论文标题
六角形和矩阵的二元定期lozenge块
Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials
论文作者
论文摘要
我们分析了一个大型常规六角形的随机润滑瓷砖模型，其基本的重量结构在水平和垂直方向上是周期性$ 2 $的周期性。这是一个确定点过程，其相关核是根据非赫米特矩阵的正交多项式表示的。该模型属于一类模型，无法应用用于研究渐近学的现有技术。我们方法的新部分包括建立矩阵值和标量值的正交多项式之间的联系。这样可以简化Duits和Kuijlaars获得的内核的双轮廓公式，从而减少Riemann-Hilbert问题的大小。证据依赖于以下事实：矩阵值重量具有生存在基础的Riemann Surface $ \ Mathcal {M} $ $ 0 $ 0 $的基础的特征值。我们认为这种独立利益的联系；只要相应的Riemann Surface $ \ Mathcal {M} $是属$ 0 $，只要相应的正交多项式可以将类似的想法用于其他矩阵的正交多项式，这是很自然的。 该方法的其余部分由两个部分组成，主要遵循Charlier，Duits，Kuijlaars和Lenells的作品的界限。首先，我们执行Deift-Zhou最陡峭的下降分析，以获得有价值的正交多项式的标量的渐近分析。主要困难是研究复杂平面中的平衡问题。其次，在双轮廓积分中取代正交多项式的渐近差异，并使用鞍点法对后者进行分析。 我们的主要结果是无序花形区域中lozenges的密度。但是，我们强调，该方法原则上可以严格计算模型中其他有意义的概率数量。
We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period $2$ in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non-Hermitian matrix valued orthogonal polynomials. This model belongs to a class of models for which the existing techniques for studying asymptotics cannot be applied. The novel part of our method consists of establishing a connection between matrix valued and scalar valued orthogonal polynomials. This allows to simplify the double contour formula for the kernel obtained by Duits and Kuijlaars by reducing the size of a Riemann-Hilbert problem. The proof relies on the fact that the matrix valued weight possesses eigenvalues that live on an underlying Riemann surface $\mathcal{M}$ of genus $0$. We consider this connection of independent interest; it is natural to expect that similar ideas can be used for other matrix valued orthogonal polynomials, as long as the corresponding Riemann surface $\mathcal{M}$ is of genus $0$. The rest of the method consists of two parts, and mainly follows the lines of a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a Deift-Zhou steepest descent analysis to obtain asymptotics for the scalar valued orthogonal polynomials. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the orthogonal polynomials are substituted in the double contour integral and the latter is analyzed using the saddle point method. Our main results are the limiting densities of the lozenges in the disordered flower-shaped region. However, we stress that the method allows in principle to rigorously compute other meaningful probabilistic quantities in the model.
