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A palm hierarchy for determinantal point processes with the Bessel kernel

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Abstract

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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References

  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.

    MATH  Google Scholar 

  2. I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, “Uniform spanning forests,” Ann. Probab. 29 (1), 1–65 (2001).

    MathSciNet  MATH  Google Scholar 

  3. A. Borodin, A. Okounkov, and G. Olshanski, “Asymptotics of Plancherel measures for symmetric groups,” J. Am. Math. Soc. 13 (3), 481–515 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. I. Bufetov, “Multiplicative functionals of determinantal processes,” Usp. Mat. Nauk 67 (1), 177–178 (2012) [Russ. Math. Surv. 67, 181–182 (2012)].

    Article  MathSciNet  MATH  Google Scholar 

  6. A. I. Bufetov, “Infinite determinantal measures,” Electron. Res. Announc. Math. Sci. 20, 12–30 (2013).

    MathSciNet  MATH  Google Scholar 

  7. A. I. Bufetov, “Quasi-symmetries of determinantal point processes,” arXiv: 1409.2068 [math.PR].

  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)].

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    Article  MATH  Google Scholar 

  10. D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes (Springer, New York, 2008), Vols. I, II.

    Book  MATH  Google Scholar 

  11. S. Ghosh, “Determinantal processes and completeness of random exponentials: The critical case,” arXiv: 1211.2435 [math.PR].

  12. S. Ghosh, “Rigidity and tolerance in Gaussian zeros and Ginibre eigenvalues: Quantitative estimates,” arXiv: 1211.3506 [math.PR].

  13. S. Ghosh and Y. Peres, “Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues,” arXiv: 1211.2381 [math.PR].

  14. J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág, “Determinantal processes and independence,” Probab. Surv. 3, 206–229 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Kallenberg, Random Measures (Akademie, Berlin, 1983).

    MATH  Google Scholar 

  17. R. Lyons, “Determinantal probability measures,” Publ. Math., Inst. Hautes étud. Sci. 98, 167–212 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Olshanski, “The quasi-invariance property for the gamma kernel determinantal measure,” Adv. Math. 226 (3), 2305–2350 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. B. Soshnikov, “Determinantal random point fields,” Usp. Mat. Nauk 55 (5), 107–160 (2000) [Russ. Math. Surv. 55, 923–975 (2000)].

    Article  MathSciNet  MATH  Google Scholar 

  23. C. A. Tracy and H. Widom, “Level spacing distributions and the Bessel kernel,” Commun. Math. Phys. 161 (2), 289–309 (1994).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Alexander I. Bufetov

  2. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, str. 1, Moscow, 127051, Russia

    Alexander I. Bufetov

  3. National Research University “Higher School of Economics,”, ul. Myasnitskaya 20, Moscow, 101000, Russia

    Alexander I. Bufetov

  4. Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot-Curie, 13453, Marseille Cedex 13, France

    Alexander I. Bufetov

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  1. Alexander I. Bufetov
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Correspondence to Alexander I. Bufetov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 105–112.

To the memory of Dmitry Victorovich Anosov

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Bufetov, A.I. A palm hierarchy for determinantal point processes with the Bessel kernel. Proc. Steklov Inst. Math. 297, 90–97 (2017). https://doi.org/10.1134/S0081543817040058

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  • DOI: https://doi.org/10.1134/S0081543817040058

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