A Note on the Pfaffian Integration Theorem

TL;DR

This paper presents two concise proofs of the Pfaffian integration theorem, which is relevant to the spectral analysis of Ginibre's Orthogonal Ensemble, enhancing theoretical understanding of random matrix spectra.

Contribution

It introduces two novel, compact proofs of the Pfaffian integration theorem, one using Fredholm Pfaffians and the other based on linear algebra.

Findings

Proofs clarify the mathematical foundation of the Pfaffian integration theorem.

The results support spectral studies of Ginibre's Orthogonal Ensemble.

The methods may facilitate further research in random matrix theory.

Abstract

Two alternative, fairly compact proofs are presented of the Pfaffian integration theorem that is surfaced in the recent studies of spectral properties of Ginibre's Orthogonal Ensemble. The first proof is based on a concept of the Fredholm Pfaffian; the second proof is purely linear-algebraic.