Random matrix ensembles for $PT$-symmetric systems

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. They are related to the split signature versions of the complex and the quaternionic numbers, respectively. We conjecture that these ensembles represent universality classes for $PT$-symmetric matrices. For the case of $2\times2$ matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.