Skip to main content
SpringerLink
Log in

Symmetric Functions and Random Partitions

  • Conference paper
Symmetric Functions 2001: Surveys of Developments and Perspectives

Part of the book series: NATO Science Series ((NAII,volume 74))

Abstract

These axe notes from the lectures I gave at the NATO ASI “Symmetric Functions 2001” at the Isaac Newton Institute in Cambridge (June 25 — July 6, 2001). Their goal is an informal introduction to asymptotic combinatorics related to partitions

In memory of Sergei Kerov (1946–2000)

Partial financial support by NSF grant DMS-0096246, Sloan foundation, and the Packard foundation is gratefully acknowledged

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413–432.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, Journal of AMS, 12 (1999), 1119–1178.

    MathSciNet  MATH  Google Scholar 

  3. J. Baik, P. Deift, K. Johansson, 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers I. Asymptotic methods and perturbation theory, Reprint of the 1978 original. Springer-Verlag, New York, 1999.

    Google Scholar 

  5. A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of the Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481–515.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Borodin and G. Olshanski, Distribution on partitions, point processes, and the hypergeometric kernel, Comm. Math. Phys. 211 (2000), no. 2, 335–358.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Britz and S. Fomin, Finite posets and Ferrers shapes, Adv. Math. 158 (2001), no. 1, 86–127.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Bressoud, Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999

    Book  MATH  Google Scholar 

  9. R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impendances, Ann. Prob. 21 (1993), 1329–1371.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Comm. Math. Phys. 222 (2001), no. 1, 147–179.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), no. 1, 117–166.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), no. 2, 297–346.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Deift, Integrable systems and combinatorial theory. Notices Amer. Math. Soc. 47 (2000), no. 6, 631–640.

    MathSciNet  MATH  Google Scholar 

  14. P. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York, 1999.

    Google Scholar 

  15. J. Gravner, C. A. Tracy, and H. Widom Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Statist. Phys. 102 (2001), no. 5-6, 1085–1132.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Johansson, Discrete orthogonal polynomials and the Plancherel measure, Ann. of Math. (2)153 (2001), no. 1, 259–296.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Johansson, Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Comm. Math. Phys. 215 (2001), no. 3, 683–705.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Johansson, Non-intersecting paths, random tilings, and random matrices, math.PR/0011250.

    Google Scholar 

  20. V. Kac, Infinite dimensional Lie algebras, Cambridge University Press.

    Google Scholar 

  21. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré, Prob. et Stat. 33 (1997), 591–618.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Kenyon, The planar dimer model with a boundary: a survey, Directions in mathematical quasicrystals, 307–328, CRM Monogr. Ser., 13, Amer. Math. Soc, Providence, RI, 2000.

    Google Scholar 

  23. B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math., 26, 1977, 206–222.

    Article  MathSciNet  MATH  Google Scholar 

  24. I. G. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, 1995.

    Google Scholar 

  25. M. L. Mehta, Random matrices, Second edition. Academic Press, Inc., Boston, MA, 1991.

    MATH  Google Scholar 

  26. F. W. J. Olver, Asymptotics and special functions, Reprint of the 1974 original, AKP Classics. A K Peters, Ltd., Wellesley, MA, 1997.

    MATH  Google Scholar 

  27. A. Okounkov, Random matrices and random permutations, IMRN, 20, 2000, 1043–1095.

    Article  MathSciNet  Google Scholar 

  28. A. Okounkov, Infinite wedge and random partitions, Selecta Math., New Ser., 7 (2001), 1–25.

    Article  MathSciNet  Google Scholar 

  29. A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random S-dimensional Young diagram, math.CO/0107056.

    Google Scholar 

  30. M. Praehofer and H. Spohn, Scale Invariance of the PNG Droplet and the Airy Process, math.PR/0105240.

    Google Scholar 

  31. B. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001.

    MATH  Google Scholar 

  32. C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math., 13, 1961, 179–191.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207 (1999), no. 3, 697–733.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. P. Stanley, Enumerative combinatorics, Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  35. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys., 159, 1994, 151–174.

    Article  MathSciNet  MATH  Google Scholar 

  36. C. A. Tracy and H. Widom, Random unitary matrices, permutations and Painlevé, Comm. Math. Phys. 207 (1999), no. 3, 665–685.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Vershik, talk at the 1997 conference on Formal Power Series and Algebraic Combinatorics, Vienna.

    Google Scholar 

  38. A. Vershik and S. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Soviet Math. Dokl., 18, 1977, 527–531.

    MATH  Google Scholar 

  39. A. Vershik and S. Kerov, Asymptotics of the maximal and typical dimension of irreducible representations of symmetric group, Func. Anal. Appl., 19, 1985, no.1.

    Google Scholar 

  40. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1944.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. U.C. Berkeley, USA

    Andrei Okounkov

Authors
  1. Andrei Okounkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Sergey Fomin

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media Dordrecht

About this paper

Cite this paper

Okounkov, A. (2002). Symmetric Functions and Random Partitions. In: Fomin, S. (eds) Symmetric Functions 2001: Surveys of Developments and Perspectives. NATO Science Series, vol 74. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0524-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-0524-1_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0774-3

  • Online ISBN: 978-94-010-0524-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation