Random Matrices, Statistical Applications, and Spin Glass Dynamics

该奖项的主要目的是研究随机矩阵统计的基础和应用。在大数据时代，经典概率论并没有为当前大数据的应用提供足够的工具。数据分析中最基本的对象之一是表示数据和噪声的大型随机矩阵。这些矩阵的特征值和特征向量的统计本质上决定了数据矩阵中包含的真实信息。在这项工作中，我们的目标是开发工具来理解大类随机矩阵的特征值和特征向量统计。除了数据矩阵之外，该研究还研究了大型随机图关联矩阵的谱特性。为了加强不同研究人员之间的思想交流，我们一直与哈佛大学数学科学与应用中心的统计系和计算机科学系联合举办研讨会。我们还共同组织了查尔斯河概率讲座（与麻省理工学院和微软研究院联合）、当前数学发展会议（与麻省理工学院联合）以及 CMSA 大数据年度会议。这些项目将让来自概率论、统计学、组合学、数学物理和计算机科学的研究人员一起工作。我们正在积极制定统计学家和计算机科学家感兴趣的数学问题。显然，这些科学家对大型随机矩阵的分析非常感兴趣，我们期待这些合作取得丰硕的成果。本研究旨在将随机矩阵统计扩展到大类矩阵模型，包括带状矩阵、稀疏随机图的邻接矩阵和重尾随机矩阵。我们的目标是了解随机矩阵统计量，特别是高斯正交系综的统计量，可以在多大程度上独立于矩阵定律。我们的工作是由 E. Wigner 的宏伟愿景推动的，他断言随机矩阵统计是高复杂性系统的普遍法则。更准确地说，我们选择以下五个项目： 1.能带矩阵的离域化和普适性。 2. Quantum独特的遍历性和数据分析。 3.稀疏随机图的边统计。 4. Levy 矩阵和重尾随机矩阵。 5. 超立方体上的自旋玻璃动力学。第一个项目旨在将当前的随机矩阵理论扩展到非平均场模型，因为大多数基本物理定律都是短程的，因此远离平均场类型。我们认为，关于非平均场带矩阵的项目 1 是关于维格纳最初设想的最重要的问题之一。项目 2 和 3 涉及随机稀疏图邻接矩阵的特征值和特征向量的边缘统计。由于 Erdos-Renyi 图的边缘统计中存在高斯到 Tracy-Widom 的转变，我们希望该项目能够阐明稀疏性在随机图中的作用。此外，我们对特征向量统计的理解将在受限的最小奇异值问题和信号恢复算法中得到应用。项目 4 旨在根据重尾随机矩阵的谱统计对相图进行分类。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格劳伯动力学的光谱间隙和对数索博列夫不等式。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

The main objective of this award is to investigate the foundation and application of random matrix statistics. In the age of large data, classical probability theory does not offer sufficient tools for current applications in large data. One of the most basic objects in data analysis is large random matrices representing data and noise. The statistics of eigenvalues and eigenvectors of these matrices essentially determine the real information contained in the data matrices. In this work, we aim to develop tools in understanding the eigenvalue and eigenvector statistics of the large class of random matrices. Besides data matrices, the research also investigates the spectral properties of the associated matrices of large random graphs. In order to enhance the exchange of ideas among different groups of researchers, we have been running seminars jointly with the Statistics and Computer Science departments at the Center of Mathematical Sciences and Applications at Harvard University. We also co-organize the Charles River Lectures in Probability (jointly with MIT and Microsoft Research), the Current Development in Mathematics conference (jointly with MIT), and the CMSA's annual conference on Big Data. These programs will bring researchers from probability theory, statistics, combinatorics, mathematical physics, and computer science to work together. We are actively formulating mathematical questions that are interesting to statisticians and computer scientists. It is clear that the analysis of large random matrices is of great interest to these scientists and we expect fruitful results from these collaborations. This research aims to extend random matrix statistics to large classes of matrix models, including band matrices, adjacency matrices of sparse random graphs, and heavy-tailed random matrices. Our goal is to understand to what extent random matrix statistics, in particular the statistics of the Gaussian orthogonal ensemble, can be shown to hold independent of the matrix law. Our work is driven by the grand vision of E. Wigner, asserting that random matrix statistics are universal laws for systems of high complexity. More precisely, we choose the following five projects: 1. Delocalization and universality of band matrices. 2. Quantum unique ergodicity and data analysis. 3. Edge statistics of sparse random graphs. 4. Levy matrices and heavy-tailed random matrices. 5. Spin glass dynamics on hypercubes. The first project aims to extend the current random matrix theory to non mean-field models since most basic physics laws are short-ranged and thus far from the mean field type. Project 1 concerning non mean-field band matrices is in our view one of the most important questions regarding Wigner's original vision. Projects 2 and 3 concern edge statistics of both eigenvalues and eigenvectors of the adjacency matrices of random sparse graphs. Due to a Gaussian to Tracy-Widom transition in the edge statistics for the Erdos-Renyi graphs, we expect this project to clarify the role of sparsity in random graphs. In addition, our understanding of eigenvector statistics will have applications in restricted minimum singular value problems and signal recovery algorithms. Project 4 aims to classify the phase diagrams of heavy-tailed random matrices according to their spectral statistics. The last project aims to develop methods to prove spectral gaps and logarithmic Sobolev inequality for the Glauber dynamics of 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.