Convergence of the largest singular value of a polynomial in independent Wigner matrices

Details Convergence of the largest singular value of a polynomial in independent Wigner matrices Convergence of the largest singular value of a polynomial in independent Wigner matrices

2011-03-24

arxiv.org/abs/1103.4825v2

Mathematics

Probability

63 pages, LaTeX, no figures. Final version, to appear in the Annals of Probability. Reorganized and simplified following refer

Scientific paper

For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form "no eigenvalues outside the support of the limiting eigenvalue distribution." We build on ideas of Haagerup-Schultz-Thorbj{\o}rnsen on the one hand and Bai-Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincar\'{e}-type inequalities, we use a variety of matrix identities and $L^p$ estimates. The Schwinger-Dyson equation controls much of the analysis.

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