与六边形镶嵌有关的矩阵值正交多项式
Matrix-valued orthogonal polynomials related to hexagon tilings
摘要：
本文研究了一类与六边形的2-周期菱形镶嵌有关的矩阵值正交多项式。一般模型依赖于许多参数。在常数和$2$周期参数值的情况下，我们表明，MVOP可以表示在复平面的封闭轮廓上的非厄米正交的标量多项式。具有常数参数的2周期六边形镶嵌模型在大尺寸极限下存在相变。这反映在MVOP的渐近行为中，因为度趋于无穷大。与标量正交多项式的连接允许我们找到的MVOP的行列式的零点的限制行为。零点趋向于具有自相交的复平面中的曲线$\widetildeΣ_0$。MVOP的各个条目的零点表现出不同的行为，我们发现在曲线$\widetildeΣ_0$上的几何条件下右上条目的极限零点分布，我们无法证明，但这是令人信服的数值证据支持。
英文摘要：
In this paper, we study a class of matrix-valued orthogonal polynomials (MVOPs) that are related to 2-periodic lozenge tilings of a hexagon. The general model depends on many parameters. In the cases of constant and $2$-periodic parameter values we show that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane. The 2-periodic hexagon tiling model with a constant parameter has a phase transition in the large size limit. This is reflected in the asymptotic behavior of the MVOP as the degree tends to infinity. The connection with the scalar orthogonal polynomials allows us to find the limiting behavior of the zeros of the determinant of the MVOP. The zeros tend to a curve $\widetildeΣ_0$ in the complex plane that has a self-intersection. The zeros of the individual entries of the MVOP show a different behavior and we find the limiting zero distribution of the upper right entry under a geometric condition on the curve $\widetildeΣ_0$ that we were unable to prove, but that is convincingly supported by numerical evidence.
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