The random matrix theory of the classical compact groups

224 Pages·2019·2.46 MB·English

by  Meckes, Elizabeth S.

THE RANDOM MATRIX THEORY OF THE CLASSICAL
COMPACT GROUPS
This is the first book to provide a comprehensive overview of foundational
results and recent progress in the study of random matrices from the classi-
cal compact groups, drawing on the subject’s deep connections to geometry,
analysis, algebra, physics, and statistics. The book sets a foundation with an
introduction to the groups themselves and six different constructions of Haar
measure.Classicalandrecentresultsarethenpresentedinadigested,accessible
form,includingthefollowing:resultsonthejointdistributionsoftheentries;
anextensivetreatmentofeigenvaluedistributions,includingtheWeylintegra-
tion formula, moment formulae, and limit theorems and large deviations for
thespectralmeasures;concentrationofmeasurewithapplicationsbothwithin
randommatrixtheoryandinhighdimensionalgeometry,suchastheJohnson–
Lindenstrauss lemma and Dvoretzky’s theorem; and results on characteristic
polynomialswithconnectionstotheRiemannzetafunction.Thisbookwillbe
ausefulreferenceforresearchersinthearea,andanaccessibleintroductionfor
studentsandresearchersinrelatedfields.
elizabeth s. meckes is Professor of Mathematics at Case Western
Reserve University. She is a mathematical probabilist specializing in random
matrixtheoryanditsapplicationstootherareasofmathematics,physics,and
statistics.ShereceivedherPhDfromStanfordUniversityin2006andreceived
theAmericanInstituteofMathematicsfive-yearfellowship.Shehasalsobeen
supported by the Clay Institute of Mathematics, the Simons Foundation, and
the US National Science Foundation. She is the author of 24 research papers
inmathematics,aswellasthetextbookLinearAlgebra,coauthoredwithMark
MeckesandpublishedbyCambridgeUniversityPressin2018.
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The Random Matrix Theory of the
Classical Compact Groups
Cambridge Tracts in Mathematics 218
ELIZABETH S. MECKES
CaseWesternReserveUniversity
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108419529
DOI:10.1017/9781108303453
©ElizabethS.Meckes2019
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andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2019
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AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Meckes,ElizabethS.,author.
Title:Therandommatrixtheoryoftheclassicalcompactgroups/
ElizabethMeckes(CaseWesternReserveUniversity).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Series:Cambridgetractsinmathematics;218|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019006045|ISBN9781108419529(hardback:alk.paper)
Subjects:LCSH:Randommatrices.|Matrices.
Classification:LCCQA196.5.M432019|DDC512.9/434–dc23
LCrecordavailableathttps://lccn.loc.gov/2019006045
ISBN978-1-108-41952-9Hardback
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Contents
Preface pageix
1 HaarMeasureontheClassicalCompactMatrixGroups 1
1.1 TheClassicalCompactMatrixGroups 1
1.2 HaarMeasure 7
1.3 LieGroupStructureandCharacterTheory 17
2 DistributionoftheEntries 31
2.1 Introduction 31
2.2 TheDensityofaPrincipalSubmatrix 38
2.3 HowMuchIsaHaarMatrixLikeaGaussianMatrix? 42
2.4 ArbitraryProjections 53
3 EigenvalueDistributions:ExactFormulas 60
3.1 TheWeylIntegrationFormula 60
3.2 DeterminantalPointProcesses 71
3.3 MatrixMoments 80
3.4 PatternsinEigenvalues:PowersofRandomMatrices 85
4 EigenvalueDistributions:Asymptotics 90
4.1 TheEigenvalueCountingFunction 90
4.2 TheEmpiricalSpectralMeasureandLinear
EigenvalueStatistics 106
4.3 PatternsinEigenvalues:Self-Similarity 113
4.4 LargeDeviationsfortheEmpiricalSpectralMeasure 117
5 ConcentrationofMeasure 131
5.1 TheConcentrationofMeasurePhenomenon 131
5.2 LogarithmicSobolevInequalitiesandConcentration 133
vii
viii Contents
5.3 TheBakry–ÉmeryCriterionandConcentrationforthe
ClassicalCompactGroups 141
5.4 ConcentrationoftheSpectralMeasure 153
6 GeometricApplicationsofMeasureConcentration 161
6.1 TheJohnson–LindenstraussLemma 161
6.2 Dvoretzky’sTheorem 165
6.3 AMeasure-TheoreticDvoretzkyTheorem 170
7 CharacteristicPolynomialsandConnectionstotheRiemann
ζ-function 181
7.1 Two-PointCorrelationsandMontgomery’sConjecture 181
7.2 TheZetaFunctionandCharacteristicPolynomials
ofRandomUnitaryMatrices 186
7.3 NumericalandStatisticalWork 197
References 205
Index 212
Preface
This book grew out of lecture notes from a mini-course I gave at the 2014
WomenandMathematicsprogramattheInstituteforAdvancedStudy.When
askedtoprovidebackgroundreadingfortheparticipants,Ifoundmyselfata
bit of a loss; while there are many excellent books that give some treatment
of Haar distributed random matrices, there was no single source that gave a
broad,andbroadlyaccessible,introductiontothesubjectinitsownright.My
goalhasbeentofillthisgap:togiveanintroductiontothetheoryofrandom
orthogonal,unitary,andsymplecticmatricesthatapproachesthesubjectfrom
manyangles,includesthemostimportantresultsthatanyonelookingtolearn
aboutthesubjectshouldknow,andtellsacoherentstorythatallowsthebeauty
ofthismany-facetedsubjecttoshinethrough.
The book begins with a very brief introduction to the orthogonal, unitary,
andsymplecticgroups–justenoughtogetstartedtalkingaboutHaarmeasure.
Section 1.2 includes six different constructions of Haar measure on the clas-
sicalgroups;Chapter1alsocontainssomefurtherinformationonthegroups,
includingsomebasicaspectsoftheirstructureasLiegroups,identificationof
theLiealgebras,anintroductiontorepresentationtheory,anddiscussionofthe
characters.
Chapter2isaboutthejointdistributionoftheentriesofaHaar-distributed
randommatrix.ThefactthatindividualentriesareapproximatelyGaussianis
classical and goes back to the late nineteenth century. This chapter includes
modern results on the joint distribution of the entries in various senses: total
variation approximation of principal submatrices by Gaussian matrices, in-
probabilityapproximationof(muchlarger)submatricesbyGaussianmatrices,
andatreatmentofarbitraryprojectionsofHaarmeasureviaStein’smethod.
Chapters 3 and 4 deal with the eigenvalues. Chapter 3 is all about exact
formulas: the Weyl integration formulas, the structure of the eigenvalue pro-
cesses as determinantal point processes with explicit kernels, exact formulas
ix
x Preface
due to Diaconis and Shahshahani for the matrix moments, and an interesting
decomposition(duetoEricRains)ofthedistributionofeigenvaluesofpowers
ofrandommatrices.
Chapter4dealswithasymptoticsfortheeigenvaluesoflargematrices:the
sinekernelmicroscopicscalinglimit,limittheoremsfortheempiricalspectral
measures and linear eigenvalue statistics, large deviations for the empirical
spectralmeasures,andaninterestingself-similaritypropertyoftheeigenvalue
distribution.
Chapters 5 and 6 are where this project began: concentration of measure
on the classical compact groups, with applications in geometry. Chapter 5
introducesthe concept of concentration of measure, the connection with log-
Sobolev inequalities, and derivations of optimal (at least up to constants)
log-Sobolev constants. The final section contains concentration inequalities
fortheempiricalspectralmeasuresofrandomunitarymatrices.
Chapter6hassomeparticularlyimpressiveapplicationsofmeasureconcen-
trationontheclassicalgroupstohigh-dimensionalgeometry.First,aproofof
thecelebratedJohnson–Lindenstrausslemmaviaconcentrationofmeasureon
theorthogonalgroup,witha(verybrief)discussionoftheroleofthelemmain
randomizedalgorithms.ThesecondsectionisdevotedtoaproofofDvoretzky’s
theorem,viaconcentrationofmeasureontheunitarygroup.Thefinalsection
givestheproofofa“measure-theoretic”Dvoretzkytheorem,showingthat,sub-
jecttosomemildconstraints,mostmarginalsofhigh-dimensionalprobability
measuresareclosetoGaussian.
Finally,chapter7givesatasteoftheintriguingconnectionbetweeneigenval-
uesofrandomunitarymatricesandzerosoftheRiemannzetafunction.There
isasectiononMontgomery’stheoremandconjectureonpaircorrelationsand
oneontheresultsofKeatingandSnaithonthecharacteristicpolynomialofa
randomunitarymatrix,whichledthemtoexcitingnewconjecturesonthezeta
side.Somenumericalevidence(andstrikingpictures)ispresented.
Haar-distributedrandommatricesappearandplayimportantrolesinawide
spectrumofsubfieldsofmathematics,physics,andstatistics,anditwouldnever
havebeenpossibletomentionthemall.Ihaveusedtheend-of-chapternotesin
parttogivepointerstosomeinterestingtopicsandconnectionsthatIhavenot
included,anddoubtlesstherearemanymorethatIdidnotmentionatall.Ihave
triedtomakethebookaccessibletoareaderwithanundergraduatebackground
in mathematics generally, with a bit more in probability (e.g., comfort with
measuretheorywouldbegood).Butbecausetherandommatrixtheoryofthe
classical compact groups touches on so many diverse areas of mathematics,
it has been my assumption in writing this book that most readers will not be
familiarwithallofthebackgroundthatcomesup.Ihavedonemybesttogive
accessible, bottom-line introductions to the areas I thought were most likely