Bulk eigenvalue fluctuations of sparse random matrices

We consider a class of sparse random matrices, which includes the adjacency matrix of Erd.s.Rényi graphs .(N,p) for p.[N..1,N..]. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization Np..p. In addition, the eigenvalues fluctuate simultaneously: the correlation of two eigenvalues of the same/different sign is asymptotically 1/-1. We also prove CLTs for the eigenvalue counting function and trace of the resolvent at mesoscopic scales.