Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

Abstract

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ⊞λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ⊞λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ.