Raphaël Ducatez

We analyse the spectrum of the (scaled) adjacency matrix A of the Erdős-Rényi graph G(N,d/N) in the critical regime d=blog N with b constant. We establish a one-to-two correspondence between vertices of degree at least 2d and nontrivial eigenvalues outside the asymptotic [−2,2]. This correspondence implies a transition at an explicit b*. For d>b∗logN the spectrum is just the [−2,2] and the eigenvectors are completely delocalized. For d<b*log N we still have delocalization in [−2,2] but another phase appears. The spectrum outside [−2,2] is not empty and the associated eigenvectors concentrate around the large degree vertices.