A PROBABILISTIC INTERPRETATION OF THE MACDONALD POLYNOMIALS

• 成果类型: Article
• 署名作者: Diaconis, Persi; Ram, Arun
• 署名单位: Stanford University; Stanford University; University of Melbourne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP674
• 发表日期: 2012
• 页码: 1861-1896
• 关键词: combinatorial formula; symmetrical group; asymptotics

摘要

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.