### Abstract

In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N^{−1/2+ε} for non-Hermitian random matrices at any point z ∈ C with ||z| − 1| > c for any c > 0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z| − 1 = o(1). Without the vanishing third moment assumption, we prove that the circular lawis valid near the spectral edge |z| − 1 = o(1) up to scale N^{−1/4+ε}.

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

Pages (from-to) | 619-660 |

Number of pages | 42 |

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), 619-660. https://doi.org/10.1007/s00440-013-0516-x

**The local circular law II : the edge case.** / Bourgade, Paul; Yau, Horng Tzer; Yin, Jun.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 159, no. 3-4, pp. 619-660. https://doi.org/10.1007/s00440-013-0516-x

}

TY - JOUR

T1 - The local circular law II

T2 - the edge case

AU - Bourgade, Paul

AU - Yau, Horng Tzer

AU - Yin, Jun

PY - 2014

Y1 - 2014

N2 - In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N−1/2+ε for non-Hermitian random matrices at any point z ∈ C with ||z| − 1| > c for any c > 0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z| − 1 = o(1). Without the vanishing third moment assumption, we prove that the circular lawis valid near the spectral edge |z| − 1 = o(1) up to scale N−1/4+ε.

AB - In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N−1/2+ε for non-Hermitian random matrices at any point z ∈ C with ||z| − 1| > c for any c > 0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z| − 1 = o(1). Without the vanishing third moment assumption, we prove that the circular lawis valid near the spectral edge |z| − 1 = o(1) up to scale N−1/4+ε.

KW - Local circular law

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=85027936897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027936897&partnerID=8YFLogxK

U2 - 10.1007/s00440-013-0516-x

DO - 10.1007/s00440-013-0516-x

M3 - Article

VL - 159

SP - 619

EP - 660

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -