The Annals of Probability

A probabilistic interpretation of the Macdonald polynomials

Persi Diaconis and Arun Ram

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Abstract

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 $k$.

Article information

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

Dates
First available in Project Euclid: 8 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1349703310

Digital Object Identifier
doi:10.1214/11-AOP674

Mathematical Reviews number (MathSciNet)
MR3025704

Zentralblatt MATH identifier
1255.05194

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

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

Citation

Diaconis, Persi; Ram, Arun. A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40 (2012), no. 5, 1861--1896. doi:10.1214/11-AOP674. https://projecteuclid.org/euclid.aop/1349703310


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