Appears in collection : 2017 - T3 - WS2 - Probabilistic techniques and quantum information theory

The properties of the limiting non commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e. g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. This talk presents an approach for the study of random matrices invariant in law by conjugation by permutation matrices, the theory of traffics. A traffic is a non commutative random variable in a space with more structure than a general non commutative probability space, so that the notion of traffic distribution is richer than the notion of non commutative distribution. It comes with a notion of independence which is able to encode the different notions of non commutative independence.