The Annals of Applied Probability

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

Yunjiang Jiang

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Abstract

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

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

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124137

Digital Object Identifier
doi:10.1214/14-AAP1031

Mathematical Reviews number (MathSciNet)
MR3325282

Zentralblatt MATH identifier
1330.60089

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

Keywords
Jack polynomials Metropolis algorithm mixing time random transposition

Citation

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. https://projecteuclid.org/euclid.aoap/1427124137


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References

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