Random Matrices, Random Schrödinger Operators, and Applications

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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.