Extremal eigenvectors of sparse random matrices

We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős-Rényi graph $\mathbf{G}(N,p)$. For $N^{-1+o(1)}\le p\le 1\mathrm{/}2$, we show that the non-trivial edge eigenvectors are asymptotically jointly normal. The main ingredient of the proof is an algorithm that directly computes the joint eigenvector distributions, without comparisons with GOE. The method is applicable in general. As an illustration, we also use it to prove the normal fluctuation in quantum ergodicity at the edge for Wigner matrices. Another ingredient of the proof is the isotropic local law for sparse matrices, which at the same time improves several existing results.