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

TL;DR

This paper proves that the largest singular value of polynomials in independent Wigner matrices converges to the operator norm of the corresponding polynomial in free semicircular variables, extending understanding of spectral behavior in random matrix theory.

Contribution

It introduces a refined linearization technique and analytical methods to establish convergence and eigenvalue distribution results for polynomials in Wigner matrices.

Findings

Largest singular value converges to the operator norm of free semicircular polynomial

No eigenvalues outside the limiting spectral support

Refined analytical techniques for spectral analysis

Abstract

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.