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Poisson statistics and localization at the spectral edge of sparse Erd\H{o}s-R\'enyi graphs by Alt, Johannes; Ducatez, Raphael; Knowles, Antti is a scholarly article available to read on EtoBox.

What is Poisson statistics and localization at the spectral edge of sparse Erd\H{o}s-R\'enyi graphs about?

We consider the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph on $N$ vertices with edge probability $d/N$. For $(\log \log N)^4 \ll d \lesssim \log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale $d \asymp \log N$, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. Together with [arXiv:2005.14180], our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of $A$. The proof relies on a three-scale rigidity argument, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.

Author

Alt, Johannes; Ducatez, Raphael; Knowles, Antti

Published

2021

Language

EN