Skip to main content
SpringerLink
Log in

Asymptotic representation theory and Riemann — Hilbert problem

  • Chapter
  • First Online:
Asymptotic Combinatorics with Applications to Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1815))

Abstract

We show how the Riemann — Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete Bessel kernel are computed as examples.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Borodin, A.:Harmonic analysis on the infinite symmetric group and the Whittaker kernel.St. Petersburg Math.J.,12,no.5,(2001)

    Google Scholar 

  2. Borodin, A.:Riemann—Hilbert problem and the discrete Bessel kernel.Intern.Math.Research Notices,no.9,467–494 (2000), math/9912093

    Google Scholar 

  3. Borodin, A.:Discrete gap probabilities and discrete Painlevé equations.Preprint (2001)

    Google Scholar 

  4. Borodin, A., Deift, P.: Fredholm determinants, Jimbo—Miwa—Ueno tau-functions, and representation theory.Preprint (2001)

    Google Scholar 

  5. Borodin, A., Okounkov, A., Olshanski, G.:Asymptotics of Plancherel measures for symmetric groups.J.Amer.Math.Soc.,13, 491–515 (2000);math/9905032

    Article  MathSciNet  Google Scholar 

  6. Borodin, A., Olshanski, G.:Point processes and the infinite symmetric group. Math.Research Lett.,5,799–816 (1998);math/9810015

    MATH  MathSciNet  Google Scholar 

  7. Borodin, A., Olshanski, G.:Distributions on partitions,point processes and the hypergeometric kernel.Comm.Math.Phys.,211, no.2, 335–358 (2000);math/9904010

    Article  MATH  MathSciNet  Google Scholar 

  8. Borodin, A., Olshanski, G.: Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and β =2 random matrix ensembles.Mathematical Sciences Research Institute Publications, 40,71–94 (2001);math/9905189

    MathSciNet  Google Scholar 

  9. Borodin, A., Olshanski, G.:Harmonic analysis on the infinite-dimensional unitary group.Preprint (2001);math/0109194

    Google Scholar 

  10. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes.Springer series in statistics, Springer, (1988)

    Google Scholar 

  11. Deift, P.: Integrable operators. In:Buslaev, V., Solomyak, M., Yafaev, D.(eds) Differential operators and spectral theory:M.Sh.Birman’ s 70th anniversary collection. American Mathematical Society Translations, ser.2,189, Providence,R.I.,AMS (1999)

    Google Scholar 

  12. Dyson, F.J.: Statistical theory of the energy levels of complex systems I, II, III.J.Math.Phys.,3,140–156, 157-165, 166-175 (1962)

    Article  MathSciNet  Google Scholar 

  13. Erdelyi, A.(ed):Higher transcendental functions,Vols.1, 2. McGraw-Hill,(1953)

    Google Scholar 

  14. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.:Differential equations for quantum correlation functions. Intern.J.Mod.Phys., B4, 1003–1037 (1990)

    Article  MathSciNet  Google Scholar 

  15. Johansson, K.:Discrete orthogonal polynomial ensembles and the Plancherel measure.Ann.Math.(2),153, no.1,259–296 (2001);math/9906120

    Google Scholar 

  16. Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group.A deformation of the regular representation.Comptes Rend.Acad. Sci.Paris, Sér.I, 316, 773–778 (1993);detailed version in preparation

    MATH  MathSciNet  Google Scholar 

  17. Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd edition.Oxford University Press (1995)

    Google Scholar 

  18. Mehta, M.L.: Random matrices, 2nd edition.Academic Press, New York (1991)

    MATH  Google Scholar 

  19. Nagao, T., Wadati, M.:Correlation functions of random matrix ensembles related to classical orthogonal polynomials. J.Phys. Soc. Japan,60, no.10,3298–3322 (1991)

    Article  MathSciNet  Google Scholar 

  20. Olshanski, G.: Point processes and the infinite symmetric group.Part V:Analysis of the matrix Whittaker kernel. Preprint (1998);math/9810014

    Google Scholar 

  21. Olshanski, G. An introduction to harmonic analysis on the infinite symmetric group.In this volume

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Einstein Drive, 08540, Princeton, NJ, USA

    Alexei Borodin

Authors
  1. Alexei Borodin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Anatoly M. Vershik

  2. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Yuri Yakubovich

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Borodin, A. (2003). Asymptotic representation theory and Riemann — Hilbert problem. In: Vershik, A.M., Yakubovich, Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics(), vol 1815. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44890-X_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-44890-X_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40312-8

  • Online ISBN: 978-3-540-44890-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation