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.
|Where?||PER 08 Phys 2.52
Chemin du Musée 3
|Contact||Department of Mathematics