Moderate deviations for the determinant of Wigner matrices

TL;DR

This paper proves moderate deviations principles and related probabilistic bounds for the log-determinant of Wigner and Gaussian random matrices, extending understanding of their fluctuation behaviors.

Contribution

It establishes a moderate deviations principle for the log-determinant of Wigner matrices matching four moments with GUE/GOE, along with Cramér-type deviations and Berry-Esseen bounds for various Gaussian ensembles.

Findings

MDP for log-determinant of Wigner matrices matching four moments with GUE/GOE

Cramér-type moderate deviations for GUE and GOE ensembles

Berry-Esseen bounds for non-symmetric and non-Hermitian Gaussian 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 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.