Local law and tracy-widom limit of sparse random matrix models

Abstract

We consider spectral properties of sparse random matrix models, which includes sparse sample covariance matrices and sparse stochastic block models. We prove a local law for the eigenvalue density up to the spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy--Widom law with an explicit formula on the deterministic shift of the spectral edge.