Towards non-Hermitian random Levy matrices

We review a new technique for calculating spectral properties of infinite non-Hermitian random matrix models, and we present an algorithm for calculating bulk spectral properties of ensembles of the type H(1) + iH(2), where H(1) and H(2) are arbitrary free (in the sense of Voiculescu) ensembles, including cases of the Levy (heavy-tailed) spectra. As a particular example, we solve analytically the ensemble C(1) + iC(2), where C(1) and C(2) are free centered random matrix ensembles of the Cauchy class.