Exact Solvability in Random Matrices and Data Sciences

随机矩阵和数据科学中的精确可解性

• 批准号: 2152588
• 负责人: Vadim Gorin
• 金额: $ 25.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152588&HistoricalAwards=false
• 关键词: Exact; Solvability; Random; Matrices; Data

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.

该项目着重于随机矩阵 - 随机数的矩形阵列。虽然主要的重点是此类矩阵的理论特性，但问题是由其他科学中的应用激发的：经济学，统计和物理学。在数据科学中，只要观测在二维上排列，例如时间和空间，就会出现矩形阵列。在量子力学中，矩阵和相关运算符出现在物理系统的建模中。当有关此类系统的可用信息量有限时，正确的建模是将矩阵作为随机。该项目为研究生提供了研究培训机会。随机矩阵及其特征值在许多研究领域都起着核心作用，包括高能物理学，增长模型，数量理论和高维统计。该项目围绕着随机矩阵的完全可解决的或可以集成的家庭，通过明确的公式，差分操作，正交多项式的作用以及其他本质上的代数技术，可以访问特征值。该项目的目的是三个方面：寻找这些家庭，对它们产生微妙的渐近结果（通常远远超出了通用系统可用的定理），并使用它们来获得对更广泛的随机矩阵和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β-元素，它们是n维分布将各种随机矩阵特征值定律统一和推广。虽然在经典环境中，参数beta采用值1、2或4，具体取决于所考虑的矩阵是真实的，复杂的还是Quaternion矩阵的，该项目强调了一个观点，在该观点中，允许Beta允许beta进行任意真实值，并应通过统计机制的术语中的统计学来指出NESF的术语，并应将其解释为NESF的术语。基金会的智力优点和更广泛的影响评论标准。