Global and Local Semicircle Laws for Random Matrices with Correlated Entries

Abstract

In this dissertation, we analyze random matrix ensembles with correlated entries, for which we derive global and local semicircle laws. The ensembles we study in the first part of the dissertation are called ”α-almost uncorrelated”, where α > 0 represents a parameter that controls the correlation decay in the ensemble. The model appears naturally when analyzing Gaussian ensembles with a uniform decay rate of all covariances, but also Curie-Weiss ensembles fit within this framework. Under varying assumptions, we show semicircle laws in probability and almost surely for periodic and non-periodic α-almost uncorrelated random band matrices, including full matrices as a special case. In addition, we formulate a theorem that shows asymptotic equivalence of empirical spectral distributions (ESDs) of periodic and non-periodic random band matrices in very general settings. In the second part of the dissertation, we derive local semicircle laws of various types for ensembles with correlated entries, which we call ”of Curie-Weiss type.” Local semicircle laws were developed in 2009 for independent and identically distributed matrix entries to give very detailed insights into the convergence mechanisms of ESDs towards the semicircle distribution. They allow high-probability statements about convergence on very small intervals. Our contribution is to give complete and rigorous proofs of these local laws (which are so far lacking in the literature), to establish a connection between various formulations of the local laws, and to derive local laws in presence of very slow correlation decay not yet covered by the literature.