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1、AnIntroductiontoRandomMatricesAnIntroductiontoRandomMatricesGregW.AndersonUniversityofMinnesotaAliceGuionnetENSLyonOferZeitouniUniversityofMinnesotaandWeizmannInstituteofSciencecopyrightinformationhereToMeredith,BenandNaomiContentsPrefacepagexiii1Introduction12RealandComplexWignermatrices
2、62.1RealWignermatrices:traces,momentsandcombinatorics62.1.1Thesemicircledistribution,Catalannumbers,andDyckpaths72.1.2Proof#1ofWigner'sTheorem2.1.1102.1.3ProofofLemma2.1.6:WordsandGraphs112.1.4ProofofLemma2.1.7:SentencesandGraphs172.1.5Someusefulapproximations212.1.6MaximaleigenvaluesandF
3、¨uredi-Koml′osenumeration232.1.7Centrallimittheoremsformoments292.2ComplexWignermatrices352.3ConcentrationforfunctionalsofrandommatricesandlogarithmicSobolevinequalities382.3.1Smoothnesspropertiesoflinearfunctionsoftheempiricalmeasure382.3.2Concentrationinequalitiesforindependentvariables
4、satisfyinglogarithmicSobolevinequalities392.3.3ConcentrationforWigner-typematrices422.4Stieltjestransformsandrecursions43viiviiiCONTENTS2.4.1GaussianWignermatrices462.4.2GeneralWignermatrices472.5JointdistributionofeigenvaluesintheGOEandtheGUE512.5.1De?nitionandpreliminarydiscussionoftheG
5、OEandtheGUE512.5.2Proofofthejointdistributionofeigenvalues542.5.3Selberg'sintegralformulaandproofof(2.5.4)592.5.4Jointdistributionofeigenvalues-alternativeformu-lation652.5.5Superpositionanddecimationrelations662.6Largedeviationsforrandommatrices712.6.1Largedeviationsfortheempiricalmeasur
6、e722.6.2Largedeviationsforthetopeigenvalue822.7Bibliographicalnotes863Hermitepolynomials,spacings,andlimitdistributionsfortheGaus-sianensembles913.1Summaryofmainresults:spacingdistributionsinthebulkandedgeofthespectrumfortheGaussianensembles913.1.1LimitresultsfortheGUE913.1.2Generalizatio
7、ns:limitformulasfortheGOEandGSE943.2HermitepolynomialsandtheGUE953.2.1TheGUEanddeterminantallaws953.2.2PropertiesoftheHermitepolynomialsandoscillatorwave-functions1003.3Thesemicirclelawrevisited1033.3.1CalculationofmomentsofLˉN1033.3.2TheHarer–ZagierrecursionandLedo
