PUB - Publikationen an der Universität Bielefeld

We briefly review the solution of three ensembles of non-Hermitian randommatrices generalizing the Wishart-Laguerre (also called chiral) ensembles.These generalizations are realized as Gaussian two-matrix models, where thecomplex eigenvalues of the product of the two independent rectangular matricesare sought, with the matrix elements of both matrices being either real,complex or quaternion real. We also present the more general case depending ona non-Hermiticity parameter, that allows us to interpolate between thecorresponding three Hermitian Wishart ensembles with real eigenvalues and themaximally non-Hermitian case. All three symmetry classes are explicitly solvedfor finite matrix size NxM for all complex eigenvalue correlations functions(and real or mixed correlations for real matrix elements). These are given interms of the corresponding kernels built from orthogonal or skew-orthogonalLaguerre polynomials in the complex plane. We then present the correspondingthree Bessel kernels in the complex plane in the microscopic large-N scalinglimit at the origin, both at weak and strong non-Hermiticity with M-N greateror equal to 0 fixed.