Delocalization transition for critical Erd\H{o}s-R\'enyi graphs

TL;DR

This paper investigates the eigenvector localization properties of critical Erdős-Rényi graphs, revealing a sharp transition from delocalized to localized eigenvectors near the spectrum edges, with implications for understanding spectral phases.

Contribution

It characterizes the delocalization transition in eigenvectors of critical Erdős-Rényi graphs, identifying a sharp phase change and describing the structure of localized eigenvectors.

Findings

Eigenvectors in the middle spectrum are fully delocalized.

Eigenvectors near the spectrum edges are localized on small vertex clusters.

The transition between phases is sharp, with a discontinuity in the localization exponent.

Abstract

We analyse the eigenvectors of the adjacency matrix of a critical Erd\H{o}s-R\'enyi graph $\mathbb G(N,d/N)$, where $d$ is of order $\log N$. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $\gamma(\mathbf w)$ of an eigenvector $\mathbf w$, defined through $\|\mathbf w\|_\infty / \|\mathbf w\|_2 = N^{-\gamma(\mathbf w)}$. Our results remain valid throughout the optimal regime $\sqrt{\log N} \ll d \leq O(\log N)$.