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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Mathematical Reviews number (MathSciNet)

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. http://projecteuclid.org/euclid.ejp/1463434879.

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