Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner Matrices

TL;DR

This paper analyzes the spectral behavior of multivariate quadratic polynomials of Wigner matrices, establishing optimal convergence rates for their operator norms and classifying edge behaviors in reducible cases.

Contribution

It provides the first precise convergence rate for operator norms of such polynomials and characterizes spectral edge behaviors, extending random matrix theory.

Findings

Operator norm converges to a deterministic limit at rate N^{-2/3+o(1)}

Spectral density exhibits square root growth at edges

Classification of edge behaviors in reducible cases

Abstract

We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension $N$ of the matrices grows to infinity, the operator norm of such polynomials $q$ converges to a deterministic limit with a rate of convergence of $N^{-2/3+o(1)}$. Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviours.