Abstract
We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős–Rényi graphs $\mathcal{G}(N,p)$ for $p\in [N^{\varepsilon -1},N^{-\varepsilon }]$. 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 $N\sqrt{p}$. 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.
Citation
Yukun He. "Bulk eigenvalue fluctuations of sparse random matrices." Ann. Appl. Probab. 30 (6) 2846 - 2879, December 2020. https://doi.org/10.1214/20-AAP1575
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