Exact Solvability in Random Matrices and Data Sciences

该项目的重点是随机矩阵-随机数的矩形阵列。虽然主要关注的是这些矩阵的理论性质，但这些问题的动机是在其他科学中的应用：经济学，统计学和物理学。在数据科学中，每当观察结果自然地排列在两个维度上（例如，时间和空间）时，就会出现矩形阵列。在量子力学中，矩阵和相关算子出现在物理系统的建模中。当关于这种系统的可用信息量有限时，适当的建模是通过将矩阵随机化。随机矩阵及其特征值在高能物理、增长模型、数论和高维统计等许多研究领域中发挥着核心作用。该项目围绕着随机矩阵的精确可解或可积族，其特征值可以通过显式公式，微分运算，正交多项式和其他基本代数技术来获得。这个项目的目标是三重的：寻找这些家庭，发展微妙的渐近结果（这通常远远超出了一般系统的定理），并使用它们获得渐近预测更广泛的类随机矩阵和相关的应用对象的兴趣。将项目的不同部分粘合在一起的中心对象是β集合，它是N维分布，统一和推广了各种随机矩阵的特征值定律。虽然在经典的上下文中，参数β取1、2或4的值，这取决于所考虑的矩阵是真实的、复数还是四元数矩阵，这个项目强调了一个观点，即允许β取任意的正的真实的值，并应解释为统计力学术语中的逆温度。这个奖项反映了NSF的法定使命，并已通过使用基金会的知识价值和更广泛的影响审查标准进行评估，认为值得支持。

The project focuses on random matrices - rectangular arrays of random numbers. While the primary focus is on theoretical properties of such matrices, the questions are motivated by applications in other sciences: economics, statistics, and physics. In data sciences, rectangular arrays appear whenever the observations are naturally arranged in two dimensions, for instance, in time and space. In quantum mechanics, matrices and related operators appear in modelling of a physical system. When the amount of available information about such a system is limited, a proper modelling is by taking the matrix to be random. This project provides research training opportunities for graduate students.Random matrices and their eigenvalues play a central role in many research areas, including high-energy physics, growth models, number theory, and high-dimensional statistics. The project revolves around exactly solvable or integrable families of random matrices, for which the eigenvalues are accessible through explicit formulas, actions of differential operations, orthogonal polynomials, and other essentially algebraic techniques. The goal of this project is three-fold: to search for these families, develop delicate asymptotic results about them (which usually go far beyond theorems available for generic systems), and use them for obtaining asymptotic predictions for much wider classes of random matrices and related objects of applied interest. The central objects gluing together different parts of the project are beta-ensembles, which are N-dimensional distributions uniting and generalizing the laws of eigenvalues of various random matrices. While in classical contexts the parameter beta takes values 1, 2, or 4, depending on whether the matrices under consideration are real, complex, or are quaternion matrices, this project emphasize a point of view in which beta is allowed to take arbitrary positive real values and should be interpreted as the inverse temperature in the terminology of statistical mechanics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.