Traffic probability for rectangular random matrices

This article develops a rectangular version of Male's theory of traffic probability, in which an algebra is equipped with a trace evaluated on arbitrary graphs whose edges are labeled by elements and whose vertices are subspaces. Using the language of traffic distributions, we characterize the asymptotic behavior of independent families of rectangular random matrices which are bi-permutation invariant. In the process, we take a tour of non-commutative probabilities and their random matrix models. Special attention is paid to rectangular random matrices with independent or exchangeable entries, including the existence and description of limiting & lowast;-distributions for a broad range of models.