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

TL;DR

This paper establishes precise convergence rates of a Metropolis algorithm based on random transpositions to the multivariate Ewens distribution, utilizing symmetric Jack polynomial theory to extend previous analyses.

Contribution

It provides sharp convergence rates for the Metropolis chain on the symmetric group, employing symmetric Jack polynomials, and explores related integrable Markov chains.

Findings

Proved sharp convergence rates to Ewens distribution

Extended analysis of random transposition Metropolis algorithms

Connected symmetric function theory with Markov chain convergence

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.