Electronic Communications in Probability

Random Strict Partitions and Determinantal Point Processes

Leonid Petrov

Full-text: Open access

Abstract

We present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice $\{1,2,\dots\}$ or on the open half-line $ in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 16, 162-175.

Dates
Accepted: 19 May 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243959

Digital Object Identifier
doi:10.1214/ECP.v15-1542

Mathematical Reviews number (MathSciNet)
MR2651548

Zentralblatt MATH identifier
1226.60072

Subjects
Primary: 60G55: Point processes
Secondary: 20C25: Projective representations and multipliers

Keywords
random strict partitions determinantal point process Macdonald kernel

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Petrov, Leonid. Random Strict Partitions and Determinantal Point Processes. Electron. Commun. Probab. 15 (2010), paper no. 16, 162--175. doi:10.1214/ECP.v15-1542. https://projecteuclid.org/euclid.ecp/1465243959


Export citation

References

  • Baik, Jihno; Deift, Percy; Johansson, Kurt. On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geom. Funct. Anal. 10 (2000), no. 4, 702-731.
  • Borodin, Alexei. Multiplicative central measures on the Schur graph. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 44-52, 290-291; translation in J. Math. Sci. (New York) 96 (1999), no. 5, 3472-3477
  • Borodin, Alexei. Riemann-Hilbert problem and the discrete Bessel kernel. Internat. Math. Res. Notices 2000, no. 9, 467-494.
  • Borodin, Alexei. Determinantal point processes (2009). arXiv preprint, arXiv:0911.1153v1 [math.PR].
  • Borodin, Alexei; Okounkov, Andrei; Olshanski, Grigori. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (2000), no. 3, 481-515 (electronic).
  • Borodin, Alexei; Olshanski, Grigori. Point processes and the infinite symmetric group. Math. Res. Lett. 5 (1998), no. 6, 799-816.
  • Borodin, Alexei; Olshanski, Grigori. Distributions on partitions, point processes, and the hypergeometric kernel. Comm. Math. Phys. 211 (2000), no. 2, 335-358.
  • Borodin, Alexei; Olshanski, Grigori. Random partitions and the gamma kernel. Adv. Math. 194 (2005), no. 1, 141-202.
  • Borodin, Alexei; Olshanski, Grigori. Markov processes on partitions. Probab. Theory Related Fields 135 (2006), no. 1, 84-152.
  • Borodin, Alexei; Olshanski, Grigori. Infinite-dimensional diffusions as limits of random walks on partitions. Probab. Theory Related Fields 144 (2009), no. 1-2, 281-318.
  • Deift, Percy. Integrable operators. Differential operators and spectral theory, 69-84, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999.
  • Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., editors. Higher transcendental functions. McGraw-Hill, New York, 1953-1955.
  • Forrester, Peter J. Log-gases and random matrices. Book in progress, http://www.ms.unimelb.edu.au/~matpjf/matpjf.html.
  • Hoffman, P. N.; Humphreys, J. F. Projective representations of the symmetric groups. Q-functions and shifted tableaux. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. xiv+304 pp. ISBN: 0-19-853556-2
  • Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint. Determinantal processes and independence. Probab. Surv. 3 (2006), 206-229 (electronic).
  • Its, A. R.; Izergin, A. G.; Korepin, V. E.; Slavnov, N. A. Differential equations for quantum correlation functions. Proceedings of the Conference on Yang-Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory. Internat. J. Modern Phys. B 4 (1990), no. 5, 1003-1037.
  • Ivanov, Vladimir. The dimension of skew shifted Young diagrams, and projective characters of the infinite symmetric group. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 115-135, 292-293; translation in J. Math. Sci. (New York) 96 (1999), no. 5, 3517-3530
  • Ivanov, Vladimir. Plancherel measure on shifted Young diagrams. Representation theory, dynamical systems, and asymptotic combinatorics, 73-86, Amer. Math. Soc. Transl. Ser. 2, 217, Amer. Math. Soc., Providence, RI, 2006.
  • Johansson, Kurt. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 (2001), no. 1, 259-296.
  • Kerov, Sergei; Olshanski, Grigori; Vershik, Anatoli. Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 8, 773-778.
  • Kerov, Sergei; Olshanski, Grigori; Vershik, Anatoli. Harmonic analysis on the infinite symmetric group. Invent. Math. 158 (2004), no. 3, 551-642.
  • Lisovyy, Oleg. Dyson's constant for the hypergeometric kernel (2009). arXiv preprint, arXiv:0910.1914v2 [math-ph].
  • Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2
  • Matsumoto, Sho. Correlation functions of the shifted Schur measure. J. Math. Soc. Japan 57 (2005), no. 3, 619-637.
  • Nazarov, M. L. Projective representations of the infinite symmetric group. Representation theory and dynamical systems, 115-130, Adv. Soviet Math., 9, Amer. Math. Soc., Providence, RI, 1992.
  • Okounkov, Andrei. SL(2) and z-measures. Random matrix models and their applications, 407-420, Math. Sci. Res. Inst. Publ., 40, Cambridge Univ. Press, Cambridge, 2001.
  • Okounkov, Andrei. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), no. 1, 57-81.
  • Okounkov, Andrei. Symmetric functions and random partitions, in: Symmetric functions 2001: Surveys of Developments and Perspectives. Proceedings of the NATO Advanced Study Institute held in Cambridge, June 25-July 6, 2001. Edited by Sergey Fomin. NATO Science Series II: Mathematics, Physics and Chemistry, 74. Kluwer Academic Publishers, Dordrecht, 2002. xiv+273 pp. ISBN: 1-4020-0773-6
  • Olshanski, Grigori. Point processes and the infinite symmetric group. Part V: Analysis of the matrix Whittaker kernel (1998). arXiv preprint.v1 [math.RT].
  • Olshanski, Grigori. The quasi-invariance property for the Gamma kernel determinantal measure (2009). arXiv preprint, arXiv:0910.0130v2 [math.PR].
  • Petrov, Leonid. Random walks on strict partitions. Zap. Nauchn. Sem. POMI, 373 (2009), 226-272 (in Russian). arXiv preprint (in English), arXiv:0904.1823v1 [math.PR].
  • Rains, Eric M. Correlation functions for symmetrized increasing subsequences (2000). arXiv preprint.v1 [math.CO].
  • Schur, I. Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrocheme lineare Substitionen. J. Reine Angew. Math. 139 (1911), 155-250.