Electronic Journal of Probability

Eigenvector statistics of sparse random matrices

Paul Bourgade, Jiaoyang Huang, and Horng-Tzer Yau

Full-text: Open access

Abstract

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 $\eta _*$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 64, 38 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1502417019

Digital Object Identifier
doi:10.1214/17-EJP81

Mathematical Reviews number (MathSciNet)
MR3690289

Zentralblatt MATH identifier
1372.05195

Subjects
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

Keywords
sparse random graphs eigenvectors isotropic local law

Rights
Creative Commons Attribution 4.0 International License.

Citation

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


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