Classical and free infinitely divisible distributions and random matrices

Abstract

We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙ_d^μ on the space of d×d Hermitian matrices such that ℙ_d^μ*ℙ_d^ν=ℙ_d^μ*ν. The spectral distribution of a random matrix with distribution ℙ_d^μ 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 $\mathbb{L}_{d}^{\mu}$ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |M_d|, when M_d is $\mathbb{L}_{d}^{\mu}$-distributed, converges in probability to Ψ(μ).