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 [Formula: see text] . We show that if [Formula: see text] 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 [Formula: see text] down to the optimal scale [Formula: see text] . The main technical achievement of our proof is a rigidity bound of accuracy [Formula: see text] for the extreme eigenvalues, which avoids the [Formula: see text] -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 [Formula: see text] .