Distributions of traffics and their free product: an asymptotic freeness theorem for random matrices and a central limit theorem

Details Distributions of traffics and their free product: an asymptotic freeness theorem for random matrices and a central limit theorem Distributions of traffics and their free product: an asymptotic freeness theorem for random matrices and a central limit theorem

2011-11-20

arxiv.org/abs/1111.4662v2

Mathematics

Probability

51 pages. Major update of the first version

Scientific paper

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.

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