Matrix models for $$\varvec{\varepsilon }$$-free independence

We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to $$\varepsilon $$ ε -free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular $$\varepsilon $$ ε arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary $$\varepsilon $$ ε may be realized in this way. As a result, we obtain a new proof that $$\mathcal {R}^\omega $$ R ω -embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.

On Products of Random Matrices

We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant. Dessins d’enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the “spectrum of stars”. The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.