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September 2012 A probabilistic interpretation of the Macdonald polynomials
Persi Diaconis, Arun Ram
Ann. Probab. 40(5): 1861-1896 (September 2012). DOI: 10.1214/11-AOP674


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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Persi Diaconis. Arun Ram. "A probabilistic interpretation of the Macdonald polynomials." Ann. Probab. 40 (5) 1861 - 1896, September 2012.


Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1255.05194
MathSciNet: MR3025704
Digital Object Identifier: 10.1214/11-AOP674

Primary: 05E05
Secondary: 60J10

Keywords: auxiliary variables , Macdonald polynomials , Markov chain , measures on partitions , Random permutations , rates of convergence

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 5 • September 2012
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