The Annals of Applied Probability

Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution

Yunjiang Jiang

Full-text: Open access


We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of Metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point at the identity. The proofs rely heavily on the theory of symmetric Jack polynomials, developed initially by Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1–18], Macdonald [Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv. Math. 77 (1989) 76–115]. This completes the analysis started by Diaconis and Hanlon in [Contemp. Math. 138 (1992) 99–117]. In the end we also explore other integrable Markov chains that can be obtained from symmetric function theory.

Article information

Ann. Appl. Probab., Volume 25, Number 3 (2015), 1581-1615.

First available in Project Euclid: 23 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 05E05: Symmetric functions and generalizations

Jack polynomials Metropolis algorithm mixing time random transposition


Jiang, Yunjiang. Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution. Ann. Appl. Probab. 25 (2015), no. 3, 1581--1615. doi:10.1214/14-AAP1031.

Export citation


  • [1] Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519–535.
  • [2] Bayer, D. and Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Ann. Appl. Probab. 2 294–313.
  • [3] Beerends, R. J. (1991). Chebyshev polynomials in several variables and the radial part of the Laplace–Beltrami operator. Trans. Amer. Math. Soc. 328 779–814.
  • [4] Bump, D. (2004). Lie Groups. Graduate Texts in Mathematics 225. Springer, New York.
  • [5] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • [6] Diaconis, P. and Ram, A. (2012). A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40 1861–1896.
  • [7] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [8] Dummit, D. S. and Foote, R. M. (1991). Abstract Algebra. Prentice Hall, Englewood Cliffs, NJ.
  • [9] Fulton, W. and Harris, J. (1991). Representation Theory: A First Course. Graduate Texts in Mathematics 129. Springer, New York.
  • [10] Goodman, R. and Wallach, N. R. (1998). Representations and Invariants of the Classical Groups. Encyclopedia of Mathematics and Its Applications 68. Cambridge Univ. Press, Cambridge.
  • [11] Hanlon, P. (1992). A Markov chain on the symmetric group and Jack symmetric functions. Discrete Math. 99 123–140.
  • [12] Heckman, G. J. and Opdam, E. M. (1987). Root systems and hypergeometric functions. I. Compos. Math. 64 329–352.
  • [13] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley, New York.
  • [14] Koike, K. (1993). On a conjecture of Stanley on Jack symmetric functions. Discrete Math. 115 211–216.
  • [15] Koornwinder, T. H. (1994). Special functions associated with root systems: Recent progress. In From Universal Morphisms to Megabytes: A Baayen Space Odyssey 391–404. Math. Centrum, Centrum Wisk. Inform., Amsterdam.
  • [16] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [17] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. Oxford Mathematical Monographs. Oxford Univ. Press, New York.
  • [18] Macdonald, I. G. (2000/2001). Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45 Art. B45a, 40 pp. (electronic).
  • [19] Morris, B. (2009). Improved mixing time bounds for the Thorp shuffle and $L$-reversal chain. Ann. Probab. 37 453–477.
  • [20] Stanley, R. P. (1989). Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 76–115.
  • [21] Diaconis, P. and Hanlon, P. (1992). Eigen-analysis for some examples of the Metropolis algorithm. Contemp. Math. 138 99–117.