Edge Universality of Sparse Random Matrices

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an explicit random shift, the optimal rigidity of extreme eigenvalues holds, provided the averaged degree grows with the size of the graph, $pN>N^\varepsilon$. We prove in the same regime, (i) Optimal rigidity holds for all eigenvalues with respect to an explicit random measure. (ii) Up to an explicit random shift, the fluctuations of the extreme eigenvalues are given the Tracy-Widom distribution.