论文标题
较高属riemann表面的关键措施
Critical measures on higher genus Riemann surfaces
论文作者
论文摘要
复杂平面中的临界度量是与外场的对数能量的鞍点。他们的本地和全球结构由Martinez-Finkelshtein和Rakhmanov描述。在本文中，我们启动了关于较高属的临界措施理论的发展，其中对数能量被能量相对于双极绿色的内核所取代。我们研究了双极绿色能量的最大值问题，其中DV是meromormormormormormormormormormormormormormormormormormor的差异。在合理的假设下，最大值问题具有解决方案，我们表明相应的平衡度量是外部场中的关键措施。在一种特殊的属中，我们能够证明，关键措施受到了meromoromoromormormormormormormormormormormormormormormormorthic差异差异的支持。 我们是由应用定期权重的六角形的随机润滑块的应用。这些模型中的相关性以矩阵有价值的正交多项式表达。矩阵正交性被解释为Riemann表面上的（部分）标量正交性。当我们在本文中概述时，关键措施理论将对相应的Riemann-Hilbert问题的渐近分析有用。
Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martinez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green's kernel. We study a max-min problem for the bipolar Green's energy with external fields Re V where dV is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann-Hilbert problem as we outline in the paper.
