Exact Solvability in Random Matrices and Data Sciences

该项目侧重于随机矩阵-随机数的矩形数组。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，只要观察结果自然地排列在二维空间中，例如在时间和空间中，矩形数组就会出现。在量子力学中，矩阵和相关的算符出现在物理系统的建模中。当关于这样一个系统的可用信息量有限时，适当的建模方法是将矩阵设为随机。本项目为研究生提供研究训练机会。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发挥着核心作用。该项目围绕精确可解或可积的随机矩阵族展开，其特征值可通过显式公式、微分运算、正交多项式和其他本质上的代数技术获得。这个项目的目标有三个方面：搜索这些族，开发关于它们的精细渐近结果（通常远远超出一般系统可用的定理），并使用它们获得更广泛类别的随机矩阵和应用兴趣相关对象的渐近预测。将项目的不同部分粘合在一起的中心对象是beta-ensembles，它是n维分布，将各种随机矩阵的特征值定律统一并推广。虽然在经典情况下，参数β的值为1、2或4，取决于所考虑的矩阵是实数、复数还是四元数矩阵，但本项目强调了一种观点，即β可以取任意正实数，并且应该被解释为统计力学术语中的逆温度。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.