Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials
Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials
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楼梯造型、不对称排除过程和 Askey-Wilson 多项式
DOI：
10.1073/pnas.0909915107
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
S. Corteel;L. Williams
我们引入了一些称为阶梯表的组合对象，它们的基数为4 nn！，并将它们与非对称排斥过程（ASEP）和Askey-Wilson多项式相联系。ASEP是20世纪60年代末引入的统计力学模型，它描述了一个相互作用的粒子系统，这些粒子在具有开放边界的n个位置的一维晶格上左右跳跃。它已被引用为蛋白质合成中的交通流和翻译的模型。在最一般的形式中，粒子可以以概率α和γ从左边进入和离开，也可以以概率β和δ从右边离开和进入。在整体中，向左跳跃的概率是向右跳跃的概率的q倍。我们的第一个结果是一个公式的平稳分布的ASEP与所有参数一般，在楼梯tableaux。我们的第二个结果是一个公式的时刻（权函数）的阿斯基威尔逊多项式，也在楼梯tableaux。自20世纪80年代以来，已经有大量的工作给出了经典正交多项式（例如Hermite，Charlier，Laguerre）的矩的组合公式;在这些多项式中，Askey-Wilson多项式是最重要的，因为它们位于经典正交多项式的层次结构的顶部。
We introduce some combinatorial objects called staircase tableaux, which have cardinality 4nn !, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials.