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Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs


He, Yukun; Knowles, Antti (2021). Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs. Probability Theory and Related Fields, 180(3/4):985-1056.

Abstract

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős-Rényi graph G(N,p) . We show that if Nε⩽Np⩽N1/3−ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916-962, 2020) on the fluctuations of the extreme eigenvalues from Np⩾N2/9+ε down to the optimal scale Np⩾Nε . The main technical achievement of our proof is a rigidity bound of accuracy N−1/2−ε(Np)−1/2 for the extreme eigenvalues, which avoids the (Np)−1 -expansions from Erdős et al. (Ann Prob 41:2279-2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543-616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587-640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np⩾Nε

Keywords: 05C50; 05C80; 15B52; 60B20

Abstract

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős-Rényi graph G(N,p) . We show that if Nε⩽Np⩽N1/3−ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916-962, 2020) on the fluctuations of the extreme eigenvalues from Np⩾N2/9+ε down to the optimal scale Np⩾Nε . The main technical achievement of our proof is a rigidity bound of accuracy N−1/2−ε(Np)−1/2 for the extreme eigenvalues, which avoids the (Np)−1 -expansions from Erdős et al. (Ann Prob 41:2279-2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543-616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587-640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np⩾Nε

Keywords: 05C50; 05C80; 15B52; 60B20

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability, Analysis
Language:English
Date:24 April 2021
Deposited On:13 Oct 2022 16:41
Last Modified:27 Jun 2024 01:40
Publisher:Springer
ISSN:0178-8051
OA Status:Hybrid
Free access at:PubMed ID. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00440-021-01054-4
PubMed ID:34720301
Project Information:
  • : FunderSNSF
  • : Grant ID200020_172623
  • : Project TitleDynamical and energetic properties of Bose-Einstein condensates
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)