Random Matrices, Random Schrödinger Operators, and Applications

本计画的主要目的是探讨随机矩阵理论的基础及其在随机薛定谔算子与统计上的应用。经典概率论一直是统计学的基石;对于当前大数据统计和人工神经网络的应用，最基本的对象是表示大数据和噪声的随机矩阵。在这个项目中，PI将开发理论工具来分析大型随机矩阵的特征值和特征向量的统计，这是数据分析的基本对象。除了数据矩阵，PI还将研究大型随机图的相关矩阵，并在数学物理中应用随机薛定谔算子。为了加强不同研究人员群体之间的思想交流，将与PI机构的统计和计算机科学部门以及该地区的其他机构联合组织研讨会和其他科学活动。这些计划将汇集来自概率论，统计学，组合学，数学物理和计算机科学的研究人员，共同研究围绕这些科学界感兴趣的大型随机矩阵分析的问题。该项目还为研究生提供研究培训机会。该奖项资助的研究目标是了解随机矩阵的基础以及数据分析和数学物理中的相关应用。其中一个具体项目旨在探索随机矩阵和随机薛定谔算子之间的联系。这个项目是PI最近关于随机带矩阵的离域和量子扩散的工作的自然延伸。这项工作表明，一个平均场模型，如高斯正交系综或高斯酉系综可以用来模拟非平均场模型，在这种情况下，带矩阵。PI预计，该项目将导致解决（在某种弱意义上）的长期开放的问题，关于随机薛定谔算子的离域。另一个项目旨在将现有的关于戴森布朗运动的理论扩展到自由卷积模型的特征向量。这个项目可以被看作是扩展戴森的布朗运动到一个非均匀设置。第三个项目是研究d-正则图的邻接矩阵的特征值和特征向量的统计性质，其中d是大于或等于3的任意度。这个项目的一个长期目标是证明第二大本征值的分布符合Tracy-Widom定律。 最后一个项目旨在开发方法来证明超立方体上自旋玻璃的Glauber动力学的光谱间隙。该奖项反映了NSF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 main objective of this project is to investigate the foundation of random matrix theory and its applications to Random Schrodinger operators and statistics. Classical probability theory has been a cornerstone to statistics; for current applications to large data statistics and artificial neural networks, the most basic objects are random matrices representing large data and noise. In this project, the PI will develop theoretical tools in analyzing the statistics of eigenvalues and eigenvectors of large random matrices, which are the fundamental objects in data analysis. Besides data matrices, the PI will also investigate the associated matrices of large random graphs and, in applications to mathematical physics, random Schrodinger operators. In order to enhance the exchange of ideas among different groups of researchers, seminars and other scientific events will be organized jointly with the Statisticsand Computer Science departments at the PI's institution and with other institutions in the area. These programs will bring together researchers from probability theory, statistics, combinatorics, mathematical physics, and computer science to work together on questions centered around the analysis of large random matrices that are interesting to these scientific communities. The project also provides research training opportunities for graduate students. The goal of the research funded by this award is to understand the foundation of random matrices and associated applications in data analysis and mathematical physics. One of the specific projects aims to explore the connection between random matrices and random Schrodinger operators. This project is a natural extension of the PI’s recent work on delocalization and quantum diffusion of random band matrices. This work shows that a mean-field model like Gaussian orthogonal ensemble or Gaussian unitary ensemble can be used to model non-mean-field models—in this case, band matrices. The PI anticipates that this project will lead to a solution (in a certain weak sense) of the long standing open problem regarding the delocalization of random Schrodinger operators. Another project aims to extend the existing theory concerning Dyson’s Brownian motion to eigenvectors of free convolution models. This project can be viewed as extending Dyson’s Brownian motion to a non-uniform setting. The third project concerns investigation of the eigenvalue and eigenvector statistics of the adjacency matrices of d-regular graphs for any degree d bigger than or equal to three. A long-term goal of this project is to show that the second largest eigenvalue distributes by the Tracy-Widom law up to a shift. A final project aims to develop methods to prove spectral gaps for the Glauber dynamics for spin glasses on hypercubes.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.