Antti Knowles: Extreme eigenvalues of sparse random graphs

Abstract: I review some recent work on the extreme eigenvalues of sparse random graphs, such as inhomogeneous Erdos-Renyi graphs. Let n denote the number of vertices and d the maximal mean degree. We establish a crossover in the behaviour of the extreme eigenvalues at the scale d = log n. For d >> log n, we prove that the extreme eigenvalues converge to the edges of the support of the asymptotic eigenvalue distribution. For d << log n, we prove that these extreme eigenvalues are governed by the largest degrees, and that they exhibit a novel behaviour, which in particular rules out their convergence to a nondegenerate point process.
​ Joint work with Florent Benaych-Georges and Charles Bordenave.​