RUI: Applications of Operator Theory to Random Matrix Theory

RUI：算子理论在随机矩阵理论中的应用

• 批准号: 9970879
• 负责人: Estelle Basor
• 金额: $ 9.55万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970879&HistoricalAwards=false
• 关键词: RUI; Applications; Operator; Theory; Random

DMS-9970879ABSTRACTAsymptotic properties of determinants for different classes of operators that depend on a parameter have been important in several branches of physics and mathematics. In random matrix theory such determinants yield formulas that compute many fundamental statistical properties. For example, the distribution function for a linear statistic, (or a certain kind of random variable) in the classical Gaussian Unitary Ensemble is described by a Fredholm determinant of a convolution operator. Other ensembles of random matrices and other statistical objects give rise to more complicated operators. In the orthogonal and symplectic ensembles, matrix-valued symbols of operators naturally occur, while non-smooth random variables lead to operators with singular symbols. In all the above cases the asymptotic expansions are generalizations of the Strong Szego Limit Theorem to various classes of operators. Many physical systems possess such complicated behavior that exact predictions become impossible. Two billiard balls that are set into motion close together on an irregularly shaped billiard table may have very different paths. The energy level of a particle of a compound nucleus in a slow nuclear reaction also has complicated unpredictable behavior.Random matrix theory provides mathematical models that allow a simulation of the energy levels of the particle or the energies of the billiard balls.One of the goals of the subject is to understand the statistical behaviorof the energy levels. Here is an example. Suppose we add all the energies together. Does the distribution of the sum of the energies change depending on our initial assumptions about our models? For a large class of models, the answer is no. The distribution of the sum of energies, after appropriate normalization is bell-shaped. These results agree with experimentally acquired data and illustrate the universality of the theories.

DMS-9970879Abstractastabstractasymptastotic在不同类别的运算符的决定因素的依赖参数的特性在物理和数学的几个分支中很重要。在随机矩阵理论中，这种决定因素产生了计算许多基本统计特性的公式。例如，经典高斯单位合奏中线性统计量的分布函数（或某种随机变量）由卷积操作员的弗雷德尔姆（Fredholm）描述。随机矩阵和其他统计对象的其他集合会导致更复杂的操作员。在正交和符号合奏中，自然发生了操作员的基质值符号，而非平滑的随机变量会导致具有单数符号的运算符。在上述所有情况下，渐近扩展是强大的Szego定理对各种运营商的概括。许多物理系统具有如此复杂的行为，确切的预测变得不可能。在不规则的台球桌上靠近运动的两个台球球可能有很大的路径。缓慢的核反应中化合物核的颗粒的能级也使无法预测的行为复杂化。随机矩阵理论提供了数学模型，允许模拟台球球的粒子或能量的能量水平。该受试者目标的一个目标是了解能量水平的统计行为。这是一个例子。假设我们将所有能量加在一起。能量总和的分布会根据我们对模型的最初假设而变化？对于大型模型，答案是否定的。适当的归一化后，能量之和的分布是钟形的。这些结果与实验获得的数据一致，并说明了理论的普遍性。