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

托普利兹矩阵的行列式在数学和物理的许多分支中都有出现。例如，他们描述了二维磁体经典Isingmodel中两个位置之间的自旋相关。最近，它们被用来描述随机矩阵的统计性质，随机矩阵的特征值是复杂系统的模型。用强塞格极限定理及其推广描述Toeplitz矩阵行列式的渐近性质。最近，这个定理在某种意义上得到了改进，它为行列式找到了一个新的恒等式，使得人们可以很好地估计行列式的zego展开式中的误差。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。这有一些应用，比如在随机矩阵理论中找到线性统计的分布，以及找到特征值的水平间隔。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，所以取而代之的是研究这些系统的平均性质。例如，在缓慢的核反应中，复合原子核的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能级的数学模型。用于研究随机矩阵的统计行为和能级的工具之一是Toeplitz矩阵的行列式。行列式是一个数字，它产生关于一组数字的重要信息。这些托普利兹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（或任何厚度）磁铁模型的性质。行列式通常很难计算。然而，有一个结果，称为强Szego极限定理，它产生了这样一个行列式的估计。最近，这种估计得到了改进，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扩展到其他类型的决定因素。这将产生关于其他类型模型中复杂系统能级的信息，Toeplitz案例中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.