Random matrices: Law of the determinant

Abstract

Let $A_{n}$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_{n}|$ satisfies a central limit theorem. More precisely,

\begin{eqnarray*}&&\sup_{x\in{\mathbf {R}}}\biggl|{\mathbf{P} }\biggl(\frac{\log(|\det A_{n}|)-({1}/{2})\log(n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf{P} }\bigl(\mathbf{N} (0,1)\le x\bigr)\biggr|\\&&\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}