Open Access
2017 Eigenvector statistics of sparse random matrices
Paul Bourgade, Jiaoyang Huang, Horng-Tzer Yau
Electron. J. Probab. 22: 1-38 (2017). DOI: 10.1214/17-EJP81


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 $\boldsymbol q$ 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 $\boldsymbol q$ down to the scale $\eta _*$.


Download Citation

Paul Bourgade. Jiaoyang Huang. Horng-Tzer Yau. "Eigenvector statistics of sparse random matrices." Electron. J. Probab. 22 1 - 38, 2017.


Received: 19 February 2017; Accepted: 6 July 2017; Published: 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1372.05195
MathSciNet: MR3690289
Digital Object Identifier: 10.1214/17-EJP81

Primary: 05C50 , 05C80 , 15B52 , 60B20

Keywords: eigenvectors , isotropic local law , sparse random graphs

Vol.22 • 2017
Back to Top