Electronic Journal of Probability

Random Walks on Trees and Matchings

Persi Diaconis and Susan Holmes

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We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on $2n$ vertices. Roughly, the results show that $(1/2) n \log n$ steps are necessary and suffice to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between trees and matchings.

Article information

Electron. J. Probab., Volume 7 (2002), paper no. 6, 17 pp.

Accepted: 2 January 2002
First available in Project Euclid: 16 May 2016

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62F10: Point estimation 62F15: Bayesian inference 65C05: Monte Carlo methods 82C80: Numerical methods (Monte Carlo, series resummation, etc.)

Markov Chain Matchings Phylogenetic Tree Fourier analysis Zonal polynomials Coagulation-Fragmentation

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Diaconis, Persi; Holmes, Susan. Random Walks on Trees and Matchings. Electron. J. Probab. 7 (2002), paper no. 6, 17 pp. doi:10.1214/EJP.v7-105. https://projecteuclid.org/euclid.ejp/1463434879

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  • Aldous, D. J. (1996), Probability Distributions on Cladograms in Random discrete structures (Aldous and Pemantle, eds.) Number 76 in IMA Series. Springer Verlag, NY, 1-18.
  • Aldous, D. J. (1998), Stochastic coalescence In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), number Extra Vol. III, 205-211.
  • Aldous, D. J.(1999), Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3-48.
  • Aldous, D. J. (2000), Mixing time for a Markov chain on cladograms.Combin. Probab. Computing 9, 191-204.
  • Aldous, D. J. (2001), Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today. Statist. Sci. 16, 23-34.
  • Aldous, D. J. and Fill, J. (2002), Reversible Markov chains and random walks on graphs.
  • Barbour, A. D., Holst, L. and Janson, S. (1992), Poisson approximation The Clarendon Press Oxford University Press, New York. Oxford Science Publications.
  • Diaconis, P. (1988), Group Representations in Probability and Statistics Institute of Mathematical Statistics, Hayward, Calif.
  • Diaconis, P. and Hanlon, P. (1992), Eigen-analysis for some examples of the Metropolis algorithm. In Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Amer. Math. Soc., Providence, RI. 99-117.
  • Diaconis, P. and Ram, A. (2000), Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. Journ. 4, 157-190.
  • Diaconis, P. and Saloff-Coste, L. (1993a), Comparison techniques for random walk on finite groups. Ann. Probab. 21, 2131-2156.
  • Diaconis, P. and Saloff-Coste, L.(1993b), Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3, 696-730.
  • Diaconis, P. and Shahshahani, M. (1981), Generating a random permutation with random transpositions. Z. W. 57, 159-179.
  • Diaconis, P. and Shahshahani, M. (1987), Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18, 208-218.
  • Diaconis, P. W.and Holmes, S. P. (1998). Matchings and phylogenetic trees. Proc. Natl. Acad. Sci. USA 95, 14600-14602 (electronic).
  • Durrett, R., Granovsky, B. L., and Gueron, S. (1999). The equilibrium behavior of reversible coagulation-fragmentation processes. J. Theoret. Probab. 12, 447-474.
  • Ewens, W. J. (1972), The sampling theory of selectively neutral alleles. Theoretical Population Biology 3, 87-112.
  • Hammersley, J. M. and Handscomb, D. C. (1964), Monte Carlo Methods. Chapman and Hall, London.
  • Holmes, S. (1999), Phylogenies: An overview. In Halloran, E. and Geisser, S., editors, Statistics and Genetics, number 81 in IMA. Springer Verlag, NY. Preprint
  • Inglis, N. F. J., Richardson, R. W., and Saxl, J. (1990), An explicit model for the complex representations of S_n. Arch. Math. (Basel) 54, 258-259.
  • Ingram, R. E. (1950), Some characters of the symmetric group. Proc. Amer. Math. Soc. 1, 358-369.
  • James, G. and Kerber, A. (1981), The representation theory of the symmetric group. Addison-Wesley, Reading, MA.
  • James, A. T. (1968), Calculation of zonal polynomial coefficients by use of the Laplace-Beltrami operator. Ann. Math. Statist 39, 1711-1718.
  • James, A. T.(1982). Analysis of variance determined by symmetry and combinatorial properties of zonal polynomials. In Statistics and probability: essays in honor of C. R. Rao, North-Holland, Amsterdam, 329-341.
  • Jerrum, M. and Sinclair, A. (1989), Approximating the permanent. SIAM Journal on Computing 18, 1149-1178.
  • Jerrum, M., Sinclair, A., and Vigoda, E. (2000), A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. Electronic Colloquium on Computational Complexity (ECCC).
  • Lovász, L. and Plummer, M. D. (1985), Matching Theory. North Holland, Amsterdam.
  • Macdonald, I. G. (1995), Symmetric functions and Hall polynomials. The Clarendon Press Oxford University Press, New York, second edition. With contributions by A. Zelevinsky, Oxford Science Publications.
  • Mayer-Wolfe, E., Zeitouni, O., and Zerner, M. (2001), Asymptotics of certain coagulation-fragmentation processes and invariant Poisson-Dirichlet measures. Technical report, Dept. of Mathematics, Stanford University, Stanford, CA94305.
  • Page, R. and Holmes, E. (2000), Molecular Evolution, A Phylogenetic Approach. Blackwell Science.
  • Pulleyblank, W. (1995), Matchings and extensions. In Graham, R. L., M. Grötschel and L. Lovász editors, Handbook of combinatorics. Vol. 1, 2, 179-232, Amsterdam. Elsevier Science B.V.
  • Roussel, S. (2000). Phénomène de cutoff pour certaines marches aléatoires sur le groupe symetrique. Colloq. Math. 86, 111-135.
  • Sagan, B. E. (2001), The symmetric group. Springer-Verlag, New York, second edition. Representations, combinatorial algorithms, and symmetric functions.
  • Saloff-Coste, L. (1997), Lectures on finite Markov chains. In Lectures on probability theory and statistics, pages 301-413, Berlin. Lecture Notes in Math., 1665, Springer-Verlag. Lectures from the 26th Summer School on Probability Theory held in Saint-Flour, Edited by P. Bernard.
  • Saxl, J. (1981), On multiplicity-free permutation representations. In Finite geometries and designs (Proc. Conf., Chelwood Gate, 1980), pages 337-353. Cambridge Univ. Press, Cambridge.
  • Scarabotti, F. (1997), Time to reach stationarity in the Bernoulli-Laplace diffusion model with many urns. Adv. Math. 18, 351-371.
  • Schröder, E. (1870), Veir combinatorische probleme. Zeit. für. Math. Phys. 15, 361-376.
  • Schweinsberg, J. (2001), An O(n2) bound for the relaxation time of a Markov chain on cladograms. Technical Report 572, Dept Statistics, UC Berkeley. Preprint
  • Serre, J.-P. (1977), Linear representations of finite groups. Springer-Verlag, New York. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42.
  • Wachs, M. (2001), Topology of matching, homology of matching, combinatorial Laplacian of the matching complex. Technical report, preprint.