RUI: Determinant Identities, Szego Type Limit Theorems, and Connections to Random Matrices

Toeplitz矩阵的行列式出现在数学和物理的许多分支中。例如，他们描述了二维磁体的经典伊辛模型中两个位置之间的钉扎关联。最近，它们被用来描述随机矩阵的统计性质，其特征值模拟复杂的系统。用强Szego极限定理及其推广刻画了Toeplitz矩阵行列式的渐近行为。最近，这个定理得到了改进，因为发现了行列式的一个新恒等式，使得人们可以找到行列式Szego展开中的误差的很好的估计。这个项目的主要目的是将这个恒等式扩展到其他类型的矩阵和算子。这具有一些应用，如在随机矩阵理论中求出线性统计量的分布，以及求出特征值的水平间隔。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，因此研究这些系统的平均性质。例如，在慢核反应中，化合物核的一个粒子的能级具有复杂的不可预测的行为。随机矩阵理论提供了允许模拟粒子的能级的数学模型。用于研究随机矩阵以及能级的统计行为的工具之一是Toeplitz矩阵的行列式。行列式是一个产生关于一组正方形数字的重要信息的数字。这些Toeplitz行列式出现在应用数学的许多分支中。一个经典的用途是研究二维(或极薄)磁体模型的特性。行列式通常很难计算。然而，有一个结果，称为强Szego极限定理，它给出了这样一个行列式的估计。最近，这个估计被改进了，以便更容易地计算误差项。该项目的一个主要目标是将这些结果推广到其他类别的决定因素。这将产生关于其他类型模型中复杂系统能级的信息，Toeplitz案例没有涉及到这些信息。

Determinants of Toeplitz matrices have arisen in many branchesof mathematics and physics. For example, they describe thespin correlation between two sites in the classical Isingmodel of a two-dimensional magnet. More recently, they havebeen used to describe statistical properties of random matrices, whose eigenvalues model complicated systems. The asymptotic behavior of determinants of Toeplitz matricesis described using the Strong Szego-Limit Theorem and itsgeneralizations. Recently, this theorem was improved in thesense that a new identity was found for the determinant thatallows one to find very good estimates for the error in theSzego expansion of the determinant. The major purpose ofthis project is to extend this identity to other classes ofmatrices and operators. This has applications such as finding the distributions of linear statistics in random matrix theory,as well as finding the level spacing of the eigenvalues. Many physical systems possess such complicated behavior thatexact predictions become impossible, so instead averageproperties of these systems are studied. For example, theenergy level of a particle of a compound nucleus in a slownuclear reaction has complicated unpredictable behavior.Random matrix theory provides mathematical models that allowa simulation of the energy levels of the particle. One of the tools that is used to study the statistical behavior of the random matrices and thus of the energy levels, is a determinant of a Toeplitz matrix. A determinant is a number that yields important information about a square array of numbers. These Toeplitz determinants occur in manybranches of applied mathematics. One classical use was tostudy the properties of models of two-dimensional (or verythin) magnets. Determinants are often hard to compute.However, there is a result, called the Strong Szego LimitTheorem, which yields an estimate for such a determinant.Recently, this estimate was improved so that error termscould be calculated more easily. One major goal of theproject is to extend these results to other classes ofdeterminants. This will yield information about the energylevels of complicated systems in other types of models, not covered by the Toeplitz case.