Moderate deviations for the determinant of Wigner matrices

Abstract

We establish a moderate deviations principle (MDP) for the log-determinant $\log |\det (M_n)|$ of a Wigner matrix $M_n$ matching four moments with either the GUE or GOE ensemble. Further we establish Cramér--type moderate deviations and Berry-Esseen bounds for the log-determinant for the GUE and GOE ensembles as well as for non-symmetric and non-Hermitian Gaussian random matrices (Ginibre ensembles), respectively.

20 pages, one missing reference added; Limit Theorems in Probability, Statistics and Number Theory, Springer Proceedings in Mathematics and Statistics, 2013