Mathematical Sciences: Application of Operator Theory to Random Matrices and Random Variables

数学科学：算子理论在随机矩阵和随机变量中的应用

• 批准号: 9623278
• 负责人: Estelle Basor
• 金额: $ 5.7万
• 依托单位: California Polytechnic State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623278&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Application; Operator; Theory

ABSTRACT Proposal: DMS-9623278 PI: Basor The main objects of study in the theory of random matrices are certain Fredholm determinants. These arise to describe the level spacing distribution functions for random matrix models. Other random variables also have distribution function given by Fredholm determinants. The primary focus of this research is to describe the distribution function of random variables for different matrix models. In the classical case the operators are convolution operators, but for other models the operators are generalizations of the convolution type. The techniques already developed to study the classical Szego limit theorems will be generalized for use in the more complicated cases. If the random variable is of a certain discontinuous type, then the operators have singular symbols. The known theory for discontinuous symbols will be extended to apply in the more general setting. This will involve extensions of the Fisher-Hartwig conjecture. Many physical systems possess such complicated behavior that exact predictions become impossible. However, it is sometimes still possible to describe the average or statistical properties of the systems. For example in slow nuclear reactions it is important to understand the average properties of the energy levels of a compound nucleus. Quantum mechanics tells us that the energy levels of the system are simulated by the eigenvalues of a large Hermitian matrix. A matrix is a square array of numbers and the eigenvalues are a set of fundamental numbers associated to the matrix. The purpose of the proposed research is to study the statistical properties of the eigenvalues for different classes of randomly generated matriaes. The theory of random matrices also occurs in other areas of mathematics and physics, such as the distribution of the zeros of the Riemann zeta function, the study of disordered conductors and chaotic systems.

摘要提案：DMS-9623278 PI：Basor 随机矩阵理论的主要研究对象是某些Fredholm行列式。这些出现描述随机矩阵模型的水平间距分布函数。其他随机变量也有分布函数由Fredholm行列式给出。本研究的主要焦点是描述不同矩阵模型的随机变量的分布函数。在经典情况下，算子是卷积算子，但对于其他模型，算子是卷积类型的推广。已经开发的技术来研究经典Szego极限定理将推广使用在更复杂的情况下。如果随机变量是某种不连续类型，则算子具有奇异符号。不连续符号的已知理论将被扩展以应用于更一般的设置。这将涉及Fisher-Hartwig猜想的扩展。 许多物理系统具有如此复杂的行为，以至于精确的预测变得不可能。然而，有时仍然可以描述系统的平均或统计特性。为 例如，在慢核反应中，理解复合核能级的平均性质是很重要的。量子力学告诉我们，系统的能级是由 一个大型厄米特矩阵的特征值。矩阵是一个数字的方阵，特征值是与矩阵相关的一组基本数字。拟议研究的目的是 研究不同类型随机生成矩阵特征值的统计特性。随机矩阵理论也出现在数学和物理的其他领域，例如黎曼zeta函数的零点分布，无序导体和混沌系统的研究。