The Random \((n-k)\)-Cycle to Transpositions Walk on the Symmetric Group

  • Alperen Y. Özdemir


We study the rate of convergence of the Markov chain on \(S_n\) which starts with a random \((n-k)\)-cycle for a fixed \(k \ge 1\), followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after \(cn + \frac{\ln k}{2}n\) steps for \(c>0\), the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the \((n-1)\)-cycle case. The upper bound relies on estimates for the difference of normalized characters.


Markov chain Convergence rate Symmetric group Defining representation Asymptotic distribution Murnaghan–Nakayama Rule 

Mathematics Subject Classification (2010)




The author would like to thank Jason Fulman for suggesting the problem and his most valuable comments.


  1. 1.
    Aigner, M.: A Course in Enumeration. Springer, Berlin (2007)zbMATHGoogle Scholar
  2. 2.
    Behrends, E.: Introduction to Markov Chains (with Special Emphasis on Rapid Mixing). Vieweg Verlag, Braunschweig/Wiesbaden (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berestycki, N., Schramm, O., Zeitouni, O.: Mixing times for random k-cycles and coalescence-fragmentation chains. Ann. Probab. 39(5), 1815–1843 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bernstein, M.: Likelihood orders for the \(p\)-cycle walks on the symmetric group. Electron. J. Comb. 25(1), 1–25 (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Billingsley, P.: Probability and Measure, 2nd edn, p. 406. Wiley, New York (1986)zbMATHGoogle Scholar
  6. 6.
    Bormashenko, O.: A coupling argument for the random transposition walk. arXiv preprint arXiv:1109.3915 (2011)
  7. 7.
    Diaconis, P.: Group Representations in Probability and Statistics. Institute of Mathematical Sciences, Lecture Notes-Monograph Series 11, Hayward, CA (1988)Google Scholar
  8. 8.
    Diaconis, P., Greene, C.: Applications of Murphy’s elements. Stanford University Technical Reports No. 335, pp. 1–22 (1989)Google Scholar
  9. 9.
    Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, S.: A Random Walk in Representations. Ph.D. dissertation, University of Pennslyvania (2014)Google Scholar
  11. 11.
    Goupil, A., Chauve, C.: Combinatorial operators for Kronecker powers of representations of \(S_n\). Sémin. Lothar. Comb. 54, B54j (2006)zbMATHGoogle Scholar
  12. 12.
    James, G.D.: The Representation Theory of the Symmetric Groups, p. 8. Springer, Berlin (1978)Google Scholar
  13. 13.
    Kuba, M., Panholzer, A.: On moment sequences and mixed poisson distribution. Probab. Surv. 13, 89–155 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Time. AMS, Providence (2009)zbMATHGoogle Scholar
  15. 15.
    Lulov, N., Pak, I.: Rapidly mixing random walks and bounds on characters of the symmetric group. J. Algebra. Comb. 16(2), 151–163 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, New York (1979)zbMATHGoogle Scholar
  17. 17.
    Sagan, B.: The Symmetric Group. Brooks/Cole Publishing Co., Belmont (1991)zbMATHGoogle Scholar
  18. 18.
    Saloff-Coste, L.: Random walks on finite groups. In: Probability on Discrete Structures, Encyclopaedia Math. Sci. 110, 263–346 (2004)Google Scholar
  19. 19.
    Takács, L.: The problem of coincidences. Arch. Hist. Exact Sci. 21(3), 229–244 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wilf, H.S.: Generatingfunctionology. Academic, New York (1990)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations