The logarithmic law of random determinant

Abstract

Consider the square random matrix $A_{n}=(a_{ij})_{n,n}$, where $\{a_{ij}:=a_{ij}^{(n)},i,j=1,\ldots,n\}$ is a collection of independent real random variables with means zero and variances one. Under the additional moment condition \[\sup_{n}\max_{1\leq i,j\leq n}\mathbb{E}a_{ij}^{4}<\infty,\] we prove Girko’s logarithmic law of $\det A_{n}$ in the sense that as $n\rightarrow\infty$ \begin{eqnarray*}\frac{\log|\det A_{n}|-(1/2)\log(n-1)!}{\sqrt{(1/2)\log n}}\stackrel{d}{\longrightarrow}N(0,1).\end{eqnarray*}