Moderate deviations for the determinant of Wigner matrices

We establish a moderate deviations principle (MDP) for the log-determinant $\log \mathrm{\mid }\det (M_n)\mathrm{\mid }$ of a Wigner matrix $M_n$ matching four moments with either the GUE or GOE ensemble. Further we establish Cram'er–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.