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March 2020 Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs
Jiaoyang Huang, Benjamin Landon, Horng-Tzer Yau
Ann. Probab. 48(2): 916-962 (March 2020). DOI: 10.1214/19-AOP1378

Abstract

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdős–Rényi graph $G(N,p)$. Tracy–Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in (Probab. Theory Related Fields 171 (2018) 543–616; Comm. Math. Phys. 314 (2012) 587–640). We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy–Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erdős–Rényi graphs are less rigid than those of random $d$-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.

Citation

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Jiaoyang Huang. Benjamin Landon. Horng-Tzer Yau. "Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs." Ann. Probab. 48 (2) 916 - 962, March 2020. https://doi.org/10.1214/19-AOP1378

Information

Received: 1 March 2018; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199865
MathSciNet: MR4089498
Digital Object Identifier: 10.1214/19-AOP1378

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

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • March 2020
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