Non-Hermitian random matrix models: Free random variable approach

Using the standard concepts of free random variables, we show that for a large class of non-Hermitian random matrix models, the support of the eigenvalue distribution follows from their Hermitian analogs using a conformal transformation. We also extend the concepts of free random variables to the class of non-Hermitian matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) J. Ginibre, J. Math. Phys. 6, 1440 (1965); V. L. Girko, Spectral Theory of Random Matrices (in Russian) (Nauka, Moscow, 1988) and Mahaux-Weidenm\"uller (chaotic resonance scattering) C. Mahaux and H. A. Weidenm\"uller, Shell-model Approach to Nuclear Reactions (North-Holland, Amsterdam, 1969).