The spectral edge of (sub-)critical Erdös-Rényi graphs.

Abstract: It is well known that the Erdős-Rényi graph on N vertices with edge probability d/N undergoes a dramatic change in behaviour when the mean degree d crosses the critical scale log(N): the degrees of the graph cease to concentrate about their means and the graph loses its homogeneity. We analyse the eigenvalues and eigenvectors of its adjacency matrix in the regime where the mean degree d is comparable to or less than the critical scale log(N). We show that the eigenvalue process near the spectral edges is asymptotically Poisson, and the intensity measure is determined by the fluctuations of the large degrees as well as the size of the 2-spheres around vertices of large degree. We conclude that in general the laws of the largest eigenvalues are not described by the classical Fisher–Tippett–Gnedenko theorem. As an application of our result, we prove that the associated eigenvectors are are exponentially localized in unique, disjoint balls. Together with the previously established complete delocalization of the eigenvectors in the middle of the spectrum, this establishes the coexistence of a delocalized and a localized phase in the critical Erdös-Rényi graph. Joint work with Johannes Alt and Raphael Ducatez.

statistical mechanicsmathematical physicsprobability

Audience: researchers in the topic

Séminaire MEGA

Series comments: Description: Monthly seminar on random matrices and random graphs