 55(5(335)), 107–160 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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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