Bulk eigenvalue fluctuations of sparse random matrices

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Original languageEnglish
Pages (from-to)2846-2879
Journal / PublicationAnnals of Applied Probability
Issue number6
Online published14 Dec 2020
Publication statusPublished - Dec 2020
Externally publishedYes


We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős–Rényi graphs G(Np) for p ∈ [Nϵ-1, N-ϵ]. 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 Np. 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.

Research Area(s)

  • CLT, random matrices, sparse Erdős–Rényi graphs