NON-GAUSSIAN NON-HERMITIAN RANDOM-MATRIX THEORY - PHASE-TRANSITION AND ADDITION FORMALISM

We apply the recently introduced method of hermitization to study in t he large N limit non-hermitian random matrices that are drawn from a l arge class of circularly symmetric non-gaussian probability distributi ons, thus extending the recent gaussian non-hermitian literature. We d evelop the general formalism for calculating the Green function and av eraged density of eigenvalues, which may be thought of as the non-herm itian analog of the method due to Brezin, Itzykson, Parisi and Zuber f or analyzing hermitian non-gaussian random matrices, We obtain an expl icit algebraic equation for the integrated density of eigenvalues. A s omewhat surprising result of that equation is that the shape of the ei genvalue distribution in the complex plane is either a disk or an annu lus, As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution, Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitian random variable s, We use this formalism to state and prove a non-abelian non-hermitia n version of the central limit theorem. (C) 1997 Elsevier Science B.V.