Spectral Statistics of Erd{\H o}s-R\'enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

TL;DR

This paper proves the universality of eigenvalue distributions, including spacing and extreme eigenvalues, for Erdős-Rényi graphs under certain density conditions, aligning their spectral statistics with those of Gaussian ensembles.

Contribution

It establishes the universality of eigenvalue spacing and extreme eigenvalues for Erdős-Rényi graphs when the edge probability satisfies specific growth conditions, extending spectral universality results.

Findings

Eigenvalue spacing in the bulk matches Gaussian orthogonal ensemble

Second largest eigenvalue distribution aligns with Gaussian orthogonal ensemble

Bulk universality extends to generalized Wigner matrices with finite moments

Abstract

We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption $p N \gg N^{2/3}$, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least $4 + \epsilon$ moments.