Local law and Tracy-Widom limit for sparse random matrices

Ji Oon Lee, Kevin Schnelli

Published 2016-05-27Version 1

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erd\H{o}s-R\'enyi graph model $G(N,p)$. We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erd\H{o}s-R\'{e}nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue for $p\gg N^{-1/3}$ with a deterministic shift of order $(Np)^{-1}$.