正交多项式在六顶点模型中的作用及其组合应用
The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
摘要：
利用正交多项式理论研究了具有畴壁边界条件的六顶点模型的配分函数和边界关联函数的Hankel行列式表示.对于模型参数的特定值，对应于交替符号矩阵（ASM）的1-、2-和3-枚举，这些多项式专门用于经典多项式（分别为连续哈恩、Meixner-Pollaczek和连续对偶哈恩）。因此，一个统一的和简化的处理ASMs枚举原来是可能的，也导致一些新的结果，如改进的3-枚举ASMs。此外，正交多项式的使用允许我们表示，对于模型的参数的通用值，（部分）非齐次模型的配分函数在齐次模型的一点边界关联函数方面。
英文摘要：
The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASMs enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.
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