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 ${\tilde{\Sigma }}_0$ 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 ${\tilde{\Sigma }}_0$ that we were unable to prove, but which is convincingly supported by numerical evidence.

在本文中，我们研究了一类与六边形的 2 周期菱形平铺相关的矩阵值正交多项式 (MVOP)。一般模型取决于许多参数。在常数和 2 周期参数值的情况下，我们表明 MVOP 可以表示为在复平面的闭合轮廓上具有非厄米正交性的标量多项式。

具有恒定参数的 2-周期六边形平铺模型在大尺寸极限内具有相变。这反映在 MVOP 的渐近行为中，因为度数趋于无穷大。与标量正交多项式的联系使我们能够找到 MVOP 行列式零点的极限行为。零点趋向于曲线${\stackrel{～}{\Sigma }}_0$ 在具有自相交的复平面中。

MVOP 的各个条目的零点表现出不同的行为，我们在曲线上的几何条件下找到了右上角条目的极限零分布 ${\stackrel{～}{\Sigma }}_0$ 我们无法证明，但有令人信服的数字证据支持。