Painlevé 函数与随机矩阵普适性理论研究

近年来，随机矩阵理论有了突破性进展，提出了Riemann-Hilbert方法和“三步法”等新理论和新方法，证明了随机矩阵普适性理论中多个重要猜想，揭示了许多新的普适性极限律。这些极限律主要通过行列式点过程表述。本课题拟系统研究几类具有普适性的行列式点过程的分析性质和渐近性质。主要研究内容包括：（1）拟研究合流超几何函数、Pearcey 函数、Painlevé 函数等核函数行列式点过程的点间隙、极值点、线性和等统计量概率分布的Painlevé 函数表示，建立Tracy-Widom型分布；（2）拟研究行列式点过程概率分布的渐近性质，包括确定Tracy-Widom型分布的尾部衰减速度、Widom-Dyson 常数等；（3）拟研究该课题中所使用的Painlevé 函数的渐近性质及其积分等整体性质。本课题研究将有助于推动随机矩阵理论、Painlevé 函数理论和渐近分析理论的深入发展。

In last twenty years, new methods termed as Riemann-Hilbert method and three steps strategy have been developed in the field of random matrix theory (RMT). Using the powerful methods, several long-standing conjectures are proved and many new universality limiting laws are discovered. The new universality laws are described by determinantal point processes. The present project is devoted to the study of the analysis and the asymptotics of several of the determinantal point processes with kernels expressed in terms of special functions, which arise in the study of universality in RMT. Particularly, the project concentrates on the following problems: (1) discovering Tracy-Widom type formulas for the gap probabilities of the determinantal point processes with confluent hypergeometric kernel, Pearcey kernel and Painleve kernels; (2) investigating the asymptotics of Tracy-Widom type distributions, including the tail and the Widom-Dyson constants; (3) studying the asymptotics and the total integral of the Painleve transcendents used in this project. The project will contribute to the development of the RMT, Painleve equations and the theory of asymptotic analysis.