Matricial approximations of higher dimensional master fields

Abstract

We study matricial approximations of master fields we constructed in a previous work. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in $\mathbb{R}, \mathbb{C}$ or $\mathbb{K}$) and letting the dimension of these blocks to tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory and operator-valued free probability appear as the natural framework in which the limiting distributions are most accurately described.