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Article Dans Une Revue Probability Theory and Related Fields Année : 2011

Norm of polynomials in large random and deterministic matrices

Camille Male

Résumé

In this article we show the convergence of the operator norm (the largest singular value) of any polynomial in some random matrices. If X_N= (X_1^(N), ...,X_p^(N)) denotes a family of N-by-N independent random matrices from the Gaussian Unitary Ensemble (GUE), then we state sufficient conditions on (possibly random) matrices Y_N =(Y_1^(N), ..., Y_q^(N)), independent with X_N, for which ||P(X_N, Y_N, Y_N^*)|| converges for all polynomial P. Limits are described by operator norms of objects from free probability theory. We give examples of diagonal matrices $\mathbf Y_N $ for which the convergence holds. The case of block matrices is also investigated. Convergence of the operator norm holds when the GUE matrices are replaced by some non-white Wishart matrices.
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Dates et versions

hal-00494600 , version 1 (23-06-2010)
hal-00494600 , version 2 (25-09-2012)

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Camille Male. Norm of polynomials in large random and deterministic matrices. Probability Theory and Related Fields, 2011, pp.41. ⟨10.1007/s00440-011-0375-2⟩. ⟨hal-00494600v1⟩
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