Combinatorics of the two-species ASEP and Koornwinder moments
Combinatorics of the two-species ASEP and Koornwinder moments
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两种 ASEP 和 Koornwinder 矩的组合
DOI：
10.1016/j.aim.2017.09.034
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发表时间：
2015
影响因子：
1.7
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
S. Corteel;Olya Mandelshtam;L. Williams
在以前的工作[12]，[13]，[14]中，第一和第三作者引入了阶梯表，他们用它来给出非对称简单排斥过程（ASEP）的平稳分布和Askey-Wilson权函数的矩的组合公式. ASEP和Askey-Wilson矩完全相关的事实是出乎意料的，这是由于[45]。ASEP是粒子在具有开放边界的N个位点的一维晶格上跳跃的模型;粒子可以在左边界和右边界进入和退出。它在1970年左右被引入[34]，[43]并被引用为蛋白质合成中的交通流和翻译的模型。同时，Askey-Wilson多项式是一族单变量正交多项式，它位于经典正交多项式的层次的顶部。因此，从以前的工作，我们有关系ASEP−−楼梯tableaux−− Askey-Wilson时刻。Askey-Wilson多项式可以被视为多元Koornwinder多项式的一元情况，多元Koornwinder多项式也称为附着在非约化仿射根系（C n ‡，C n）上的麦克唐纳多项式。这是自然的，然后问一个是否可以推广之间的关系ASEP，阿斯基威尔逊时刻，和楼梯tableaux，在这样一种方式，Koornwinder时刻取代阿斯基威尔逊时刻。在[15]中，我们在Koornwinder矩和两种ASEP之间建立了精确的联系，ASEP是ASEP的推广，其中有两种具有不同“权重”的粒子。在这篇文章中，我们介绍了菱形楼梯tableaux，并表明，我们有关系2-物种ASEP−−菱形楼梯tableaux−− Koornwinder时刻。特别是，我们给出的固定分布的两种ASEP和Koornwinder时刻的公式，在菱形楼梯tableaux。下载：下载高分辨率图像（75 KB）下载：下载全尺寸图像
Abstract In previous work [12],[13],[14], the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey–Wilson weight function. The fact that the ASEP and Askey–Wilson moments are related at all is unexpected, and is due to [45]. The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34],[43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey–Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP−− staircase tableaux−− Askey–Wilson moments. The Askey–Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system (C n∨, C n). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey–Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey–Wilson moments. In [15], we made a precise link between Koornwinder moments and the two-species ASEP, a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux, and show that we have the relationship 2-species ASEP−− rhombic staircase tableaux−− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux. Download: Download high-res image (75KB) Download: Download full-size image