Construction of a free L\'evy process as high-dimensional limit of a Brownian motion on the Unitary group

Abstract

It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free L\'evy processes as high-dimensional limits of classical matricial L\'evy processes. We will focus here on one specific such construction, discussing and generalizing a work done previously by Biane, who has shown that the (classical) Brownian motion on the Unitary group U\left(d\right) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free L\'evy process on the dual group (in the sense of Voiculescu) U\langle n\rangle. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of the which Biane's result constitutes the case n=1.