A palm hierarchy for determinantal point processes with the Bessel kernel

  • Alexander I. Bufetov


The main result of this note shows that Palm distributions of the determinantal point process governed by the Bessel kernel with parameter s are equivalent to the determinantal point process governed by the Bessel kernel with parameter s + 2. The Radon–Nikodym derivative is explicitly computed as a multiplicative functional on the space of configurations.


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  1. 1.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th print. (U.S. Gov. Print. Off., Washington, D.C., 1972), US Dept. Commerce, Natl. Bur. Stand. Appl. Math. Ser. 55.zbMATHGoogle Scholar
  2. 2.
    I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, “Uniform spanning forests,” Ann. Probab. 29 (1), 1–65 (2001).MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Borodin, A. Okounkov, and G. Olshanski, “Asymptotics of Plancherel measures for symmetric groups,” J. Am. Math. Soc. 13 (3), 481–515 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Borodin and E. M. Rains, “Eynard–Mehta theorem, Schur process, and their Pfaffian analogs,” J. Stat. Phys. 121 (3–4), 291–317 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. I. Bufetov, “Multiplicative functionals of determinantal processes,” Usp. Mat. Nauk 67 (1), 177–178 (2012) [Russ. Math. Surv. 67, 181–182 (2012)].MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. I. Bufetov, “Infinite determinantal measures,” Electron. Res. Announc. Math. Sci. 20, 12–30 (2013).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. I. Bufetov, “Quasi-symmetries of determinantal point processes,” arXiv: 1409.2068 [math.PR].Google Scholar
  8. 8.
    A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I: Construction of infinite determinantal measures,” Izv. Ross. Akad. Nauk, Ser. Mat. 79 (6), 18–64 (2015) [Izv. Math. 79, 1111–1156 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. I. Bufetov, “Action of the group of diffeomorphisms on determinantal measures,” Usp. Mat. Nauk 70 (5), 175–176 (2015) [Russ. Math. Surv. 70, 953–954 (2015)].CrossRefzbMATHGoogle Scholar
  10. 10.
    D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes (Springer, New York, 2008), Vols. I, II.CrossRefzbMATHGoogle Scholar
  11. 11.
    S. Ghosh, “Determinantal processes and completeness of random exponentials: The critical case,” arXiv: 1211.2435 [math.PR].Google Scholar
  12. 12.
    S. Ghosh, “Rigidity and tolerance in Gaussian zeros and Ginibre eigenvalues: Quantitative estimates,” arXiv: 1211.3506 [math.PR].Google Scholar
  13. 13.
    S. Ghosh and Y. Peres, “Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues,” arXiv: 1211.2381 [math.PR].Google Scholar
  14. 14.
    J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág, “Determinantal processes and independence,” Probab. Surv. 3, 206–229 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. R. Its, A. G. Izergin, V. E. Korepin, and N. A. Slavnov, “Differential equations for quantum correlation functions,” Int. J. Mod. Phys. B 4 (5), 1003–1037 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    O. Kallenberg, Random Measures (Akademie, Berlin, 1983).zbMATHGoogle Scholar
  17. 17.
    R. Lyons, “Determinantal probability measures,” Publ. Math., Inst. Hautes étud. Sci. 98, 167–212 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Olshanski, “The quasi-invariance property for the gamma kernel determinantal measure,” Adv. Math. 226 (3), 2305–2350 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Shirai and Y. Takahashi, “Fermion process and Fredholm determinant,” in Proc. Second ISAAC Congr., Fukuoka (Japan), 1999 (Kluwer, Dordrecht, 2000), Vol. 1, pp. 15–23.CrossRefGoogle Scholar
  20. 20.
    T. Shirai and Y. Takahashi, “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
  21. 21.
    T. Shirai and Y. Takahashi, “Random point fields associated with certain Fredholm determinants. II: Fermion shifts and their ergodic and Gibbs properties,” Ann. Probab. 31 (3), 1533–1564 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. B. Soshnikov, “Determinantal random point fields,” Usp. Mat. Nauk 55 (5), 107–160 (2000) [Russ. Math. Surv. 55, 923–975 (2000)].MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    C. A. Tracy and H. Widom, “Level spacing distributions and the Bessel kernel,” Commun. Math. Phys. 161 (2), 289–309 (1994).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussia
  3. 3.National Research University “Higher School of Economics,”MoscowRussia
  4. 4.Aix-Marseille Université, CNRS, Centrale MarseilleMarseille Cedex 13France

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