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

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.

DOI: https://doi.org/10.17323/1609-4514-2006-6-4-629-655

Full text: http://www.ams.org/.../abst6-4-2006.html
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Bibliographic databases:

MSC: 60C05, 33C45
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Citation: Alexei Borodin, Grigori Olshanskii, “Meixner polynomials and random partitions”, Mosc. Math. J., 6:4 (2006), 629–655

Citation in format AMSBIB
\Bibitem{BorOls06} \by Alexei~Borodin, Grigori~Olshanskii \paper Meixner polynomials and random partitions \jour Mosc. Math.~J. \yr 2006 \vol 6 \issue 4 \pages 629--655 \mathnet{http://mi.mathnet.ru/mmj263} \crossref{https://doi.org/10.17323/1609-4514-2006-6-4-629-655} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2291156} \zmath{https://zbmath.org/?q=an:1126.60006} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208596000002} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Strahov E., “Matrix kernels for measures on partitions”, J. Stat. Phys., 133:5 (2008), 899–919
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
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
5. Strahov E., “The $z$-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel”, Adv. Math., 224:1 (2010), 130–168
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
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
8. Olshanski G., “The quasi-invariance property for the Gamma kernel determinantal measure”, Adv. Math., 226:3 (2011), 2305–2350
9. Petrov L., “Pfaffian stochastic dynamics of strict partitions”, Electron. J. Probab., 16:82 (2011), 2246–2295
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
11. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106
12. J. Math. Sci. (N. Y.), 190:3 (2013), 451–458
13. Olshanski G., “Laguerre and Meixner Orthogonal Bases in the Algebra of Symmetric Functions”, Int. Math. Res. Notices, 2012, no. 16, 3615–3679
14. Petrov L., “Sl(2) Operators and Markov Processes on Branching Graphs”, J. Algebr. Comb., 38:3 (2013), 663–720
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
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
17. Borodin A., Olshanski G., “The Asep and Determinantal Point Processes”, Commun. Math. Phys., 353:2 (2017), 853–903
18. Borodin A., “Stochastic Higher Spin Six Vertex Model and Macdonald Measures”, J. Math. Phys., 59:2 (2018), 023301