The logarithmic law of random determinant

TL;DR

This paper proves a logarithmic law describing the asymptotic distribution of the determinant of large random matrices with independent entries, showing it converges to a normal distribution after proper normalization.

Contribution

It establishes Girko's logarithmic law for the determinant of random matrices with independent entries under finite fourth moment conditions.

Findings

The normalized log-determinant converges in distribution to a standard normal.

The result holds for matrices with independent, mean-zero, variance-one entries.

The proof relies on moment conditions and asymptotic analysis.

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*}