Annals of Probability

Mixing times for random k-cycles and coalescence-fragmentation chains

Nathanaël Berestycki, Oded Schramm, and Ofer Zeitouni

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Let $\mathcal{S}_{n}$ be the permutation group on n elements, and consider a random walk on $\mathcal{S}_{n}$ whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of $\mathcal{S}_{n}$.

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Ann. Probab., Volume 39, Number 5 (2011), 1815-1843.

First available in Project Euclid: 18 October 2011

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Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J27: Continuous-time Markov processes on discrete state spaces

Mixing times coalescence cutoff phenomena random cycles random transpositions


Berestycki, Nathanaël; Schramm, Oded; Zeitouni, Ofer. Mixing times for random k -cycles and coalescence-fragmentation chains. Ann. Probab. 39 (2011), no. 5, 1815--1843. doi:10.1214/10-AOP634.

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  • [1] Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Seminar on Probability, XVII. Lecture Notes in Math. 986 243–297. Springer, Berlin.
  • [2] Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. Eur. Math. Soc., Zürich.
  • [3] Arratia, R. and Tavaré, S. (1992). The cycle structure of random permutations. Ann. Probab. 20 1567–1591.
  • [4] Barbour, A. (1990). Comments on “Poisson approximations and the Chen–Stein method,” by R. Arratia, L. Goldstein and L. Gordon. Statist. Sci. 5 425–427.
  • [5] Berestycki, N. and Durrett, R. (2006). A phase transition in the random transposition random walk. Probab. Theory Related Fields 136 203–233.
  • [6] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • [7] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [8] Durrett, R. (2004). Probability: Theory and Examples, 3rd ed. Duxbury Press, Belmont, CA.
  • [9] Flatto, L., Odlyzko, A. M. and Wales, D. B. (1985). Random shuffles and group representations. Ann. Probab. 13 154–178.
  • [10] Karoński, M. and Łuczak, T. (1997). The number of connected sparsely edged uniform hypergraphs. Discrete Math. 171 153–167.
  • [11] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [12] Lulov, N. and Pak, I. (2002). Rapidly mixing random walks and bounds on characters of the symmetric group. J. Algebraic Combin. 16 151–163.
  • [13] Lulov, N. A. M. (1996). Random Walks on the Symmetric Group Generated by Conjugacy Classes. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Harvard Univ.
  • [14] Roichman, Y. (1999). Characters of the symmetric group: Formulas, estimates, and applications. In Emerging Applications of Number Theory (D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds.). IMA Volumes on Applied Mathematics 109 525–545. Springer, New York.
  • [15] Roichman, Y. (1996). Upper bound on the characters of the symmetric groups. Invent. Math. 125 451–485.
  • [16] Roussel, S. (1999). Marches aléatoires sur e groupe symétrique. Thèse de doctorat, Toulouse.
  • [17] Roussel, S. (2000). Phénomène de cutoff pour certaines marches aléatoires sur le groupe symétrique. Colloq. Math. 86 111–135.
  • [18] Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 263–346. Springer, Berlin.
  • [19] Saloff-Coste, L. and Zúñiga, J. (2008). Refined estimates for some basic random walks on the symmetric and alternating groups. ALEA Lat. Am. J. Probab. Math. Stat. 4 359–392.
  • [20] Schramm, O. (2005). Compositions of random transpositions. Israel J. Math. 147 221–243.
  • [21] Vershik, A. M. and Kerov, S. V. (1981). Asymptotic theory of the characters of a symmetric group. Funktsional. Anal. i Prilozhen. 15 15–27 (in Russian).