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1、TopicsinrandommatrixtheoryTerenceTaoDepartmentofMathematics,UCLA,LosAngeles,CA90095E-mailaddress:tao@math.ucla.eduToGarthGaudry,whosetmeontheroad;Tomyfamily,fortheirconstantsupport;Andtothereadersofmyblog,fortheirfeedbackandcontributions.ContentsPrefaceixAcknowledgmentsxC
2、hapter1.Preparatorymaterial1x1.1.Areviewofprobabilitytheory2x1.2.Stirling'sformula41x1.3.EigenvaluesandsumsofHermitianmatrices45Chapter2.Randommatrices65x2.1.Concentrationofmeasure66x2.2.Thecentrallimittheorem93x2.3.Theoperatornormofrandommatrices124x2.4.Thesemicircularla
3、w159x2.5.Freeprobability183x2.6.Gaussianensembles217x2.7.Theleastsingularvalue246x2.8.Thecircularlaw263Chapter3.Relatedarticles277x3.1.BrownianmotionandDysonBrownianmotion278x3.2.TheGolden-Thompsoninequality297viiviiiContentsx3.3.TheDysonandAirykernelsofGUEviasemiclassica
4、lanalysis305x3.4.ThemesoscopicstructureofGUEeigenvalues313Bibliography321Index329PrefaceInthespringof2010,Itaughtatopicsgraduatecourseonrandommatrixtheory,thelecturenotesofwhichthenformedthebasisforthistext.Thiscoursewasinspiredbyrecentdevelopmentsinthesubject,particularl
5、ywithregardtotherigorousdemonstrationofuniversallawsforeigenvaluespacingdistributionsofWignermatri-ces(seetherecentsurvey[Gu2009b]).Thiscoursedoesnotdirectlydiscusstheselaws,butinsteadfocusesonmorefoundationaltopicsinrandommatrixtheoryuponwhichthemostrecentworkhasbeenbase
6、d.Forinstance,the rstpartofthecourseisdevotedtobasicprobabilistictoolssuchasconcentrationofmeasureandthecentrallimittheorem,whicharethenusedtoestablishbasicresultsinran-dommatrixtheory,suchastheWignersemicirclelawonthebulkdistributionofeigenvaluesofaWignerrandommatrix,ort
7、hecir-cularlawonthedistributionofeigenvaluesofaniidmatrix.Otherfundamentalmethods,suchasfreeprobability,thetheoryofdeter-minantalprocesses,andthemethodofresolvents,arealsocoveredinthecourse.ThistextbeginsinChapter1withareviewoftheaspectsofprob-abilitytheoryandlinearalgebr
8、aneededforthetopicsofdiscussion,butassumessomeexistingfamiliaritywithbothtopics,aswillasa rst-ye
