We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability . We determine the full region of delocalization by determining the critical values of down to which delocalization persists: for all eigenvectors are completely delocalized, and for 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.
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.
"The completely delocalized region of the Erdős-Rényi graph." Electron. Commun. Probab. 27 1 - 9, 2022. https://doi.org/10.1214/22-ECP450