Open Access
Translator Disclaimer
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


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.


Download Citation

Gérard Ben Arous. Paul Bourgade. "Extreme gaps between eigenvalues of random matrices." Ann. Probab. 41 (4) 2648 - 2681, July 2013.


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

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

Primary: 11M50 , 15B52 , 60B20

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

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • July 2013
Back to Top