Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials
Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials
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S 曲线的确定及其在非厄米正交多项式理论中的应用
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
E. Medina
中科院分区：
文献类型：
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作者：
G. Álvarez;L. Alonso;E. Medina
本文讨论了复平面上沿合适路径的非厄米正交多项式中关于指数权值的s曲线的确定问题。已知相应的复平衡势可以写成合适的黎曼曲面上的阿贝尔积分的组合，其分支点可以作为问题的主要参数。这些分支点的方程可以用阿贝尔微分的周期来表示，并且有几种等价形式。我们选择其中的一种形式，用解析和数值相结合的方法研究了正交多项式的渐近零密度和随机矩阵模型的渐近特征值密度的相结构。作为一种应用，我们给出了标准三次模型的阶段和关键过程的完整描述。
This paper deals with the determination of the S-curves in the theory of non-Hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic and numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.