CAREER: Random Matrices and Many-Body Systems

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The investigator's project is dedicated to a study of random matrix theory. It is a very active field, since many questions in other fields can be represented mathematically as matrix problems. For example: the spectra of heavy atoms in physics, statistical estimation of covariance matrices in statistics, capacity of quantum channels in quantum communication, conducting properties of semiconductors (Anderson models). This project aims to study the basic properties of some random matrix ensembles, which will shed some light on the nature of the highly correlated many-body system. In particular, for understanding the conducting properties of semiconductors and other disordered systems, as explained by the Anderson model in quantum physics, one main project in this research program is designed to provide rigorous proof that similar (conducting) property appears in the random matrix models. There are some problems in this program designed for training graduate and undergraduate students who have an interest in probability theory. The investigator will also run workshops designed to foster the upward professional development of graduate and undergraduate students working in fields related to random matrix theory as well as to stimulate interaction across barriers between these disciplines. The investigator will focus on the identification of the eigenvalue densities, the local eigenvalue statistics and the eigenvector distributions of many different classes of random matrix ensembles. The ensembles considered in this project include anisotropic matrices, band matrices, i.i.d. random matrices, and general non-Hermitian matrices. One main project will be focused on the localization-delocalization conjecture of random band matrices, which is very similar to the metal-insulator phase transition for disordered quantum systems (i.e., the so called Anderson conjecture in quantum physics). Different from regular Wigner matrices, random band matrices are not a mean-field type ensemble. Hence this ensemble is more like the realistic models of quantum many-body systems, which involve quantum states with more geometric structure. Another main project is a study of non-Hermitian matrices (e.g., i.i.d. random matrices). The study on this type of ensembles is much weaker than the study on random Hermitian matrices, since the regular spectrum method can not be applied. The investigator will apply some new ideas on these type of matrices. In this proposal, some projects are also designed for training graduate and undergraduate students, which include proving the universality phenomenon for more general random matrix ensembles.