### Abstract

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n^{ -4/3}, has a limiting density proportional to x^{3k-1e} -x^{3}. Concerning the largest gaps, normalized by n/v log n, they converge in L^{p} to a constant for allp >0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

Original language | English (US) |
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Pages (from-to) | 2648-2681 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 41 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

- Eigenvalues statistics
- Extreme spacings
- Gaussian unitary ensemble
- Negative association property
- Random matrices

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Extreme gaps between eigenvalues of random matrices.** / Arous, Gérard Ben; Bourgade, Paul.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 41, no. 4, pp. 2648-2681. https://doi.org/10.1214/11-AOP710

}

TY - JOUR

T1 - Extreme gaps between eigenvalues of random matrices

AU - Arous, Gérard Ben

AU - Bourgade, Paul

PY - 2013

Y1 - 2013

N2 - This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n -4/3, has a limiting density proportional to x3k-1e -x3. Concerning the largest gaps, normalized by n/v log n, they converge in Lp to a constant for allp >0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

AB - This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n -4/3, has a limiting density proportional to x3k-1e -x3. Concerning the largest gaps, normalized by n/v log n, they converge in Lp to a constant for allp >0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

KW - Eigenvalues statistics

KW - Extreme spacings

KW - Gaussian unitary ensemble

KW - Negative association property

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=84881529077&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84881529077&partnerID=8YFLogxK

U2 - 10.1214/11-AOP710

DO - 10.1214/11-AOP710

M3 - Article

AN - SCOPUS:84881529077

VL - 41

SP - 2648

EP - 2681

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -