Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs (2005.02254v2)

Published 5 May 2020 in math.PR, math-ph, and math.MP

Abstract: We consider a class of sparse random matrices which includes the adjacency matrix of the Erd\H{o}s-R\'enyi graph $\mathcal{G}(N,p)$. We show that if $N^{{\varepsilon}} \leq Np \leq N^{{1/3-\varepsilon}$} 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 [19] on the fluctuations of the extreme eigenvalues from $Np \geq N^{2/9 + \varepsilon}$ down to the optimal scale $Np \geq N^{{\varepsilon}$.} 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 [9,19,24]. Our result is the last missing piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue fluctuations of sparse random matrices for $Np \geq N^{{\varepsilon}$.}