Fluctuations of extreme eigenvalues of sparse Erd\u0151s\u2013R\xe9nyi graphs

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We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph {{mathcal {G}}}(N,p). We show that if N^{varepsilon } leqslant Np leqslant N^{1/3-varepsilon } 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 geqslant N^{2/9 + varepsilon } down to the optimal scale Np geqslant N^{varepsilon }. The main technical achievement of our proof is a rigidity bound of accuracy N^{-1/2-varepsilon } (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 geqslant N^{varepsilon }.