The norm of polynomials in large random and deterministic matrices

Let ${{bf X}_N =(X_1^{(N)}, ldots, X_p^{(N)})}$ be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices ${{bf Y}_N =(Y_1^{(N)}, ldots, Y_q^{(N)})}$ , possibly random but independent of X _N , for which the operator norm of ${P({bf X}_N, {bf Y}_N, {bf Y}_N^*)}$ converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y _N and of the polynomials P, we get for a large class of matrices the ??no eigenvalues outside a neighborhood of the limiting spectrum?? phenomena. We give examples of diagonal matrices Y _N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.