The norm of polynomials in large random and deterministic matrices

63 pagesInternational audienceLet $\mathbf X_N= (X_1^{(N)} \etc X_p^{(N)})$ be a family of $N \times N$ independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices $\mathbf Y_N =(Y_1^{(N)} \etc Y_q^{(N)})$, possibly random but independent of $\mathbf X_N$, for which the operator norm of $P(\mathbf X_N,\mathbf Y_N, \mathbf Y_N^*)$ 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 $\mathbf Y_N$ 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 $\mathbf 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