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July 2013 Extreme gaps between eigenvalues of random matrices
Gérard Ben Arous, Paul Bourgade
Ann. Probab. 41(4): 2648-2681 (July 2013). DOI: 10.1214/11-AOP710

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 $k$th smallest gap, normalized by a factor $n^{-4/3}$, has a limiting density proportional to $x^{3k-1}e^{-x^{3}}$. Concerning the largest gaps, normalized by $n/\sqrt{\log n}$, they converge in ${\mathrm{L} }^{p}$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

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Gérard Ben Arous. Paul Bourgade. "Extreme gaps between eigenvalues of random matrices." Ann. Probab. 41 (4) 2648 - 2681, July 2013. https://doi.org/10.1214/11-AOP710

Information

Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1282.60008
MathSciNet: MR3112927
Digital Object Identifier: 10.1214/11-AOP710

Subjects:
Primary: 11M50 , 15B52 , 60B20

Keywords: Eigenvalues statistics , extreme spacings , Gaussian unitary ensemble , negative association property , random matrices

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • July 2013
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