### Abstract

The circular law asserts that the spectralmeasure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z| − 1| ≥ τ for arbitrarily small τ > 0, the circular law is valid around z up to scale N^{−1/2+ε} for any ε > 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

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

Pages (from-to) | 545-595 |

Number of pages | 51 |

Journal | Probability Theory and Related Fields |

Volume | 159 |

Issue number | 3-4 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Local circular law
- Universality

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Probability Theory and Related Fields*,

*159*(3-4), 545-595. https://doi.org/10.1007/s00440-013-0514-z

**Local circular law for random matrices.** / Bourgade, Paul; Yau, Horng Tzer; Yin, Jun.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 159, no. 3-4, pp. 545-595. https://doi.org/10.1007/s00440-013-0514-z

}

TY - JOUR

T1 - Local circular law for random matrices

AU - Bourgade, Paul

AU - Yau, Horng Tzer

AU - Yin, Jun

PY - 2014

Y1 - 2014

N2 - The circular law asserts that the spectralmeasure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z| − 1| ≥ τ for arbitrarily small τ > 0, the circular law is valid around z up to scale N−1/2+ε for any ε > 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

AB - The circular law asserts that the spectralmeasure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z| − 1| ≥ τ for arbitrarily small τ > 0, the circular law is valid around z up to scale N−1/2+ε for any ε > 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

KW - Local circular law

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=85028168981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028168981&partnerID=8YFLogxK

U2 - 10.1007/s00440-013-0514-z

DO - 10.1007/s00440-013-0514-z

M3 - Article

AN - SCOPUS:85028168981

VL - 159

SP - 545

EP - 595

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -