复平面上一类离散旋转对称测度的正交多项式
Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane
摘要：
我们得到了多项式$p_n(λ)$，$λ\in\mathbb{C}$，关于复平面上的测度正交的强渐近性，形式为$$ e^{-N(|λ|^{2s}-tλ^s-\overline{tλ}^s)}dA(λ), $$，其中$s$是正整数，$t$是复参数，$dA$表示平面上的面积测度.这一问题的根源在于正规矩阵模型。研究了$p_n(λ)$在极限$n,N\to\infty$中的渐近行为，使得$n/N\to T$为常数.这种渐近行为有两个区别制度根据拓扑结构的限制支持的正常矩阵模型的特征值分布。如果$0<|t|^2<T/s$，特征值分布支持是复平面的单连通紧集，而对于$|t|^2>T/s$，特征值分布支持由$s$连通分量组成。相应地，正交多项式的极限零分布的支撑由包含在每个连通分量中的闭合轮廓组成。我们的渐近分析是通过减少多项式的平面正交条件的等值系统的轮廓积分正交条件。利用Deift-Zhou非线性最速下降法，从相应的Riemann-Hilbert问题得到了正交多项式的强渐近性.
英文摘要：
We obtain the strong asymptotics of polynomials $p_n(λ)$, $λ\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|λ|^{2s}-tλ^s-\overline{tλ}^s)}dA(λ), $$ where $s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for the area measure in the plane. Such problem has its origin from normal matrix models. We study the asymptotic behaviour of $p_n(λ)$ in the limit $n,N\to\infty$ in such a way that $n/N\to T$ constant. Such asymptotic behaviour has two distinguished regimes according to the topology of the limiting support of the eigenvalue distribution of the normal matrix model. If $0<|t|^2<T/s$, the eigenvalue distribution support is a simply connected compact set of the complex plane, while for $|t|^2>T/s$ the eigenvalue distribution support consists of $s$ connected components. Correspondingly the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to an equivalent system of contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann--Hilbert problem by the Deift--Zhou nonlinear steepest descent method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
