The norm of polynomials in large random and deterministic matrices |
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Authors: | Camille Male |
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Affiliation: | 1. Ecole Normale Sup??rieure de Lyon, Unit?? de Math??matiques pures et appliqu??es, UMR 5669, 46 all??e d??Italie, 69364, Lyon Cedex 07, France
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Abstract: | 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. |
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