A probabilistic interpretation of the Macdonald polynomials
A probabilistic interpretation of the Macdonald polynomials
复制标题
麦克唐纳多项式的概率解释
DOI：
10.1214/11-aop674
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发表时间：
2010
影响因子：
2.3
通讯作者：
Arun Ram
中科院分区：
文献类型：
--
作者：
P. Diaconis;Arun Ram
双参数麦克唐纳多项式是代数组合学和表示理论的中心对象。我们给出了一个带有特征函数（麦克唐纳多项式的系数展开为幂和多项式时的特征函数）的k分区上的马尔可夫链。马尔可夫链具有平稳分布，一个新的双参数测度族的分区，逆麦克唐纳权（重标）。排列环上的均匀分布和eowens抽样公式是特例。马尔可夫链是统计物理中辅助变量算法的一个版本。麦克唐纳多项式的性质允许对运行时间进行尖锐的分析。在自然情况下，对于任意大的k，有限的步数就足够了。
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.