随机矩阵理论与Painlevé方程若干问题研究

The Riemann-Hilbert approach is a recent and significant joint piece of complex analysis and asymptotic analysis. It is desirable, as can be seen from the ongoing applications of the approach, that the correlation kernels of Painlevé type, and the associated weights with strong singularities, should be investigated. Therefore, we propose an investigation in a systematic manner of the following problems: 1. The correlation kernels and related quantities of the strongly sigular unitary ensembles in random matrix theory, using the Painlevé transcedentals and double scaling limit techniques, and applying to problems in statistic physics, such as Wigner time delay. 2. The analytic and asymptotic properties of the Painlevé equations and Painlevé transcedentals, in particular the connection formulas and the dependence of solutions on parameters. 3. Asymptotic problems of discrete Painlevé equations and relevant difference equations, the possible relation with their continuous counterparts, and the relation with discrete Riemann-Hilbert analysis. The study of the Painlevé equations and Painlevé transcedentals will play a central role in solving these problems. Also, Riemann-Hilbert analysis is the main theme of the problems and connects them all together. The proposed investigation on these interrelated problems would deepen the well understanding of the Painlevé transcedentals, and would pave the way to various further applications to fields such as random matrix theory and difference equations.

近年来Riemann-Hilbert分析已成为复分析和渐近分析的一个重要结合点。随着这一分析方法的发展和应用的深入，关于在随机矩阵理论中Painlevé型关联核和相应奇异权的研究已成为一个紧迫的问题。有鉴于此，本项目拟以RH分析为主要工具，以问题为导向探讨以下三个方面： 1. 随机矩阵理论中强奇性酉系综无穷维极限关联核及相关问题，以双尺度分析和Painlevé函数为工具，并应用于Wigner time delay等统计物理问题。2. Painlevé方程和Painlevé函数的解析性质和渐近性质，特别是相关的连接公式及对参数的依赖关系等问题。 3. 离散Painlevé方程及相关差分方程的渐近问题，与连续Painlevé方程及离散RH分析的联系。Painlevé方程的研究和应用在上述相互关联的问题中占重要地位。Riemann-Hilbert分析是贯穿其中的主线。

近年来Riemann-Hilbert分析已成为复分析和渐近分析的一个重要结合点。随着这一分析方法的发展和应用的深入，关于在随机矩阵理论中Painlevé型关联核和相应奇异权的研究已成为一个紧迫的问题。有鉴于此，本项目拟以RH分析为主要工具，以问题为导向探讨以下三个方面： 1. 随机矩阵理论中强奇性酉系综无穷维极限关联核及相关问题，以双尺度分析和Painlevé函数为工具，并应用于Wigner time delay等统计物理问题。2. Painlevé方程和Painlevé函数的解析性质和渐近性质，特别是相关的连接公式及对参数的依赖关系等问题。 3. 离散Painlevé方程及相关差分方程的渐近问题，与连续Painlevé方程及离散RH分析的联系。Painlevé方程的研究和应用在上述相互关联的问题中占重要地位。Riemann-Hilbert分析是贯穿其中的主线。

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