Random Matrices, Statistical Applications, and Spin Glass Dynamics

Harvard University, Cambridge MA

Investigators

Abstract

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.