Entrywise limit theorems of eigenvectors and their one-step refinement for sparse random graphs

We establish finite-sample Berry-Esseen theorems for the entrywise limits of the eigenvectors and their one-step refinement for sparse random graphs. For the entrywise limits of the eigenvectors, the average expected degree is allowed to grow at the rate $\Omega(\log n)$, where $n$ is the number of vertices, and for the entrywise limits of the one-step refinement of the eigenvectors, we require the expected degree to grow at the rate $\omega(\log n)$. The one-step refinement is shown to have a smaller entrywise covariance than the eigenvectors in spectra. The key technical contribution towards the development of these limit theorems is a sharp finite-sample entrywise eigenvector perturbation bound. In particular, the existed error bounds on the two-to-infinity norms of the higher-order remainders are not sufficient when the graph average expected degree is proportional to $\log n$. Our proof relies on a decoupling strategy using a ``leave-one-out'' construction of auxiliary matrices.