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

摘要

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution boxed plus(lambda). 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 boxed plus(lambda)-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 boxed plus(lambda)-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio lambda.