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Journal of Algebraic Combinatorics

, Volume 16, Issue 2, pp 151–163 | Cite as

Rapidly Mixing Random Walks and Bounds on Characters of the Symmetric Group

  • Nathan Lulov
  • Igor Pak
Article

Abstract

We investigate mixing of random walks on Sn and An generated by permutations of a given cycle structure. The approach follows methods developed by Diaconis, which requires certain estimates on characters of the symmetric group and uses combinatorics of Young tableaux. We conclude with conjectures and open problems.

random walks on groups symmetric group Young tableaux 

References

  1. 1.
    D. Aldous and P. Diaconis, “Strong uniform times and finite random walks,” Advances in Applied Math. 8 (1987), 69–97.MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph in preparation, 1996.Google Scholar
  3. 3.
    G. Andrews, The Theory of Partitions, Addison-Wesley, New York, 1976.MATHGoogle Scholar
  4. 4.
    P. Diaconis, Group Representations in Probability and Statistics, IMS, Hayward, California, 1988.MATHGoogle Scholar
  5. 5.
    P. Diaconis, “The cutoff phenomenon in finite Markov chains,” Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 1659–1664.MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Diaconis and M. Shahshahani, “Generating a random permutation with random transpositions,” Z. Wahr. verw. Gebiete, 57 (1981), 159–179.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Y. Dvir, Covering Properties of Permutation Groups, Lecture Notes in Math. 1112, Springer-Verlag, 1985.Google Scholar
  8. 8.
    S. Fomin and N. Lulov, “On the number of rim hook tableaux,” Zap. Nauchn. Sem. POMI 223 (1995), 219–226 (available also from http://www.math.lsa.umich.edu/~fomin/Papers/).MATHGoogle Scholar
  9. 9.
    G. James and A. Kerber, “The representation theory of the symmetric group,” in Encyclopedia of Mathematics and its Applications, Vol. 16, G.-C. Rota (Ed.), Addison-Wesley, Reading, Mass. 1981.Google Scholar
  10. 10.
    M. Liebeck and A. Shalev, “Diameters of finite simple groups: Sharp bounds and applications,” Ann. of Math. 154(2) (2001), 383–406.MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Lovász and P. Winkler, Mixing Times, AMS DIMACS Series, Vol. 41, pp. 189–204, 1998.Google Scholar
  12. 12.
    A. Lubotzky, Cayley Graphs: Eigenvalues, Expanders and Random Walks, LMS Lecture Note Ser., 218, Cambridge University Press, Cambridge, 1995.MATHGoogle Scholar
  13. 13.
    N. Lulov, “RandomWalks on the Symmetric Group Generated by Conjugacy Classes,” Ph.D. Thesis, Harvard University, 1996.Google Scholar
  14. 14.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, London, 1979.MATHGoogle Scholar
  15. 15.
    I. Pak, “RandomWalks on Permutation: Strong Uniform Time Approach,” Ph.D. Thesis, Harvard University, 1997.Google Scholar
  16. 16.
    I. Pak and V.H. Vu, “On mixing of certain random walks, cutoff phenomenon and sharp threshold of random matroid processes,” Discrete Appl. Math. 110 (2001), 251–272.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Y. Roichman, “Upper bound on characters of the symmetric groups,” Invent. Math. 125 (1996), 451–485.MathSciNetCrossRefGoogle Scholar
  18. 18.
    R.P. Stanley, “Factorization of permutations into n-cycles,” Discrete Math. 37 (1981), 255–262.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Nathan Lulov
    • 1
  • Igor Pak
    • 2
  1. 1.DIMACS CenterRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsMITCambridgeUSA

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