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

By Greg W. Anderson. 2011

79 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/89c5193d7434524f6f22270bee6530738821de6d

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, unknown): “, [2, 24].”
Free Probability and Random Matrices (2014, methodology): “In 2011, based on the Schur complement, Anderson [1] devel-opped a selfadjoint version of the linearization trick, which turns out to be the right tool in our context.”
Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem (2013, background): “As already mentioned in the introduction, the subordination result for the operatorvalued free additive convolution stated in Section 2 provides in combination with Anderson’s version of the linearization trick presented in Section 3 an algorithm to calculate the distribution of any selfadjoint poly”
Universality and Sharp Matrix Concentration Inequalities (2022, result): “References [1] G. W. Anderson.”
Eigenvalues of random lifts and polynomials of random permutation matrices (2018, methodology): “It was developed simultaneously in various areas of mathematics, and applied in operator algebras by Haagerup and Thorbjørnsen and subsequently improved by Anderson [6].”
Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method (2015, methodology): “This is based on a linearization trick introduced in [1].”