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

提案：dms -9970328首席研究员：Kenneth d.t.r。摘要本文主要研究Riemann-Hilbert问题及其渐近分析新技术在(1)随机矩阵理论、(2)逼近理论与正交多项式、(3)可积系统中的应用。在随机矩阵理论中，McLaughlin提出研究状态的极限密度表现出“非典型”行为时的特征值统计，以明确地说明随机矩阵局部统计中的非普遍行为。他进一步建议研究随机矩阵的耦合族，这出现在量子场论的平面逼近研究中，以及统计力学中出现的图着色问题中。其他方向包括新的渐近问题有关的体积缩放极限。在近似理论中，McLaughlin建议继续研究平衡测度的支持性质，以进一步理解外场V与相关平衡测度的支持之间的关系。此外，还研究了关于弗洛伊德权的正交多项式的点向收敛性。提出的研究是研究吉布斯现象对弗洛伊德权值的推广，并通过这种多项式获得近似的收敛速度。在可积系统中，McLaughlin（与A. Kuijlaars）提出将近似理论的知识与可积系统的奇异极限结合起来，研究Korteweg-de Vries方程的小色散极限下无限格振荡的开始。另一个项目（与S. Kamvissis和P. Miller合作）着眼于一类特殊的初始数据，用于聚焦非线性薛定谔方程的半经典极限。随机矩阵理论最近已经成为许多数学和物理领域的交汇点。这门学科之所以重要，原因之一是尤金·维格纳（Eugene Wigner）在20世纪50年代提出的著名主张，他提出“中子共振能级”可以用“随机矩阵的特征值”来建模。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，他们观察到只有在这些特殊速度下，中子才会被大部分原子“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遍量，就像爱因斯坦的相对论认为真空中的光速是一个普遍常数一样。多年来，维格纳的猜想成为数学和物理学的一个基本问题：这种普遍行为是否可以通过将共振能级视为通过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Wigner猜想的证明。麦克劳克林目前的一部分研究致力于更深入地理解这种基本的普遍行为。在其他方向上，McLaughlin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的许多不同领域。该奖项由数学科学部的分析项目和物理部的数学物理项目共同资助。

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.