Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

TL;DR

This paper provides an elementary derivation of the Pfaffian point process describing real eigenvalues in the real Ginibre ensemble, revealing hidden symmetries and integral representations for correlation functions.

Contribution

It introduces a new elementary derivation of the Pfaffian point process for real eigenvalues, highlighting hidden symplectic symmetry and integral formulas for correlation functions.

Findings

Revealed hidden symplectic symmetry in eigenvalue statistics

Derived integral representation for K-point correlation functions

Connected eigenvalue statistics to symmetric space integrals

Abstract

An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, uses the averages of products of characteristic polynomials. This derivation reveals a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues and an integral representation for the $K$-point correlation function for any $K\in \mathbb{N}$ in terms of an asymptotically exact integral over the symmetric space $U(2K)/USp(2K)$.