Convergence of unitary matrix integrals

We introduce the unitary Schwinger-Dyson equation associated to a selfadjoint polynomial potential V. The V=0 case corresponds to the free product state, so the Schwinger-Dyson equation can be considered as a deformation of free probability. We show that the solutions of this equation are unique for small V’s and correspond to a large N limit of a multi-matrix model. This technique allows to show that a wide class of unitary and orthogonal multi-matrix models converge asymptotically. We also give a combinatorial and analytic description of the limit, solving a series of open questions raised in theoretical physics in the late 70’s. This is joint work with A. Guionnet and E. Maurel-Segala.