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1、ContentsPrefaceixAcknowledgmentsxChapter1.Preparatorymaterial1x1.1.Areviewofprobabilitytheory2x1.2.Stirling'sformula41x1.3.EigenvaluesandsumsofHermitianmatrices45Chapter2.Randommatrices65x2.1.Concentrationofmeasure66x2.2.Thecentrallimittheorem93x2.3.Theoperatornormofrando
2、mmatrices124x2.4.Thesemicircularlaw159x2.5.Freeprobability184x2.6.Gaussianensembles217x2.7.Theleastsingularvalue247x2.8.Thecircularlaw263Chapter3.Relatedarticles279x3.1.BrownianmotionandDysonBrownianmotion280x3.2.TheGolden-Thompsoninequality299viiviiiContentsx3.3.TheDyson
3、andAirykernelsofGUEviasemiclassicalanalysis307x3.4.ThemesoscopicstructureofGUEeigenvalues315Bibliography323Index331PrefaceInthewinterof2010,Itaughtatopicsgraduatecourseonrandommatrixtheory,thelecturenotesofwhichthenformedthebasisforthistext.Thiscoursewasinspiredbyrecentde
4、velopmentsinthesubject,particularlywithregardtotherigorousdemonstrationofuniversallawsforeigenvaluespacingdistributionsofWignermatri-ces(seetherecentsurvey[Gu2009b]).Thiscoursedoesnotdirectlydiscusstheselaws,butinsteadfocusesonmorefoundationaltopicsinrandommatrixtheoryupo
5、nwhichthemostrecentworkhasbeenbased.Forinstance,the rstpartofthecourseisdevotedtobasicprobabilistictoolssuchasconcentrationofmeasureandthecentrallimittheorem,whicharethenusedtoestablishbasicresultsinran-dommatrixtheory,suchastheWignersemicirclelawonthebulkdistributionofei
6、genvaluesofaWignerrandommatrix,orthecir-cularlawonthedistributionofeigenvaluesofaniidmatrix.Otherfundamentalmethods,suchasfreeprobability,thetheoryofdeter-minantalprocesses,andthemethodofresolvents,arealsocoveredinthecourse.ThistextbeginsinChapter1withareviewoftheaspectso
7、fprob-abilitytheoryandlinearalgebraneededforthetopicsofdiscussion,butassumessomeexistingfamiliaritywithbothtopics,aswellasa rst-yeargraduate-levelunderstandingofmeasuretheory(ascoveredforinstanceinmybooks[Ta2011,Ta2010]).IfthistextisusedixxPrefacetogiveagraduatecourse,the
8、nChapter1canlargelybeassignedasreadingmaterial(orreviewedasnecessary),withthelecturesthenbeginni
