The Annals of Probability

A probabilistic interpretation of the Macdonald polynomials

Persi Diaconis and Arun Ram

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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 cycles of 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 $k$.

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Ann. Probab., Volume 40, Number 5 (2012), 1861-1896.

First available in Project Euclid: 8 October 2012

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Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Macdonald polynomials random permutations measures on partitions auxiliary variables Markov chain rates of convergence


Diaconis, Persi; Ram, Arun. A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40 (2012), no. 5, 1861--1896. doi:10.1214/11-AOP674.

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