Random matrices, operators, and analytic functions

本课题的重点是研究随机矩阵理论中的各种极限。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值。方阵的特征值描述了相应的线性变换如何在特定的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取的方阵增长序列的特征值的标度性质。这是一个高度活跃的概率领域，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，包括麦迪逊实验数学实验室的项目，获奖者将参与威斯康星大学研究生和本科生概率课程的开发。该项目建立在获奖者和合作者最近关于随机矩阵极限的随机算子表示及其特征多项式的结果之上。其思想是将随机矩阵表示为离散微分算子，并通过算子本身的极限来研究其特征值的极限。该方法还提供了利用泛函和随机分析的框架来分析随机矩阵特征多项式极限的工具。该项目研究了作为这种极限得到的随机解析函数的各种性质。它还研究了适合随机算子方法的随机矩阵模型的各种变形和推广，目的是发现新的和有趣的极限对象，并更多地了解原始模型。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 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.