Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 64, 38 pp.
Eigenvector statistics of sparse random matrices
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  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 $\eta _*$.
Electron. J. Probab., Volume 22 (2017), paper no. 64, 38 pp.
Received: 19 February 2017
Accepted: 6 July 2017
First available in Project Euclid: 11 August 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 05C80: Random graphs [See also 60B20] 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices
Bourgade, Paul; Huang, Jiaoyang; Yau, Horng-Tzer. Eigenvector statistics of sparse random matrices. Electron. J. Probab. 22 (2017), paper no. 64, 38 pp. doi:10.1214/17-EJP81. https://projecteuclid.org/euclid.ejp/1502417019