  We review our recent results on pseudo-hermitian random matrix theory which
were hitherto presented in various conferences and talks. (Detailed accounts of
our work will appear soon in separate publications.) Following an introduction
of this new type of random matrices, we focus on two specific models of
matrices which are pseudo-hermitian with respect to a given indefinite metric
B. Eigenvalues of pseudo-hermitian matrices are either real, or come in
complex-conjugate pairs. The diagrammatic method is applied to deriving
explicit analytical expressions for the density of eigenvalues in the complex
plane and on the real axis, in the large-N, planar limit. In one of the models
we discuss, the metric B depends on a certain real parameter t. As t varies,
the model exhibits various "phase transitions" associated with eigenvalues
flowing from the complex plane onto the real axis, causing disjoint eigenvalue
support intervals to merge. Our analytical results agree well with presented
numerical simulations.
