CONVERGENCE OF THE LARGEST SINGULAR VALUE OF A POLYNOMIAL IN INDEPENDENT WIGNER MATRICES

• 成果类型: Article
• 署名作者: Anderson, Greg W.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP739
• 发表日期: 2013
• 页码: 2103-2181
• 关键词: noncommutative polynomials; eigenvalues

摘要

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-Thorbjornsen 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 Poincare-type inequalities, we use a variety of matrix identities and L-p estimates. The Schwinger-Dyson equation controls much of the analysis.