Delocalization Transition for Critical Erdős–Rényi Graphs

Abstract

AbstractWe analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph $${\mathbb {G}}(N,d/N)$$ G ( N , d / N ) , where d is of order $$\log N$$ 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 $$\gamma (\varvec{\mathrm {w}})$$ γ ( w ) of an eigenvector $$\varvec{\mathrm {w}}$$ w , defined through $$\Vert \varvec{\mathrm {w}} \Vert _\infty / \Vert \varvec{\mathrm {w}} \Vert _2 = N^{-\gamma (\varvec{\mathrm {w}})}$$ ‖ w ‖ ∞ / ‖ w ‖ 2 = N - γ ( w ) . Our results remain valid throughout the optimal regime $$\sqrt{\log N} \ll d \leqslant O(\log N)$$ log N ≪ d ⩽ O ( log N ) .