  In a previous paper (called "Rectangular random matrices. Related
covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free
convolution with ratio $\lambda$. 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 $\lambda$.
