The Sixth Moment of Random Determinants

TL;DR

This paper calculates the sixth moment of the determinant for large random matrices with independent, mean-zero entries from any distribution, revealing its asymptotic behavior as matrix size grows.

Contribution

It provides the first explicit calculation of the sixth moment and its asymptotics for determinants of asymmetric random matrices with arbitrary entry distributions.

Findings

Explicit sixth moment formula for finite n

Asymptotic behavior of the sixth moment as n approaches infinity

Applicable to matrices with entries from any distribution with mean zero

Abstract

In this paper, we determine the sixth moment of the determinant of an asymmetric $n \times n$ random matrix where the entries are drawn independently from an arbitrary distribution $\Omega$ with mean $0$. Furthermore, we derive the asymptotic behavior of the sixth moment of the determinant as the size of the matrix tends to infinity.