Bulk eigenvalue fluctuations of sparse random matrices

Abstract

We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős–Rényi graphs $\mathcal{G}(N,p)$ for $p\in [N^{\varepsilon -1},N^{-\varepsilon }]$. 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 $N\sqrt{p}$. 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.