Addition of Free Unitary Random Matrices

We consider a new class of non–hermitian random matrices, namely the ones which have the form of sums of freely independent terms which involve unitary matrices. After deriving some general identities describing additive properties of unitary matrices, we solve three particular models: CUE plus CUE, CUE plus . . . plus CUE (i. e. the sum of an arbitrary number of CUE matrices), and CUE plus a real constant times GUE. By solution of a given model we mean here calculating the borderline of the eigenvalues’ two– dimensional domain, as well as the eigenvalues’ density inside the domain. We confirm numerically all the results. The described method allows to deal with a variety of models of the considered form. PACS numbers: 02.50.Cw, 05.40.Ca, 05.45.Pq, 05.70.Fh, 11.15.Pg.