Rectangular random matrices. Related convolution

Abstract

We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymtotically free with amalgamation over a subalgebra. Therefore we can define a "rectangular free convolution", linearized by cumulants and by an analytic integral transform, called the "rectangular R-transform".