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Mosc. Math. J., 2006, Volume 6, Number 4, Pages 629–655 (Mi mmj263)  

This article is cited in 18 scientific papers (total in 18 papers)

Meixner polynomials and random partitions

Alexei Borodina, Grigori Olshanskiib

a Mathematics, Caltech, Pasadena, CA, U.S.A.
b Dobrushin Mathematics Laboratory, Institute for Information Transmission Problems, Moscow, RUSSIA

Abstract: The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the discrete hypergeometric kernel by a new method, based on a relationship between the z-measures and the Meixner orthogonal polynomial ensemble. In another paper (Prob. Theory Rel. Fields 135 (2006), 84–152) we apply the same approach to a dynamical model related to the z-measures.

Key words and phrases: Random partitions, random Young diagrams, determinantal point processes, correlation functions, correlation kernels, orthogonal polynomial ensembles, Meixner polynomials, Krawtchouk polynomials.


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MSC: 60C05, 33C45
Received: June 16, 2006

Citation: Alexei Borodin, Grigori Olshanskii, “Meixner polynomials and random partitions”, Mosc. Math. J., 6:4 (2006), 629–655

Citation in format AMSBIB
\by Alexei~Borodin, Grigori~Olshanskii
\paper Meixner polynomials and random partitions
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 4
\pages 629--655

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    This publication is cited in the following articles:
    1. G. I. Olshanskii, “Difference Operators and Determinantal Point Processes”, Funct. Anal. Appl., 42:4 (2008), 317–329  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Strahov E., “Matrix kernels for measures on partitions”, J. Stat. Phys., 133:5 (2008), 899–919  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Borodin A., Olshanski G., “Infinite-dimensional diffusions as limits of random walks on partitions”, Probab. Theory Related Fields, 144:1-2 (2009), 281–318  crossref  mathscinet  zmath  isi  elib  scopus
    4. Wong R., Zhao Yuqiu, “Asymptotics of orthogonal polynomials via the Riemann–Hilbert approach”, Acta Math. Sci. Ser. B Engl. Ed., 29:4 (2009), 1005–1034  crossref  mathscinet  zmath  isi  scopus
    5. Strahov E., “The $z$-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel”, Adv. Math., 224:1 (2010), 130–168  crossref  mathscinet  zmath  isi  elib  scopus
    6. Strahov E., “$Z$-measures on partitions related to the infinite Gelfand pair $(S(2\infty),H(\infty))$”, J. Algebra, 323:2 (2010), 349–370  crossref  mathscinet  zmath  isi  elib  scopus
    7. A. I. Aptekarev, D. N. Tulyakov, “Asymptotic regimes in saturation zones for the C-D-kernels for an ensemble of Meixner orthogonal polynomials”, Russian Math. Surveys, 66:1 (2011), 173–175  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Olshanski G., “The quasi-invariance property for the Gamma kernel determinantal measure”, Adv. Math., 226:3 (2011), 2305–2350  crossref  mathscinet  zmath  isi  elib  scopus
    9. Petrov L., “Pfaffian stochastic dynamics of strict partitions”, Electron. J. Probab., 16:82 (2011), 2246–2295  crossref  mathscinet  zmath  isi  elib
    10. Szabo R.J., Tierz M., “Two-dimensional Yang-Mills theory, Painlevé equations and the six-vertex model”, J. Phys. A, 45:8 (2012), 085401  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106  mathnet  crossref  mathscinet  zmath  elib
    12. J. Math. Sci. (N. Y.), 190:3 (2013), 451–458  mathnet  crossref  mathscinet
    13. Olshanski G., “Laguerre and Meixner Orthogonal Bases in the Algebra of Symmetric Functions”, Int. Math. Res. Notices, 2012, no. 16, 3615–3679  crossref  mathscinet  zmath  isi  elib  scopus
    14. Petrov L., “Sl(2) Operators and Markov Processes on Branching Graphs”, J. Algebr. Comb., 38:3 (2013), 663–720  crossref  mathscinet  zmath  isi  elib  scopus
    15. Genest V.X., Miki H., Vinet L., Zhedanov A., “The Multivariate Meixner Polynomials as Matrix Elements of So(D, 1) Representations on Oscillator States”, J. Phys. A-Math. Theor., 47:4 (2014), 045207  crossref  mathscinet  zmath  isi  elib  scopus
    16. A. I. Aptekarev, D. N. Tulyakov, “The Saturation Regime of Meixner Polynomials and the Discrete Bessel Kernel”, Math. Notes, 98:1 (2015), 180–184  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    17. Borodin A., Olshanski G., “The Asep and Determinantal Point Processes”, Commun. Math. Phys., 353:2 (2017), 853–903  crossref  zmath  isi  scopus
    18. Borodin A., “Stochastic Higher Spin Six Vertex Model and Macdonald Measures”, J. Math. Phys., 59:2 (2018), 023301  crossref  mathscinet  zmath  isi  scopus
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