The distributions of traffics and their free product

Camille Male

Published 2011-11-20, updated 2015-05-11Version 5

Traffics are defined as elements of Voiculescu's non commutative spaces (called non commutative random variables), for which we specify more structure. We define a new notion of free product in that context. It is weaker than Voiculescu's free product and encodes the independence of complex random variables. This free product models the limits of independent random matrices invariant by conjugation by permutation matrices. We generalize known theorems of asymptotic freeness (for Wigner, unitary Haar, uniform permutation and deterministic matrices) and present examples of random matrices that converges in non commutative law and are not asymptotically free in the sense of Voiculescu. Our approach provides some additional applications. Firstly, the convergence in distribution of traffics is related to two notions of convergence of graphs, namely the weak local convergence of Benjamini and Schramm and the convergence of graphons of Lovasz. These connections give descriptions of the limiting eigenvalue distributions of large graphs with uniformly bounded degree and random matrices with variance profile. Moreover, we prove a new central limit theorems for the normalized sum of non commutative random variables. It interpolates Voiculescu's and de Moivre-Laplace central limit theorems.