Journal
ELECTRONIC JOURNAL OF PROBABILITY
Volume 22, Issue -, Pages 1-38Publisher
UNIV WASHINGTON, DEPT MATHEMATICS
DOI: 10.1214/17-EJP81
Keywords
sparse random graphs; eigenvectors; isotropic local law
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Funding
- NSF [DMS-1513587, DMS-1307444, DMS-1606305]
- Simons Investigator award
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1606305] Funding Source: National Science Foundation
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We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdos-Renyi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combined with an isotropic local law for Green's function. As an auxiliary result, we prove that for the eigenvector flow of Dyson Brownian motion with general initial data, the eigenvectors are asymptotically jointly normal in the direction q after time eta* << t >> r, if in a window of size r, the initial density of states is bounded below and above down to the scale eta*, and the initial eigenvectors are delocalized in the direction q down to the scale eta*.
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