TY - JOUR
T1 - Bulk eigenvalue fluctuations of sparse random matrices
AU - HE, Yukun
PY - 2020/12
Y1 - 2020/12
N2 - We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős–Rényi graphs G(N, p) 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 N√p. 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.
AB - We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős–Rényi graphs G(N, p) 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 N√p. 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.
KW - CLT
KW - random matrices
KW - sparse Erdős–Rényi graphs
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U2 - 10.1214/20-AAP1575
DO - 10.1214/20-AAP1575
M3 - 21_Publication in refereed journal
VL - 30
SP - 2846
EP - 2879
JO - Annals of Applied Probability
JF - Annals of Applied Probability
SN - 1050-5164
IS - 6
ER -