Extreme gaps between eigenvalues of random matrices

Gérard Ben Arous; Paul Bourgade

doi:10.1214/11-AOP710

Extreme gaps between eigenvalues of random matrices

Gérard Ben Arous, Paul Bourgade

Mathematics

Research output: Contribution to journal › Article › peer-review

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 -x³. 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)
Pages (from-to)	2648-2681
Number of pages	34
Journal	Annals of Probability
Volume	41
Issue number	4
DOIs	https://doi.org/10.1214/11-AOP710
State	Published - 2013

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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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.

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KW - Extreme spacings

KW - Gaussian unitary ensemble

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KW - Random matrices

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Extreme gaps between eigenvalues of random matrices

Abstract

Keywords

ASJC Scopus subject areas

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