Classical and free infinitely divisible distributions and random matrices

We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each ∗-infinitely divisible distribution μ a distribution Pμd on the space of d×d hermitian matrices such that Pμd ∗ Pνd = P μ∗ν d . The spectral distribution of a random matrix with distribution Pμd converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko-Pastur distribution. In an analogous way, for every d ≥ 1, we associate to each ∗-infinitely divisible distribution μ a distribution Lμd on the space of complex (non-hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |Md|, when Md is Lμd -distributed, converges in probability to Ψ(μ).