General Eigenvalue Correlations for the Real Ginibre Ensemble

TL;DR

This paper simplifies the derivation of eigenvalue distributions in the real Ginibre ensemble, revealing deep connections between real and complex eigenvalues through Pfaffian and quaternion determinant structures.

Contribution

It provides a simplified derivation of eigenvalue correlations in the real Ginibre ensemble using Pfaffian and quaternion determinants, including a universal symplectic kernel for all dimensions.

Findings

Explicit Pfaffian generating functional derived

Universal symplectic kernel valid for real and complex eigenvalues

Numerical results illustrating eigenvalue correlations

Abstract

We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian generating functional for $n$-point densities. This produces a simple free-fermion diagram expansion for the correlations leading to quaternion determinants in each order n. All will explicitly be given with the help of a very simple symplectic kernel for even dimension $N$. The kernel is valid both for complex and real eigenvalues and describes a deep connection between both. A slight modification by an artificial additional Grassmannian solves also the more complicated odd-$N$ case. As illustration we present some numerical results in the space $\mathbb{C}^n$ of complex eigenvalue $n$-tuples.