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 involving unitary matrices. To deal with them, we exploit the recently developed quaternion technique. After having derived 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 GUE. By solution of a given model we mean here to calculate the borderline of the eigenvalues' two-dimensional domain, as well as the eigenvalues' density function inside the domain. We confirm numerically all the results, obtaining very good agreement.