The norm of polynomials in large random and deterministic matrices

Abstract

Let XN = (X1(N), . . . , Xp(N)) be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices YN = (Y1(N), . . . , Yq(N)), possibly random but independent of XN, for which the operator norm of P(XN, YN, YN*) 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 YN 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 YN 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. © 2011 Springer-Verlag.