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.

在物理学和数学的几个分支中，依赖于参数的不同类算子的行列式的渐近性质一直很重要。在随机矩阵理论中，这些行列式产生计算许多基本统计性质的公式。例如，在经典的高斯酉包络中，线性统计量（或某种随机变量）的分布函数由卷积算子的Fredholm行列式描述。随机矩阵和其他统计对象的其他集合产生更复杂的运算符。在正交和辛系综中，算子的矩阵值符号自然出现，而非光滑随机变量导致算子具有奇异符号。在所有上述情况下，渐近展开式都是强Szego极限定理对各类算子的推广。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。在不规则形状的台球桌上运动的两个台球可能有非常不同的路径。在慢核反应中，复合核粒子的能级也有复杂的不可预测的行为。随机矩阵理论提供了数学模型，可以模拟粒子的能级或台球的能量。本课程的目标之一是了解能级的统计行为。下面是一个例子。假设我们把所有的能量加在一起。能量之和的分布是否随我们对模型的初始假设而变化？对于一大类模型，答案是否定的。经过适当的归一化后，能量之和的分布是钟形的。这些结果与实验数据一致，说明了理论的普适性。