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

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.