General eigenvalue correlations for the real Ginibre ensemble

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 for both complex and real eigenvalues and describes a deep connection between both. A slight modification by an artificial additional Grassmannian also solves the more complicated odd-N case. As illustration, we present some numerical results in the space of complex eigenvalue n-tuples.