Random Matrix Theory and Applications

Harvard University, Cambridge MA

Investigators

Abstract

Classical probability theory was built largely upon modeling systems with no or weak correlations; random matrix statistics, on the other hand, provide essentially the only known general laws for highly correlated systems. The central question for random matrix theory is to determine to what extent random matrix statistics prevail in random systems. Mathematically, this question is generally referred to as the universality conjecture, which asserts that random matrix statistics are independent of the distributions of the matrix elements. The investigator and collaborators previously studied special cases of this conjecture. This project will develop this direction further and will study applications of the universality conjecture to statistics, combinatorics, mathematical physics, and computer science, bringing together researchers from probability theory and these other fields. The project is likely to result in fruitful collaborations with statisticians and computer scientists on the analysis of large random matrices. The project will also support the training through research involvement of mathematics graduate students working on cutting-edge topics in probability theory. Wigner's vision that the local spectral statistics of large correlated quantum systems are modeled by random matrix statistics is a groundbreaking idea in probability theory and statistical physics. The investigator and collaborators previously proved the universality of the local spectral statistics for both Wigner and beta ensembles. These results are important first steps in understanding Wigner's vision. The project's goal is to extend this understanding to random matrices that have significant physical properties. The following four projects will be addressed: 1. Optimal relaxation time for Dyson Brownian motion. 2. Universality of spectral statistics of d-regular graphs for small d, which is a highly important topic in statistics, computer science, and large data analysis. 3. Quantum unique ergodicity and universality of band matrices. This project seeks to show that quantum unique ergodicity and universality appear essentially concurrently. 4. Local density fluctuation and linear statistics of Coulomb gases.