Extremal eigenvectors of sparse random matrices

TL;DR

This paper proves that edge eigenvectors of certain sparse random matrices are asymptotically jointly normal, introduces a novel algorithm for eigenvector distribution computation, and improves existing results on sparse matrix laws.

Contribution

It presents a new algorithm for directly computing joint eigenvector distributions in sparse matrices, applicable beyond the studied case.

Findings

Edge eigenvectors are asymptotically jointly normal for sparse Erdős-Rényi graphs.

The method applies to quantum ergodicity at the edge for Wigner matrices.

Improves isotropic local law results for sparse matrices.

Abstract

We consider a class of sparse random matrices, which includes the adjacency matrix of Erd\H{o}s-R\'enyi graph ${\bf G}(N,p)$. For $N^{-1+o(1)}\leq p\leq 1/2$, 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.