  Models of disorder with a direction (constant imaginary vector-potential) are
considered. These non-Hermitian models can appear as a result of computation
for models of statistical physics using transfer matrix technique or describe
non-equilibrium processes. Eigenenergies of non-Hermitian Hamiltonians are not
necessarily real and a joint probability density function of complex
eigenvalues can characterize basic properties of the systems. This function is
studied using the supersymmetry technique and a supermatrix $\sigma$-model is
derived. The $\sigma$-model differs from already known by a new term. The
zero-dimensional version of the $\sigma$-model turns out to be the same as that
obtained recently for ensembles of random weakly non-Hermitian or asymmetric
real matrices. Using a new parametrization for the supermatrix $Q$ the density
of complex eigenvalues is calculated in $0D$ for both the unitary and
orthogonal ensembles. The function is drastically different in these two cases.
It is everywhere smooth for the unitary ensemble but has a $\delta$-functional
contribution for the orthogonal one. This anomalous part means that a finite
portion of eigenvalues remains real at any degree of the non-Hermiticity. All
details of the calculations are presented.
