Universality in random matrices and integrable systems: asymptotic analysis via Riemann-Hilbert and d-bar methods

随机矩阵和可积系统的普遍性：通过 Riemann-Hilbert 和 d-bar 方法进行渐近分析

• 批准号: 0800979
• 负责人: Kenneth T-R McLaughlin
• 金额: $ 43.07万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-04-15 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800979&HistoricalAwards=false
• 关键词: Universality; random; matrices; integrable; systems

McLaughlin's research concerns the development of new methods for the asymptotic analysis of Riemann-Hilbert problems and d-bar problems, and to apply the full body of such techniques to cutting edge (as well as classical) problems in a range of fields including random matrix theory, nonlinear partial differential equations, orthogonal polynomials, and asymptotic combinatorics. One driving feature is the exploration of universal behavior in these areas. McLaughlin's previous work in nonlinear partial differential equations has concerned the study of violent oscillations in somewhat specialized ``integrable'' nonlinear partial differential equations. Very recently, conjectures have emerged, analogous to the famous universality conjectures in random matrix theory, which posit that the microscopic generation of these oscillations is universal over entire families of nonlinear partial differential equations. In random matrix theory, universality refers to the phenomenon that eigenvalue statistics of random matrices tend, as the size of the matrices grow to infinity, to behave in a universal way, independent of the exact form of the probabilistic laws used to generate the random matrices. McLaughlin proposes to extend the current realm of universality substantially, in nonlinear partial differential equations and random matrix theory, and to probe the limits of universality: what conditions on V are sufficient to see a deviation from the typical universal behavior? A variety of research projects are also proposed whose aim is the training of graduate students.The understanding and eventual control of complicated phenomena is a primary goal of scientific research. Through this fundamental goal we enhance our ability for technological advancement. Physical models for complex nonlinear phenomena often boil down to the study of partial differential equations in parameter regimes where their solutions exhibit singularly wild behavior. In other instances, statistical theories with great amount of randomness are developed to understand complex phenomena. ``Universality'' refers to robustness of certain phenomena, and to the counter-intuitive prevalence of the same phenomena across a wide array of different physical situations and models. Two examples: (1) Waves in the ocean can organize themselves into "trains" transporting energy, and analogous trains are also observed in laser beams propagating in optical fibers. (2) Statistical fluctuations of nuclear resonance levels measured in the 1950s led to a new type of universality, and the same statistical behavior has been observed in a wide variety of situations modeled with randomness, as far-flung as the statistics of spacings between parked cars! Scientists' ability to predict dramatic behavior through the analysis of such general nonlinear partial differential equations, or statistical theories, is limited. However, there is a class of canonical models for a wide variety of physical settings. Their singular behavior is a guide for the understanding of some complicated phenomena in nature. Some of these are partial differential equations, others are statistical models, but the unifying feature of these models is that researchers are making great progress in their analysis. McLaughlin's research involves the detailed rigorous analysis of these models; he (with collaborators) is developing methods to understand, predict, and control their behavior. The broader impacts of this research program stem from the emergence of universality as a new paradigm in nonlinear science: probing its range of applicability is fundamental in emerging areas as well as established ones. Another very important aspect of the proposed research is the continued training of graduate students in these rapidly developing areas.

McLaughlin的研究涉及Riemann-Hilbert问题和d-bar问题渐近分析的新方法的发展，并将这些技术的整体应用于包括随机矩阵理论，非线性偏微分方程，正交多项式和渐近组合在内的一系列领域的前沿（以及经典）问题。一个驱动特性是探索这些领域的普遍行为。McLaughlin先前在非线性偏微分方程方面的工作涉及在某种程度上专门的“可积”非线性偏微分方程中对剧烈振荡的研究。最近，出现了一些猜想，类似于随机矩阵理论中著名的普适性猜想，它假设这些振荡的微观产生在整个非线性偏微分方程族中是普适性的。在随机矩阵理论中，普适性指的是随机矩阵的特征值统计随着矩阵的大小增长到无穷大，倾向于以一种普遍的方式表现，而与用于生成随机矩阵的概率律的确切形式无关。McLaughlin建议在非线性偏微分方程和随机矩阵理论中大幅扩展当前的普适性领域，并探索普适性的极限：V上的什么条件足以看到偏离典型的普适性行为？本文还提出了以培养研究生为目的的各种研究项目。理解并最终控制复杂现象是科学研究的首要目标。通过这一根本目标，我们增强了技术进步的能力。复杂非线性现象的物理模型通常归结为对参数体系中的偏微分方程的研究，在这些体系中，偏微分方程的解表现出奇异的野性行为。在其他情况下，发展具有大量随机性的统计理论是为了理解复杂的现象。“普适性”指的是某些现象的稳健性，以及在一系列不同的物理情况和模型中相同现象的反直觉普遍性。两个例子：(1)海洋中的波浪可以组织成传递能量的“火车”，在光纤中传播的激光束中也可以观察到类似的火车。(2) 20世纪50年代测量的核共振水平的统计波动导致了一种新型的普遍性，并且在各种随机建模的情况下都观察到相同的统计行为，远到停车间隔的统计！科学家通过分析这种一般的非线性偏微分方程或统计理论来预测戏剧性行为的能力是有限的。然而，有一类规范模型适用于各种各样的物理环境。它们独特的行为是理解自然界一些复杂现象的指南。其中一些是偏微分方程，另一些是统计模型，但这些模型的统一特点是研究人员在分析方面取得了很大进展。McLaughlin的研究涉及对这些模型的详细严格的分析；他（与合作者）正在开发理解、预测和控制它们行为的方法。本研究项目的广泛影响源于普遍性作为非线性科学的新范式的出现：探索其适用范围在新兴领域和已建立的领域都是至关重要的。拟议研究的另一个非常重要的方面是在这些快速发展的领域继续培养研究生。