A central limit theorem for the determinant of a Wigner matrix

We establish a central limit theorem for the log-determinant $\log \mathrm{\mid }\det (M_n)\mathrm{\mid }$ 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 $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)\R$whenonematchesmomentswithGUE,and$N(\log \sqrt{n!} - 1/4 \log n, 1/4 \log n)\R$ when one matches moments with GOE.