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

摘要：在随机矩阵理论中，研究的主要对象是某些Fredholm行列式。这些是用来描述随机矩阵模型的水平间距分布函数的。其他随机变量也具有由Fredholm行列式给出的分布函数。本研究的主要重点是描述不同矩阵模型下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。已经开发的用于研究经典Szego极限定理的技术将被推广用于更复杂的情况。如果随机变量是某种不连续类型，则算子具有奇异符号。已知的不连续符号理论将被扩展到更一般的情况下。这将涉及到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。例如，在慢核反应中，了解复合原子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厄米矩阵的特征值来模拟的。矩阵是数字的正方形数组，特征值是与矩阵相关的一组基本数字。本研究的目的是研究不同类型的随机生成矩阵的特征值的统计性质。随机矩阵理论也出现在数学和物理的其他领域，如黎曼ζ函数的零点分布，无序导体和混沌系统的研究。

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.