Riemann-Hilbert 方法和随机矩阵谱分析中的 Painleve 渐近

In 1990s, a powerful new technique, termed the Riemann-Hilbert (RH) approach, to obtain asymptotics in the theory of random matrices (RMT) is developed by Deift and Zhou. In the past decade, remarkable progress has been made in RMT via the RH approach. In the ensembles with coalescing critical points, singular behavior appears, which beaks the classical universality. To investigate such singular phenomenon, one needs to develop an uniform approach to resolve the coalescing of multiple critical points in the RH method, which is also of great importance in asymptotics. . Recently, the proposer has resolved the coalescing of the soft edge with the singular point of the weight function in the RH method with certain Painleve XXXIV transcendents. The basic idea of this work maybe developed to apply to other multi-critical coalescing problems. . In this investigate, via an asymptotic study of the model of modified Jacobi unitary ensemble with coalescing critical points, we aim to obtain the Painleve generalization of the classical Bessel kernel and the multi-critical development of the Tracy-Widom formula, by developing an uniform treatment of the RH method to resolve the coalescing of critical points.. The research will certainly contribute to the development of the RH method, the spectral analysis of random matrices and the Painleve transcendents.

Riemann-Hilbert （RH）方法开创了随机矩阵谱分析的新途径，取得了很多深刻的结果。随机矩阵奇点重合系综中，经典谱分布普适性规律不再成立表现出奇异行为，引起了国内外学者的注意；利用RH方法研究奇点重合系综谱分析，需要发展奇点重合一致渐近，这是渐近分析中的重要问题。目前，申请人利用Painleve XXXIV函数很好地解决了RH 方法中软端点与权函数奇点重合问题，为课题研究提供了基础。本课题拟以奇点重合摄动 Jacobi 酉系综奇异行为研究为应用背景，以特殊函数特别是 Painleve 超越函数为主要工具，发展 RH 方法多奇点重合一致渐近，以获得随机矩阵经典 Bessel 核的 Painleve 型发展及经典 Tracy-Widom 公式的多奇点重合发展。该课题研究有助于推动RH 方法、大维随机矩阵谱分析理论及Painleve 超越函数解析理论的深入发展。

Riemann-Hilbert 方法是渐近分析领域的重要进展，开辟了无穷维随机矩阵谱分析新途径。随机矩阵理论中，奇异系综特征值统计行为不满足经典普适性律，引起国内外学者注意。本课题提出以奇点重合摄动 Jacobi 酉系综奇异行为研究为应用背景，以特殊函数特别是Painleve超越函数为主要工具，发展 Riemann-Hilbert 方法多奇点重合一致渐近，以获得随机矩阵经典Bessel核的Painleve型发展及经典 Tracy-Widom 公式的多奇点重合发展。. 做为主要结果， 我们建立了 Painleve III 方程和 Painleve V 方程解析解表示的新普适性类；得到推广的 Tracy-Widom 公式的 Painlevé III 解析解表示; 证明了关于Painlevé II 方程经典Hastings-Mcleod 解解析性质的Novokshenov 猜测。 这些结果发表于《Communication in Mathematical Physics》《Constructive Approximation》《Journal of Approximation Theory》等杂志。. 这些结果在《Physics Rieview E》, 《Int. Math. Res. Notices》,《Journal of Approximation Theory》 等杂志上被他人多次引用，特别地2015年 M. Atkin, T. Claeys 和 F. Mezzadri 推广我们的结果得到高阶Painlevé III 型普适性类。

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