The limit of the operator norm for random matrices with a variance profile

TL;DR

This paper investigates the asymptotic behavior of the operator norm of various classes of random matrices with a variance profile, establishing convergence results under finite moment conditions.

Contribution

It provides general convergence results for the operator norm of symmetric and non-symmetric random matrices with diverse variance profiles, including band and Gram matrices, under minimal moment assumptions.

Findings

Operator norm converges to the spectral edge of the limiting distribution.

Finite 4th moment suffices for convergence in probability.

Finite 4+ε moment ensures almost sure convergence.

Abstract

In this work we study symmetric random matrices with variance profile satisfying certain conditions. We establish the convergence of the operator norm of these matrices to the largest element of the support of the limiting empirical spectral distribution. We prove that it is sufficient for the entries of the matrix to have finite only the $4$-th moment or the $4+\epsilon$ moment in order for the convergence to hold in probability or almost surely respectively. Our approach determines the behaviour of the operator norm for random symmetric or non-symmetric matrices whose variance profile is given by a step or a continuous function, random band matrices whose bandwidth is proportional to their dimension, random Gram matrices, triangular matrices and more.