Spectral statistics of Erdős–Rényi graphs I: Local semicircle law
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Overview
Theoretical proof demonstrates a local semicircle law in Erdős–Rényi random graphs, indicating complete eigenvector delocalization down to optimal microscopic spectral scales.
Key Points
To establish the local Wigner semicircle law and determine the localization properties of eigenvectors for adjacency matrices of Erdős–Rényi random graphs.
Analyzed the ensemble of rescaled adjacency matrices of Erdős–Rényi random graphs with $N$ vertices and connection probability $p$.
Evaluated eigenvalue densities on spectral intervals larger than $N^{-1}$ (up to logarithmic factors) under the scaling condition that $pN \to \infty$ at least logarithmically in $N$.
Proved that the local density of eigenvalues converges to the Wigner semicircle law across spectral windows down to scale $N^{-1}$ with logarithmic corrections.
Demonstrated that all eigenvectors are completely delocalized, with their $\ell^{\infty}$-norms bounded by order $N^{-1/2}$ with very high probability.