Open Access
2022 The completely delocalized region of the Erdős-Rényi graph
Johannes Alt, Raphaël Ducatez, Antti Knowles
Author Affiliations +
Electron. Commun. Probab. 27: 1-9 (2022). DOI: 10.1214/22-ECP450


We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability dN. We determine the full region of delocalization by determining the critical values of dlogN down to which delocalization persists: for dlogN>1log41 all eigenvectors are completely delocalized, and for dlogN>1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.

Funding Statement

The authors acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 715539_RandMat and the Marie Sklodowska-Curie grant agreement No. 895698. Funding from the Swiss National Science Foundation through the NCCR SwissMAP grant is also acknowledged. J.A. and A.K. acknowledge support from the National Science Foundation under Grant No. DMS-1928930 during their participation in the program “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems” hosted by the Mathematical Sciences Research Institute in Berkeley, California during the Fall semester of 2021.


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Johannes Alt. Raphaël Ducatez. Antti Knowles. "The completely delocalized region of the Erdős-Rényi graph." Electron. Commun. Probab. 27 1 - 9, 2022.


Received: 13 September 2021; Accepted: 22 January 2022; Published: 2022
First available in Project Euclid: 3 February 2022

arXiv: 2109.03227
MathSciNet: MR4375917
Digital Object Identifier: 10.1214/22-ECP450

Primary: 05C80 , 15B52 , 60B20

Keywords: eigenvector delocalization , Local law , random graph , sparse random matrix

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