A Markov chain on the symmetric group and Jack symmetric functions

Diaconis and Shahshahani studied a Markov chain W[function of (italic small f)](1) whose states are the elements of the symmetric group S[function of (italic small f)]. In W[function of (italic small f)](1), you move from a permutation [pi] to any permutation of the form [pi](i, j) with equal probability. In this paper we study a deformation W[function of (italic small f)]([alpha]) of this Markov chain which is obtained by applying the Metropolis algorithm to W[function of (italic small f)](1). The stable distribution of W[function of (italic small f)]([alpha]) is [alpha][function of (italic small f)]-c([pi]) where c([pi]) denotes the number of cycles of [pi]. Our main result is that the eigenvectors of the transition matrix of W[function of (italic small f)]([alpha]) are the Jack symmetric functions. We use facts about the Jack symmetric functions due to Macdonald and Stanley to obtain precise estimates for the rate of convergence of W[function of (italic small f)]([alpha]) to its stable distribution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30109/1/0000481.pd

Stein's Method, Jack Measure, and the Metropolis Algorithm

The one parameter family of Jack(alpha) measures on partitions is an important discrete analog of Dyson's beta ensembles of random matrix theory. Except for special values of alpha=1/2,1,2 which have group theoretic interpretations, the Jack(alpha) measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we recover a known central limit theorem on the distribution of character ratios of random representations of the symmetric group on transpositions. The case alpha=2 gives a new central limit theorem for random spherical functions of a Gelfand pair. The proof uses Stein's method and has interesting ingredients: an intruiging construction of an exchangeable pair, properties of Jack polynomials, and work of Hanlon relating Jack polynomials to the Metropolis algorithm.Comment: very minor revisions; fix a few misprints and update bibliograph

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

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.Comment: Published at http://dx.doi.org/10.1214/14-AAP1031 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A probabilistic interpretation of the Macdonald polynomials

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $\beta$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of this work, with significantly fewer results and a different title, is available as arXiv:1201.180