Symplectic Structure of the Real Ginibre Ensemble

Abstract

We give a simple derivation of all $n$-point densities for the eigenvalues of the real Ginibre ensemble with even dimension $N$ as quaternion determinants. A very simple symplectic kernel governs both, the real and complex correlations. 1-and-2-point correlations are discussed in more detail. Scaling forms for large dimension $N$ are derived.