Correlated Random Matrices

Abstract

In this thesis, we investigate the appearance of random matrix statistics for correlated random matrices. The Wigner-Dyson-Mehta Universality Conjecture regarding the statistics on eigenvalue differences is one of the central questions in random matrix theory. Much mathematical progress has been made of matrices with i.i.d. (independent, identically distributed) entries, but it is believed that this conjecture should hold for models with a more exotic correlation structure. This thesis derives random matrix statistics for many correlated models of interest, such as edge universality for correlated Gaussian models with optimal correlation decay $d^{-2 -\epsilon}$ and nearly deterministic matrices with entries derived from the skew-shift dynamical transformation.