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December 2020 Bulk eigenvalue fluctuations of sparse random matrices
Yukun He
Ann. Appl. Probab. 30(6): 2846-2879 (December 2020). DOI: 10.1214/20-AAP1575

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

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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

Information

Received: 1 May 2019; Revised: 1 November 2019; Published: December 2020
First available in Project Euclid: 14 December 2020

Digital Object Identifier: 10.1214/20-AAP1575

Subjects:
Primary: 05C50, 05C80, 15B52, 60B20

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 6 • December 2020
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