Classical and free infinitely divisible distributions and random matrices

Abstract

We construct a random matrix model for the bijection \Psi between clas- sical and free infinitely divisible distributions: for every d\geq1, we associate in a quite natural way to each *-infinitely divisible distribution \mu a distribution P_d^{\mu} on the space of d\times d Hermitian matrices such that P_d^{\mu}P_d^{\nu}=P_d^{\mu*\nu}. The spectral distribution of a random matrix with distribution P_d^{\mu} converges in probability to \Psi (\mu) when d tends to +\infty. 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\geq1, we associate to each *-infinitely divisible distribution \mu, a distribution L_d^{\mu} on the space of complex (non-Hermitian) d\times d random matrices. If \mu is symmetric, the symmetrization of the spectral distribution of |M_d|, when M_d is L_d^{\mu}-distributed, converges in probability to \Psi(\mu).