Asymptotic freeness over the diagonal of large random matrices

Ponente(s): Camille Male Pierre, B. Au, G. Cébron, A. Dahlqvist, F. Gabriel

I will discuss the problem of computing the eigenvalues distribution of polynomials in random matrices, in the limit where the size of the matrices goes to infinity. In this context, Voiculescu's Free Probability Theory gives analytic tools to consider this question when the random matrices are in "generic position", in particular when they are invariant by conjugation by unitary matrices. Here we work under a much weaker assumption, assuming only that the random matrices are invariant in law by conjugation by permutation matrices. This requires a more general method, known as Traffic Probability Theory. Since recently, with this approach we were only able to give a combinatorial description for the moments of the limit eigenvalues distribution. More recently, we discovered that freeness in the sense of traffics implies Voiculescu's notion of freeness with amalgamation over the diagonal. In particular, this yields new numerical methods to compute limiting eigenvalues distributions.