Riemann-Hilbert Problems in Random Matrix Theory, Approximation Theory, and Integrable Systems

随机矩阵理论、逼近理论和可积系统中的黎曼-希尔伯特问题

• 批准号: 9970328
• 负责人: Kenneth T-R McLaughlin
• 金额: $ 7.98万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970328&HistoricalAwards=false
• 关键词: Riemann; Hilbert; Problems; Random; Matrix

Proposal: DMS-9970328Principal Investigator: Kenneth D. T.-R. McLaughlinAbstract: The proposed research concerns applications of Riemann-Hilbert problems and new techniques developed for their asymptotic analysis to classical problems in (1) random matrix theory, (2) approximation theory and orthogonal polynomials, and (3) integrable systems. In random matrix theory, McLaughlin proposes to study eigenvalue statistics in situations when the limiting density of states exhibits "atypical" behavior in order to illustrate explicitly nonuniversal behavior in local statistics of random matrices. He further proposes to study coupled families of random matrices, which appear in the study of the planar appoximation to quantum field theory as well as in graph coloring problems that arise in statistical mechanics. Other directions include new asymptotic problems pertaining to the bulk scaling limit. In approximation theory, McLaughlin proposes to continue work on support properties of equilibrium measures, to gain further understanding of the relationship between the external field V and the support of the associated equilibrium measure. In addition, an investigation of pointwise convergence of polynomials orthogonal with respect to Freud weights is proposed. The proposed research is to study the generalization of Gibbs's phenomenon to Freud weights and to obtain a rate of convergence of approximation by such polynomials. In integrable systems, McLaughlin proposes (with A. Kuijlaars) to combine knowledge from approximation theory and singular limits of integrable systems, to investigate the onset of infinite genus oscillations in the small dispersion limit of the Korteweg-de Vries equation. Another project (with S. Kamvissis and P. Miller) looks at a special class of initial data for the semi-classical limit of the focusing nonlinear Schrodinger equation.Random matrix theory has recently been the meeting ground for many areas of mathematics and physics. One reason for the subject's importance is the famous claim of Eugene Wigner who, in the 1950s, proposed that "neutron resonance levels" could be modeled by the "eigenvalues of a random matrix." Neutron resonance levels are special numerical values for the velocities of neutrons: when physicists fired neutrons at a large quantity of atoms with heavy nuclei, they observed that the neutrons were "trapped" by the bulk of atoms only for these special velocities. Wigner's conjecture asserts: the statistical distribution of the distance between these resonance levels is a universal quantity, just as the speed of light in a vacuum is, according to Einstein's theory of relativity, a universal constant. Over the years, Wigner's conjecture became a fundamental issue in mathematics as well as physics: Can this universal behavior be modeled by thinking of the resonance levels as "typical" eigenvalues of matrices chosen at random by some simple mathematical rules? For a large class of random matrix models, McLaughlin and his collaborators discovered a mathematically precise description of the statistics of eigenvalues of random matrices, thus providing a proof of Wigner's conjecture. A part of McLaughlin's current research is devoted to a deeper understanding of this fundamental universal behavior. In other directions, McLaughlin is working to develop further the techniques mentioned above, because of their application to many different areas of mathematical analysis and physics. This award is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Mathematical Physics Program in the Division of Physics.

提案：DMS-9970328主要研究者：Kenneth D. T.- R. McLaughlin摘要：本论文的主要研究内容是将Riemann-Hilbert问题及其渐近分析的新方法应用于（1）随机矩阵理论、（2）逼近理论和正交多项式、（3）可积系统中的经典问题。在随机矩阵理论中，McLaughlin提出研究极限态密度表现出“非典型”行为的情况下的特征值统计，以明确地说明随机矩阵局部统计中的非普适行为。他进一步建议研究耦合家庭的随机矩阵，这出现在研究的平面appoximation量子场论以及在图形着色问题，出现在统计力学。其他方向包括新的渐近问题有关的批量缩放限制。在近似理论中，麦克劳克林建议继续研究平衡测度的支撑性质，以进一步理解外场V与相关平衡测度的支撑之间的关系。此外，研究了关于Freud权正交多项式的逐点收敛性。建议的研究是研究吉布斯现象的推广到弗洛伊德权重，并获得这样的多项式的近似的收敛速度。在可积系统中，McLaughlin提出（与A. Kuijlaars）的方法，结合联合收割机的逼近理论和可积系统的奇异极限，研究了Korteweg-de弗里斯方程在小色散极限下的无穷亏格振荡.另一个项目（与S。Kamvissis和P.米勒）研究了一类特殊的初始数据，用于求解聚焦非线性薛定谔方程的半经典极限。这个课题如此重要的一个原因是尤金维格纳的著名论断，他在20世纪50年代提出“中子共振能级”可以用“随机矩阵的本征值”来模拟。中子共振能级是中子速度的特殊数值：当物理学家向大量重核原子发射中子时，他们观察到中子仅在这些特殊速度下被大量原子“捕获”。维格纳猜想断言：这些共振能级之间距离的统计分布是一个普适量，正如根据爱因斯坦的相对论，真空中的光速是一个普适常数。多年来，维格纳猜想成为数学和物理学中的一个基本问题：这种普遍的行为是否可以通过将共振能级视为由一些简单的数学规则随机选择的矩阵的“典型”特征值来建模？对于一大类随机矩阵模型，McLaughlin和他的合作者发现了随机矩阵特征值统计的数学精确描述，从而提供了维格纳猜想的证明。麦克劳克林目前的研究的一部分是致力于更深入地了解这一基本的普遍行为。在其他方面，麦克劳克林正在努力进一步发展上述技术，因为它们应用于数学分析和物理学的许多不同领域。该奖项由数学科学部的分析计划和物理部的数学物理计划共同资助。