Random Matrix Theory

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This project aims to study the basic properties of the random matrices whose entries are basically independent. The main objective is 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 Wigner matrix, generalized Wigner matrix, band matrix, Erdos-Renyi adjacency matrix, non-Hermitian (i.i.d. matrix), covariance matrix, generalized covariance matrix, finite rank perturbation of Wigner or covariance matrix, the sum or/and product of random matrix and deterministic matrix. Some open problems will be studied in this project, e.g. Universality of covariance matrix, local circular law, localization-delocalization transition of the eigenstates of band matrix, distribution of the largest eigenvalue of correlation matrix and etc.The rigorous analysis on the local statistics of random matrix will shed some light on the nature of the highly correlated many body system. The works on covariance matrix will bring collaboration between mathematicians and statistician to better understand wireless communication, economic phenomenon and population genetics. The works on band matrix aims to establish the conducting properties of semiconductors and other disordered systems.