Eigenvector statistics of sparse random matrices

We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdős-Rényi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combined with an isotropic local law for Green’s function. As an auxiliary result, we prove that for the eigenvector flow of Dyson Brownian motion with general initial data, the eigenvectors are asymptotically jointly normal in the direction q after time η_∗ ≪ t ≪ r, if in a window of size r, the initial density of states is bounded below and above down to the scale η_∗, and the initial eigenvectors are delocalized in the direction q down to the scale η_∗.