Random matrices, operators, and analytic functions

随机矩阵、运算符和解析函数

• 批准号: 2246435
• 负责人: Benedek Valko
• 金额: $ 21万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246435&HistoricalAwards=false
• 关键词: Random; matrices; operators; analytic; functions

The focus of this project is the study of various limits in random matrix theory. Matrices are one of the most basic mathematical objects. They encode multivariate linear functions and can serve as approximations to more complicated multivariate functions. The eigenvalues of a square matrix describe how the corresponding linear transformation shrinks or stretches the space in certain special directions. One of the central problems in random matrix theory is to understand the scaling properties of the eigenvalues of a growing sequence of randomly chosen square matrices. This is a highly active area of probability, with a number of interesting connections to other fields within mathematics. Graduate and undergraduate students will be involved in the research, including projects at the Madison Experimental Mathematics Lab, and the awardee is involved in developing the graduate and undergraduate probability curriculum at the University of Wisconsin. The project builds on recent results of the awardee and collaborators on random operator representations of limits of random matrices and their characteristic polynomials. The idea is to represent random matrices as certain discretized differential operators and to study the limit of their eigenvalues via the limit of the operators themselves. This approach also provides tools to analyze the limits of characteristic polynomials of random matrices using the framework of functional and stochastic analysis. The project investigates various properties of the random analytic functions obtained as such limits. It also studies various deformations and generalizations of random matrix models that are amenable to the random operator approach, with the goals of discovering new and interesting limit objects and learning more about the original models.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.

本项目的重点是随机矩阵理论中各种极限的研究。 矩阵是最基本的数学对象之一。它们编码多元线性函数，可以作为更复杂的多元函数的近似。方阵的特征值描述了相应的线性变换如何在某些特定方向上收缩或拉伸空间。随机矩阵理论的核心问题之一是理解随机选择的方阵的增长序列的特征值的标度性质。这是一个非常活跃的概率领域，与数学中的其他领域有许多有趣的联系。研究生和本科生将参与研究，包括在麦迪逊实验数学实验室的项目，获奖者参与开发研究生和本科生概率课程在威斯康星州的大学。该项目建立在获奖者和合作者关于随机矩阵及其特征多项式极限的随机算子表示的最新成果之上。 其思想是将随机矩阵表示为某些离散化的微分算子，并通过算子本身的极限来研究其特征值的极限。该方法也提供了工具，分析的限制，使用功能和随机分析的框架随机矩阵的特征多项式。该项目调查的各种性质的随机解析函数获得这样的限制。该奖项还研究了随机矩阵模型的各种变形和推广，这些变形和推广适用于随机算子方法，目的是发现新的和有趣的极限对象，并了解更多关于原始模型的信息。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。