The norm of polynomials in large random and deterministic matrices
| Autor: | Male, C. |
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| Rok vydání: | 2010 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of X_N, for which the operator norm of P(X_N, Y_N, 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. Comment: 41 pages, with an appendix by D. Shlyakhtenko |
| Databáze: | arXiv |
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