奇异幺正系综Riemann-Hilbert分析

Riemann-Hilbert method is a powerful new technique to derive asymptotics in the theory of Random matrices. In certain critical ensembles, the classical universality breaks and some non-classical universality classes appear. In the grant by the proposer(2013-2015), some critical ensembles associated with real weights were studied and new type of universal classes defined by real analytic solutions to Painlevé III and Painlevé V equations were obtained; [Xu,e.t.c., Comm. Math.Phys., 332(2014)].. In this proposal, the statistics of the eigenvalues of the critical ensembles with complex weights are to be studies. By exploring the analytic properties of the Painlevé transcendent and developing uniform treatment of the Riemann-Hilbert method with coalescing of singularities, we study the universality of critical ensemble with coalescing of singularities and the Tracy-Widom formulas for the new universality classes involved Painlevé functions and the applications to describe the phase transitions in the Wigner time-delay model and the Spin glass model.

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。申请人青年基金项目（2013年-2015年），给出实权奇异幺正系综特征值Riemann-Hilbert分析，得到Painlevé III 和Painlevé V 方程实值解析解描述的新普适性类；[徐帅侠等, Comm. Math. Phys., 332(2014)] 。. 本课题拟进一步发展 Painlevé 超越函数复平面解析理论和 Riemann-Hilbert 方法多奇点耦合一致渐近，进而研究复权奇异幺正系综非经典型普适性的区域解析 Painlevé 函数表示及其Tracy-Widom 型公式等公开问题，给出 Wigner 时滞模型和Spin glass 模型相变的Painlevé函数表示。

普适性是随机矩阵理论的基本性质。本项目研究了几类奇异幺正系综的特征值的极限分布。 揭示了多个Painleve 函数解析解表示的新普适性类，推广了经典的普适性。建立了第二类和第三类耦合Painleve方程解析解表示的Tracy-Widom 型分布，给出了分布的尾部衰减渐近展开, 解决了展开中的常数估计问题。. 这些结果发表于 Communications in Mathematical Physics、SIAM Journal on Mathematical Analysis、Annales Henri Poincare 等国际期刊。

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