Traffic distributions and independence: permutation invariant random matrices and the three notions of independence

The distributions of traffics are defined and are applied for families of larges random matrices, random groups and infinite random rooted graphs with uniformly bounded degree. There are constructed by adding axioms in Voiculescu’s definition of ∗-distribution of non commutative random variables. The convergence in distribution of traffics generalizes Benjamini, Schramm, Aldous, Lyons’ weak local convergence of random graphs. We introduce a notion of freeness of traffics, which contains both the classical notion of independence and Voiculescu’s notion of freeness. We prove an asymptotic freeness theorem for families of matrices invariant by permutation, which enlarges the class of large random matrices for which we can predict the empirical eigenvalues distribution. We prove a central limit theorem for the sum of free traffics, and interpret the limit as the (traffic)-convolution of a gaussian commutative random variable and a semicircular non commutative random variable. We make a connection between the freeness of traffics and the natural free product of random graphs, combination of the statistical independence and of the geometric free product. Overview of the article 0.1 Motivation from random matrix theory In all this article, when we consider an N by N matrix HN , we implicitly mean that we consider a sequence of matrices (HN )N>1, such that HN is N by N . Following the random matrix theory terminology, the empirical eigenvalues distribution of an N by N matrix HN is the probability measure