Delocalization transition for critical Erdős-Rényi graphs

Abstract

We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi 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 $\Vert \mathbf{w}{\Vert }_{\infty }/\Vert \mathbf{w}{\Vert }_2=N^{-\gamma (\mathbf{w})}$. Our results remain valid throughout the optimal regime $\sqrt{\log N}\ll d\le O(\log N)$.