Extreme gaps between eigenvalues of random matrices

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)2648-2681
Number of pages34
JournalAnnals of Probability
Volume41
Issue number4
DOIs
StatePublished - 2013

Fingerprint

Random Matrices
Extremes
Limiting
Eigenvalue
Unitary matrix
Riemann zeta function
Ensemble
Directly proportional
Converge
Zero
Eigenvalues

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.

In: Annals of Probability, Vol. 41, No. 4, 2013, p. 2648-2681.

Research output: Contribution to journalArticle

@article{57f0573f2e4e4979a48a005f32f5af16,
title = "Extreme gaps between eigenvalues of random matrices",
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 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.",
keywords = "Eigenvalues statistics, Extreme spacings, Gaussian unitary ensemble, Negative association property, Random matrices",
author = "Arous, {G{\'e}rard Ben} and Paul Bourgade",
year = "2013",
doi = "10.1214/11-AOP710",
language = "English (US)",
volume = "41",
pages = "2648--2681",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "4",

}

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 -