Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph G(N, p). We show that if N^ε ≤ Np ≤ N^1/3-ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from Np ≥ N^2/9+ε down to the optimal scale Np ≥ N^ε. The main technical achievement of our proof is a rigidity bound of accuracy N^-1/2-ε (Np)^-1/2 for the extreme eigenvalues, which avoids the (Np)^-1-expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np ≥ N^ε.