On the second moment of the determinant of random symmetric, Wigner, and Hermitian matrices

TL;DR

This paper provides a unified analytic approach to compute the second moment of the determinant for various classes of random matrices, including symmetric, Wigner, and Hermitian matrices, extending previous results.

Contribution

It introduces a new analytic combinatorics method to analyze the second moment of determinants across different random matrix ensembles.

Findings

Derived explicit formulas for the second moment of Hermitian matrices.

Extended previous results to include Hermitian matrices with specific entry distributions.

Unified the analysis of second moments across symmetric, Wigner, and Hermitian matrices.

Abstract

In this paper, we analyze the second moment of the determinant of random symmetric, Wigner, and Hermitian matrices. Using analytic combinatorics techniques, we determine the second moment of the determinant of Hermitian matrices whose entries on the diagonal are i.i.d and whose entries above the diagonal are i.i.d. and have real expected values. Our results extend previous work analyzing the second moment of the determinant of symmetric and Wigner matrices, providing a unified approach for this analysis.