Eigenvalue Distribution In The Self-Dual Non-Hermitian Ensemble

We consider an ensemble of self-dual matrices with arbitrary complex entries. This ensemble is closely related to a previously defined ensemble of anti-symmetric matrices with arbitrary complex entries. We study the two-level correlation functions numerically. Although no evidence of non-monotonicity is found in the real space correlation function, a definite shoulder is found. On the analytical side, we discuss the relationship between this ensemble and the $\beta=4$ two-dimensional one-component plasma, and also argue that this ensemble, combined with other ensembles, exhausts the possible universality classes in non-hermitian random matrix theory. This argument is based on combining the method of hermitization of Feinberg and Zee with Zirnbauer's classification of ensembles in terms of symmetric spaces.