Moments of the characteristic polynomial in the three ensembles of random matrices

TL;DR

This paper derives explicit determinant and pfaffian formulas for the moments of the characteristic polynomial in orthogonal, unitary, and symplectic random matrix ensembles, revealing new identities between these expressions.

Contribution

It provides novel determinant and pfaffian representations for moments across three classical random matrix ensembles and uncovers surprising identities linking these formulas.

Findings

Explicit formulas for moments as determinants or pfaffians

New identities relating determinants and pfaffians in Gaussian ensembles

Comparative analysis of expressions across different ensembles

Abstract

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n\times n$ determinant and an $m\times m$ determinant and another between the pfaffian of a $2n\times 2n$ anti-symmetric matrix and a sum of $m\times m$ determinants.