A central limit theorem for the determinant of a Wigner matrix

Abstract

We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments with GUE, and $N(\log \sqrt{n!} - 1/4 \log n, 1/4 \log n)_\R$ when one matches moments with GOE.