### 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 q after time η_{∗} ≪ t ≪ r, if in a window of size r, the initial density of states is bounded below and above down to the scale η_{∗}, and the initial eigenvectors are delocalized in the direction q down to the scale η_{∗}.

Original language | English (US) |
---|---|

Article number | 64 |

Journal | Electronic Journal of Probability |

Volume | 22 |

DOIs | |

State | Published - 2017 |

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### Keywords

- Eigenvectors
- Isotropic local law
- Sparse random graphs

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Journal of Probability*,

*22*, [64]. https://doi.org/10.1214/17-EJP81

**Eigenvector statistics of sparse random matrices.** / Bourgade, Paul; Huang, Jiaoyang; Yau, Horng Tzer.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 22, 64. https://doi.org/10.1214/17-EJP81

}

TY - JOUR

T1 - Eigenvector statistics of sparse random matrices

AU - Bourgade, Paul

AU - Huang, Jiaoyang

AU - Yau, Horng Tzer

PY - 2017

Y1 - 2017

N2 - 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 q after time η∗ ≪ t ≪ r, if in a window of size r, the initial density of states is bounded below and above down to the scale η∗, and the initial eigenvectors are delocalized in the direction q down to the scale η∗.

AB - 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 q after time η∗ ≪ t ≪ r, if in a window of size r, the initial density of states is bounded below and above down to the scale η∗, and the initial eigenvectors are delocalized in the direction q down to the scale η∗.

KW - Eigenvectors

KW - Isotropic local law

KW - Sparse random graphs

UR - http://www.scopus.com/inward/record.url?scp=85027300123&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027300123&partnerID=8YFLogxK

U2 - 10.1214/17-EJP81

DO - 10.1214/17-EJP81

M3 - Article

VL - 22

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 64

ER -