Charlier Christophe
Department of Mathematics Royal Institute of Technology (KTH) Stockholm Sweden.
Stud Appl Math. 2021 Jan;146(1):3-80. doi: 10.1111/sapm.12339. Epub 2020 Oct 2.
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 (OPs). 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 OPs. 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 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 OPs, as long as the corresponding Riemann surface 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 OPs. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the OPs 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.
我们分析了一个大的正六边形的随机菱形平铺模型，其基础权重结构在水平和垂直方向上均具有周期为2的周期性。这是一个行列式点过程，其相关核由非厄米矩阵值正交多项式（OPs）表示。该模型属于一类现有渐近性研究技术无法应用的模型。我们方法的新颖之处在于建立了矩阵值OPs和标量值OPs之间的联系。这通过减小黎曼-希尔伯特问题的规模，简化了Duits和Kuijlaars得到的核的双轮廓公式。证明依赖于矩阵值权重具有位于亏格为0的基础黎曼曲面上的特征值这一事实。我们认为这种联系具有独立的研究价值；可以预期，只要相应的黎曼曲面亏格为0，类似的想法可用于其他矩阵值OPs。该方法的其余部分由两部分组成，主要遵循Charlier、Duits、Kuijlaars和Lenells先前工作的思路。首先，我们进行Deift-Zhou最速下降分析以获得标量值OPs的渐近性。主要困难在于研究复平面中的一个平衡问题。其次，将OPs的渐近性代入双轮廓积分，并使用鞍点方法对后者进行分析。我们的主要结果是无序花形区域中菱形的极限密度。然而，我们强调该方法原则上允许严格计算模型中的其他有意义的概率量。
