Eigenvector Statistics of Sparse Random Matrices

TL;DR

This paper proves that the eigenvectors of sparse random matrices, such as Erdős-Rényi graphs, become jointly normal as the graph size grows, under certain degree conditions, using Dyson Brownian motion analysis.

Contribution

It introduces a novel method combining Dyson Brownian motion and local laws to analyze eigenvector distributions in sparse random matrices.

Findings

Eigenvectors are asymptotically jointly normal for large sparse graphs.

The methodology applies to eigenvector flows with general initial data.

Eigenvector delocalization is established in the specified regimes.

Abstract

We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erd\H{o}s-R\'enyi 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, combining 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 $\eta_*\ll t\ll r$, if in a window of size $r$, the initial density of states is bounded below and above down to the scale $\eta_*$, and the initial eigenvectors are delocalized in the direction $q$ down to the scale $\eta_*$.