Random matrices: Law of the determinant

TL;DR

This paper proves that the logarithm of the determinant of a large random matrix with independent entries follows a normal distribution, establishing a central limit theorem with explicit convergence rate.

Contribution

It establishes a central limit theorem for the log-determinant of random matrices with subexponential entries, providing the first such result with explicit convergence rate.

Findings

Logarithm of the determinant converges to a normal distribution.

Explicit rate of convergence in the CLT.

Applicable to matrices with subexponential tail entries.

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*}