### 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 - Aug 20 2013 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty