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

Abstract

In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $λ\in [0,1]$, the rectangular free convolution with ratio $λ$. Here, we investigate the related notion of infinite divisiblity, which happens to be closely related to the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of distributions infinitely divisible with respect to this convolution, which preserves limit theorems. We give an interpretation of this correspondance in term of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws (i.e. uniform distributions on their singular values) going from the symmetric classical infinitely divisible distributions to their images by the previously mentioned bijection when the dimensions go from one to infinity in a ratio $λ$.