Edith Frisk Gärtner: Local law for the Gaussian ensembles

Abstract.

In the realm of random matrix theory, the Gaussian ensembles, characterized by matrices whose entries are centered Gaussian random variables, are among the most extensively studied distributions. This thesis investigates the distribution of eigenvalues of matrices within these ensembles. The main result is a local law, demonstrating convergence in \(q\)-th mean of the Stieltjes transform of the empirical eigenvalue measure to the Stieltjes transform of the deterministic semicircular measure as the matrix size \(N\) tends to infinity.