A probabilistic interpretation of the Macdonald polynomials
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 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.
Cite
@article{arxiv.1007.4779,
title = {A probabilistic interpretation of the Macdonald polynomials},
author = {Persi Diaconis and Arun Ram},
journal= {arXiv preprint arXiv:1007.4779},
year = {2010}
}