Research in Random Matrices and Integrable Systems

随机矩阵和可积系统研究

• 批准号: 9732687
• 负责人: Harold Widom
• 金额: $ 21.57万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732687&HistoricalAwards=false
• 关键词: Research; Random; Matrices; Integrable; Systems

Proposal: DMS-9732687 Principal Investigator: Harold Widom Abstract: Random matrix theory has had remarkably wide applicability: the spacing distributions arising in random matrix theory have over the past few years been shown to have deep applications in number theory; there are applications in numerical analysis and computational complexity where condition numbers of random matrices are important; random matrix theory has motivated developments in the Riemann-Hilbert method, which in turn finds applications to a variety of problems in integrable systems and inverse scattering. In physics the applications range from many-body systems (both atomic and nuclear), to quantum chaos to quantum transport in mesoscopic systems. Four areas for research are specified. The first is related to the fact that in certain random matrix ensembles the measure describing the eigenvalue distribution is the Gibbs measure for charges interacting via a potential at inverse temperature beta equal to one, two or four (corresponding to orthogonal, unitary and symplectic ensembles, respectively). The limiting spacing distributions for these ensembles are now quite well understood but the methods are applicable to these values of beta only. The question for general beta, while quite difficult, is mathematically interesting and quite important in statistical physics. A new approach looks promising and we intend to pursue it. The second area of research is the question of universality of the limiting distribution of the largest eigenvalue in matrix ensembles. This would be analogous to the universality of the Gaussian distribution for sums of independent random variables, the central limit theorem. Thirdly, we propose to study the order statistics of the spacings between eigenvalues (which is different from the spacing distributions between consecutive eigenvalues mentioned above). For example, what is the probability distribution for the largest or smallest spacing? There are known results for independent random variabl es but, apparently, none yet for for random matrices, whose eigenvalues are far from independent. Finally, we hope to complete earlier work on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on the so-called critical curves, where the asymptotics will take a very different form. The theory of Wiener-Hopf operators and operator determinants should play a decisive role in this investigation. In the 1950s Eugene Wigner, in his now classic study of highly excited states of large nuclei of atoms, introduced a mathematical model to describe the spacing between these states. This model goes under the name of random matrix theory. Since Wigner's pioneering work, it has been shown that the mathematics of random matrices has far-reaching applications to condensed matter physics, atomic physics and the new area of quantum chaos. In mathematics itself, random matrix theory has surfaced in a multitude of different contexts. It is natural to ask why the subject has such wide applicability. In probability theory the bell-shaped curve is pervasive because of a theorem which says roughly that when one adds quantities which are random and independent, the sum follows the bell-shaped curve regardless of the distribution of the quantities themselves. The distribution functions of random matrix theory appear to have a similar universality for a class of problems in which there is a high degree of dependence in the underlying processes. In the present project the mathematics of random matrix theory will be further developed with an eye toward possible applications.

建议：DMS-9732687主要研究者：Harold Widom摘要：随机矩阵理论的适用性非常广泛：在过去几年中，随机矩阵理论中引起的间距分布已被证明在数字理论中具有深刻的应用；在数值分析和计算复杂性中有应用，其中随机矩阵的条件数量很重要。随机矩阵理论激发了Riemann-Hilbert方法中的发展，这反过来又找到了在可集成系统和逆散射中的各种问题上的应用。在物理学中，应用范围从多体系统（原子和核）到量子混乱到介质系统中的量子转运。指定了四个研究领域。第一个与以下事实有关：在某些随机矩阵组合中，描述特征值分布的度量是Gibbs测量通过在反向温度β相互作用的电荷相互作用的Gibbs度量等于一个，两个或四个或四个或四个（分别对应于正交，单一和符号合奏的））。这些集合的限制间距分布现在已经充分理解了，但是这些方法仅适用于这些beta的值。一般Beta的问题虽然很困难，但在数学上很有趣，而且在统计物理学中非常重要。一种新的方法看起来很有希望，我们打算追求它。研究的第二个领域是矩阵集合中最大特征值的限制分布的普遍性问题。这将类似于高斯分布的普遍性对于独立随机变量的总和，即中心限制定理。第三，我们建议研究特征值之间间距的顺序统计（这与上述连续特征值之间的间距分布不同）。例如，最大或最小间距的概率分布是什么？独立的随机变量有已知的结果，但显然，对于随机矩阵而言，其特征值远非独立。最后，我们希望通过确定所谓的临界曲线上的渐近学来完成对周期性TODA方程解决方案的渐近问题的早期工作，在这种临界曲线上，渐近学将采取截然不同的形式。维纳 - 霍普（Wiener-Hopf）操作员和操作员决定因素的理论在这项调查中应发挥决定性作用。 在1950年代，尤金·维格纳（Eugene Wigner）在他现在对大型原子核激发态的经典研究中，引入了一种数学模型，以描述这些状态之间的间距。该模型以随机矩阵理论的名义进行。自Wigner的开创性工作以来，已经表明，随机矩阵的数学在凝结物理学，原子理和量子混乱的新领域中具有深远的应用。在数学本身中，随机矩阵理论已在多种不同的情况下浮出水面。自然要问为什么受试者具有如此广泛的适用性。在概率理论中，钟形曲线无处不在，因为一个定理大致说，当一个人添加随机和独立的数量时，总和遵循钟形曲线，而不管数量本身的分布如何。对于一类问题，随机矩阵理论的分布函数似乎具有相似的普遍性，在这些问题中，在基础过程中具有很高的依赖性。在本项目中，随机矩阵理论的数学将进一步开发，以注重可能的应用。