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首页 An Introduction to Random Matrices

An Introduction to Random Matrices.pdf

An Introduction to Random Matri…

raowy2002
2012-09-17 0人阅读 举报 0 0 0 暂无简介

简介:本文档为《An Introduction to Random Matricespdf》,可适用于工程科技领域

ThispageintentionallyleftblankCAMBRIDGESTUDIESINADVANCEDMATHEMATICSEditorialBoardBBOLLOB´AS,WFULTON,AKATOK,FKIRWAN,PSARNAK,BSIMON,BTOTAROAnIntroductiontoRandomMatricesThetheoryofrandommatricesplaysanimportantroleinmanyareasofpuremathematicsandemploysavarietyofsophisticatedmathematicaltools(analytical,probabilisticandcombinatorial)Thisdiversearrayoftools,whileattestingtothevitalityofthefield,presentsseveralformidableobstaclestothenewcomer,andeventheexpertprobabilistThisrigorousintroductiontothebasictheoryissufficientlyselfcontainedtobeaccessibletograduatestudentsinmathematicsorrelatedsciences,whohavemasteredprobabilitytheoryatthegraduatelevel,buthavenotnecessarilybeenexposedtoadvancednotionsoffunctionalanalysis,algebraorgeometryUsefulbackgroundmaterialiscollectedintheappendicesandexercisesarealsoincludedthroughouttotestthereader’sunderstandingEnumerativetechniques,stochasticanalysis,largedeviations,concentrationinequalities,disintegrationandLiealgebrasallareintroducedinthetext,whichwillenablereaderstoapproachtheresearchliteraturewithconfidencegregwandersonisProfessorofMathematicsattheUniversityofMinnesotaaliceguionnetisCNRSResearchDirectorattheEcoleNormaleSupe´rieureinLyon(ENSLyon)oferzeitouniisProfessorofMathematicsatboththeUniversityofMinnesotaandtheWeizmannInstituteofScienceinRehovot,IsraelCAMBRIDGESTUDIESINADVANCEDMATHEMATICSEditorialBoard:BBolloba´s,WFulton,AKatok,FKirwan,PSarnak,BSimon,BTotaroAllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPressForacompleteserieslistingvisit:wwwcambridgeorgseriessSeriesaspcode=CSAMAlreadypublishedAJBerrickMEKeatingAnintroductiontoringsandmoduleswithKtheoryinviewSMorosawaetalHolomorphicdynamicsAJBerrickMEKeatingCategoriesandmoduleswithKtheoryinviewKSatoLe´vyprocessesandinfinitelydivisibledistributionsHHidaModularformsandGaloiscohomologyRIorioVIorioFourieranalysisandpartialdifferentialequationsRBleiAnalysisinintegerandfractionaldimensionsFBorceuxGJanelidzeGaloistheoriesBBolloba´sRandomgraphs(ndEdition)RMDudleyRealanalysisandprobability(ndEdition)TSheilSmallComplexpolynomialsCVoisinHodgetheoryandcomplexalgebraicgeometry,ICVoisinHodgetheoryandcomplexalgebraicgeometry,IIVPaulsenCompletelyboundedmapsandoperatoralgebrasFGesztesyHHoldenSolitonequationsandtheiralgebrogeometricsolutions,ISMukaiAnintroductiontoinvariantsandmoduliGTourlakisLecturesinlogicandsettheory,IGTourlakisLecturesinlogicandsettheory,IIRABaileyAssociationschemesJCarlson,SMu¨llerStachCPetersPeriodmappingsandperioddomainsJJDuistermaatJACKolkMultidimensionalrealanalysis,IJJDuistermaatJACKolkMultidimensionalrealanalysis,IIMCGolumbicANTrenkTolerancegraphsLHHarperGlobalmethodsforcombinatorialisoperimetricproblemsIMoerdijkJMrcˇunIntroductiontofoliationsandLiegroupoidsJKolla´r,KESmithACortiRationalandnearlyrationalvarietiesDApplebaumLe´vyprocessesandstochasticcalculus(stEdition)BConradModularformsandtheRamanujanconjectureMSchechterAnintroductiontononlinearanalysisRCarterLiealgebrasoffiniteandaffinetypeHLMontgomeryRCVaughanMultiplicativenumbertheory,IIChavelRiemanniangeometry(ndEdition)DGoldfeldAutomorphicformsandLfunctionsforthegroupGL(n,R)MBMarcusJRosenMarkovprocesses,Gaussianprocesses,andlocaltimesPGilleTSzamuelyCentralsimplealgebrasandGaloiscohomologyJBertoinRandomfragmentationandcoagulationprocessesEFrenkelLanglandscorrespondenceforloopgroupsAAmbrosettiAMalchiodiNonlinearanalysisandsemilinearellipticproblemsTTaoVHVuAdditivecombinatoricsEBDaviesLinearoperatorsandtheirspectraKKodairaComplexanalysisTCeccheriniSilberstein,FScarabottiFTolliHarmonicanalysisonfinitegroupsHGeigesAnintroductiontocontacttopologyJFarautAnalysisonLiegroups:AnIntroductionEParkComplextopologicalKtheoryDWStroockPartialdifferentialequationsforprobabilistsAKirillov,JrAnintroductiontoLiegroupsandLiealgebrasFGesztesyetalSolitonequationsandtheiralgebrogeometricsolutions,IIEdeFariaWdeMeloMathematicaltoolsforonedimensionaldynamicsDApplebaumLe´vyprocessesandstochasticcalculus(ndEdition)TSzamuelyGaloisgroupsandfundamentalgroupsAnIntroductiontoRandomMatricesGREGWANDERSONUniversityofMinnesotaALICEGUIONNETEcoleNormaleSupe´rieuredeLyonOFERZEITOUNIUniversityofMinnesotaandWeizmannInstituteofScienceCAMBRIDGEUNIVERSITYPRESSCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo,Delhi,Dubai,TokyoCambridgeUniversityPressTheEdinburghBuilding,CambridgeCBRU,UKFirstpublishedinprintformatISBNISBN©GWAnderson,AGuionnetandOZeitouniInformationonthistitle:wwwcambridgeorgThispublicationisincopyrightSubjecttostatutoryexceptionandtotheprovisionofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPressCambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsforexternalorthirdpartyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriatePublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwwwcambridgeorgeBook(EBL)HardbackToMeredith,BenoitandNaomiContentsPrefacepagexiiiIntroductionRealandcomplexWignermatricesRealWignermatrices:traces,momentsandcombinatoricsThesemicircledistribution,CatalannumbersandDyckpathsProof#ofWigner’sTheoremProofofLemma:wordsandgraphsProofofLemma:sentencesandgraphsSomeusefulapproximationsMaximaleigenvaluesandFu¨redi–Komlo´senumerationCentrallimittheoremsformomentsComplexWignermatricesConcentrationforfunctionalsofrandommatricesandlogarithmicSobolevinequalitiesSmoothnesspropertiesoflinearfunctionsoftheempiricalmeasureConcentrationinequalitiesforindependentvariablessatisfyinglogarithmicSobolevinequalitiesConcentrationforWignertypematricesStieltjestransformsandrecursionsviiviiiCONTENTSGaussianWignermatricesGeneralWignermatricesJointdistributionofeigenvaluesintheGOEandtheGUEDefinitionandpreliminarydiscussionoftheGOEandtheGUEProofofthejointdistributionofeigenvaluesSelberg’sintegralformulaandproofof()Jointdistributionofeigenvalues:alternativeformulationSuperpositionanddecimationrelationsLargedeviationsforrandommatricesLargedeviationsfortheempiricalmeasureLargedeviationsforthetopeigenvalueBibliographicalnotesHermitepolynomials,spacingsandlimitdistributionsfortheGaussianensemblesSummaryofmainresults:spacingdistributionsinthebulkandedgeofthespectrumfortheGaussianensemblesLimitresultsfortheGUEGeneralizations:limitformulasfortheGOEandGSEHermitepolynomialsandtheGUETheGUEanddeterminantallawsPropertiesoftheHermitepolynomialsandoscillatorwavefunctionsThesemicirclelawrevisitedCalculationofmomentsof¯LNTheHarer–ZagierrecursionandLedoux’sargumentQuickintroductiontoFredholmdeterminantsThesetting,fundamentalestimatesanddefinitionoftheFredholmdeterminantDefinitionoftheFredholmadjugant,FredholmresolventandafundamentalidentityCONTENTSixGapprobabilitiesatandproofofTheoremThemethodofLaplaceEvaluationofthescalinglimit:proofofLemmaAcomplement:determinantalrelationsAnalysisofthesinekernelGeneraldifferentiationformulasDerivationofthedifferentialequations:proofofTheoremReductiontoPainleve´VEdgescaling:proofofTheoremVagueconvergenceofthelargesteigenvalue:proofofTheoremSteepestdescent:proofofLemmaPropertiesoftheAiryfunctionsandproofofLemmaAnalysisoftheTracy–WidomdistributionandproofofTheoremThefirststandardmovesofthegameThewrinkleinthecarpetLinkagetoPainleve´IILimitingbehavioroftheGOEandtheGSEPfaffiansandgapprobabilitiesFredholmrepresentationofgapprobabilitiesLimitcalculationsDifferentialequationsBibliographicalnotesSomegeneralitiesJointdistributionofeigenvaluesintheclassicalmatrixensemblesIntegrationformulasforclassicalensemblesManifolds,volumemeasuresandthecoareaformulaxCONTENTSAnintegrationformulaofWeyltypeApplicationsofWeyl’sformulaDeterminantalpointprocessesPointprocesses:basicdefinitionsDeterminantalprocessesDeterminantalprojectionsTheCLTfordeterminantalprocessesDeterminantalprocessesassociatedwitheigenvaluesTranslationinvariantdeterminantalprocessesOnedimensionaltranslationinvariantdeterminantalprocessesConvergenceissuesExamplesStochasticanalysisforrandommatricesDyson’sBrownianmotionAdynamicalversionofWigner’sTheoremDynamicalcentrallimittheoremsLargedeviationboundsConcentrationofmeasureandrandommatricesConcentrationinequalitiesforHermitianmatriceswithindependententriesConcentrationinequalitiesformatriceswithdependententriesTridiagonalmatrixmodelsandtheβensemblesTridiagonalrepresentationofβensemblesScalinglimitsattheedgeofthespectrumBibliographicalnotesFreeprobabilityIntroductionandmainresultsNoncommutativelawsandnoncommutativeprobabilityspacesCONTENTSxiAlgebraicnoncommutativeprobabilityspacesandlawsC∗probabilityspacesandtheweak*topologyW∗probabilityspacesFreeindependenceIndependenceandfreeindependenceFreeindependenceandcombinatoricsConsequenceoffreeindependence:freeconvolutionFreecentrallimittheoremFreenessforunboundedvariablesLinkwithrandommatricesConvergenceoftheoperatornormofpolynomialsofindependentGUEmatricesBibliographicalnotesAppendicesALinearalgebrapreliminariesAIdentitiesandboundsAPerturbationsfornormalandHermitianmatricesANoncommutativematrixLpnormsABriefreviewofresultantsanddiscriminantsBTopologicalpreliminariesBGeneralitiesBTopologicalvectorspacesandweaktopologiesBBanachandPolishspacesBSomeelementsofanalysisCProbabilitymeasuresonPolishspacesCGeneralitiesCWeaktopologyDBasicnotionsoflargedeviationsETheskewfieldHofquaternionsandmatrixtheoryoverFEMatrixterminologyoverFandfactorizationtheoremsxiiCONTENTSEThespectraltheoremandkeycorollariesEAspecializedresultonprojectorsEAlgebraforcurvaturecomputationsFManifoldsFManifoldsembeddedinEuclideanspaceFProofofthecoareaformulaFMetrics,connections,curvature,Hessians,andtheLaplace–BeltramioperatorGAppendixonoperatoralgebrasGBasicdefinitionsGSpectralpropertiesGStatesandpositivityGvonNeumannalgebrasGNoncommutativefunctionalcalculusHStochasticcalculusnotionsReferencesGeneralconventionsandnotationIndexPrefaceThestudyofrandommatrices,andinparticularthepropertiesoftheireigenvalues,hasemergedfromtheapplications,firstindataanalysisandlaterasstatisticalmodelsforheavynucleiatomsThus,thefieldofrandommatricesowesitsexistencetoapplicationsOvertheyears,however,itbecameclearthatmodelsrelatedtorandommatricesplayanimportantroleinareasofpuremathematicsMoreover,thetoolsusedinthestudyofrandommatricescamethemselvesfromdifferentandseeminglyunrelatedbranchesofmathematicsAtthispointintime,thetopichasevolvedenoughthatthenewcomer,especiallyifcomingfromthefieldofprobabilitytheory,facesaformidableandsomewhatconfusingtaskintryingtoaccesstheresearchliteratureFurthermore,thebackgroundexpectedofsuchanewcomerisdiverse,andoftenhastobesupplementedbeforeaseriousstudyofrandommatricescanbeginWebelievethatmanypartsofthefieldofrandommatricesarenowdevelopedenoughtoenableonetoexposethebasicideasinasystematicandcoherentwayIndeed,suchatreatise,gearedtowardtheoreticalphysicists,hasexistedforsometime,intheformofMehta’ssuperbbookMehOurgoalinwritingthisbookhasbeentopresentarigorousintroductiontothebasictheoryofrandommatrices,includingfreeprobability,thatissufficientlyselfcontainedtobeaccessibletograduatestudentsinmathematicsorrelatedscienceswhohavemasteredprobabilitytheoryatthegraduatelevel,buthavenotnecessarilybeenexposedtoadvancednotionsoffunctionalanalysis,algebraorgeometryAlongtheway,enoughtechniquesareintroducedthatwehopewillallowreaderstocontinuetheirjourneyintothecurrentresearchliteratureThisprojectstartedasnotesforaclassonrandommatricesthattwoofus(GAandOZ)taughtintheUniversityofMinnesotainthefallof,andnotesforacourseintheprobabilitysummerschoolinStFlourtaughtbyAGinthexiiixivPREFACEsummerofThecommentsofparticipantsinthesecourses,andinparticularABandyopadhyay,HDong,KHoffmanCredner,AKlenke,DStantonandPMZamfir,wereextremelyusefulAsthesenotesevolved,wetaughtfromthemagainattheUniversityofMinnesota,theUniversityofCaliforniaatBerkeley,theTechnionandtheWeizmannInstitute,andreceivedmoremuchappreciatedfeedbackfromtheparticipantsinthosecoursesFinally,whenexpandingandrefiningthesecoursenotes,wehaveprofitedfromthecommentsandquestionsofmanycolleaguesWewouldlikeinparticulartothankGBenArous,FBenaychGeorges,PBiane,PDeift,ADembo,PDiaconis,UHaagerup,VJones,MKrishnapur,YPeres,RPinsky,GPisier,BRider,DShlyakhtenko,BSolel,ASoshnikov,RSpeicher,TSuidan,CTracy,BViragandDVoiculescufortheirsuggestions,correctionsandpatienceinansweringourquestionsorexplainingtheirworktousOfcourse,anyremainingmistakesandunclearpassagesarefullyourresponsibilityMINNEAPOLIS,MINNESOTALYON,FRANCEREHOVOT,ISRAELGREGANDERSONALICEGUIONNETOFERZEITOUNIIntroductionThisbookisconcernedwithrandommatricesGiventheubiquitousrolethatmatricesplayinmathematicsanditsapplicationinthesciencesandengineering,itseemsnaturalthattheevolutionofprobabilitytheorywouldeventuallypassthroughrandommatricesThereality,however,hasbeenmorecomplicated(andinteresting)Indeed,thestudyofrandommatrices,andinparticularthepropertiesoftheireigenvalues,hasemergedfromtheapplications,firstindataanalysis(intheearlydaysofstatisticalsciences,goingbacktoWishartWis),andlaterasstatisticalmodelsforheavynucleiatoms,beginningwiththeseminalworkofWignerWigStillmotivatedbyphysicalapplications,attheablehandsofWigner,Dyson,Mehtaandcoworkers,amathematicaltheoryofthespectrumofrandommatricesbegantoemergeintheearlys,andlinkswithvariousbranchesofmathematics,includingclassicalanalysisandnumbertheory,wereestablishedWhilemuchprogresswasinitiallyachievedusingenumerativecombinatorics,gradually,sophisticatedandvariedmathematicaltoolswereintroduced:Fredholmdeterminants(inthes),diffusionprocesses(inthes),integrablesystems(inthesandearlys),andtheRiemann–Hilbertproblem(inthes)allmadetheirappearance,aswellasnewtoolssuchasthetheoryoffreeprobability(inthes)Thiswidearrayoftools,whileattestingtothevitalityofthefield,presents,however,severalformidableobstaclestothenewcomer,andeventotheexpertprobabilistIndeed,whilemuchoftherecentresearchusessophisticatedprobabilistictools,itbuildsonlayersofcommonknowledgethat,intheaggregate,fewpeoplepossessOurgoalinthisbookistopresentarigorousintroductiontothebasictheoryofrandommatricesthatwouldbesufficientlyselfcontainedtobeaccessibletograduatestudentsinmathematicsorrelatedscienceswhohavemasteredprobabilitytheoryatthegraduatelevel,buthavenotnecessarilybeenexposedtoadvancednotionsoffunctionalanalysis,algebraorgeometryWithsuchreadersinmind,weINTRODUCTIONpresentsomebackgroundmaterialintheappendices,thatnoviceandexpertalikecanconsultmostmaterialintheappendicesisstatedwithoutproof,althoughthedetailsofsomespecializedcomputationsareprovidedKeepinginmindourstatedemphasisonaccessibilityovergenerality,thebookisessentiallydividedintotwopartsInChaptersand,wepresentaselfcontainedanalysisofrandommatrices,quicklyfocusingontheGaussianensemblesandculminatinginthederivationofthegapprobabilitiesatandtheTracy–WidomlawThesechapterscanbereadwithverylittlebackgroundknowledge,andareparticularlysuitableforanintroductorystudyInthesecondpartofthebook,Chaptersand,weusemoreadvancedtechniques,requiringmoreextensivebackground,toemphasizeandgeneralizecertainaspectsofthetheory,andtointroducethetheoryoffreeprobabilitySowhatisarandommatrix,andwhatquestionsareweabouttostudyThroughout,letF=RorF=C,andsetβ=intheformercaseandβ=inthelatter(InSection,wewillalsoconsiderthecaseF=H,theskewfieldofquaternions,seeAppendixEfordefinitionsanddetails)LetMatN(F)denotethespaceofNbyNmatriceswithentriesinF,andletH(β)Ndenotethesubsetofselfadjointmatrices(ie,realsymmetricifβ=andHermitianifβ=)

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