Lectures on Random Matrices

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1、LecturesonRandomMatricesFraydounRezakhanlouDepartmetofMathematics,UCBerkeleyDecember19,2010Chapter1:IntroductionChapter2:WignerEnsembleandSemicircleLawChapter3:GaussianEnsemblesGOEandGUEChapter4:CorrelationsandEdgeDistributionsforGaussianEnsemblesChapter5:D

2、ysonBrownianMotionChapter6:Universality11IntroductionDisorderedsystemsinquantummechanicsaremodeledbySchr•odingeroperatorswithran-dompotentials.Asaclassicalexample,considertheAndersonmodeldescribingelectronpropagationinadisorderedenvironment.TheassociatedSch

3、r•odingeroperatorisoftheformH=
+V,wherethepotentialVisrandomandtheparameterrepresentsthestrengthofdisorder.TheenergylevelsoftheelectronaregivenbytheeigenvaluesoftheoperatorHandduetotherandomnessofthepotential,wearemostlyinterestedintheirstatisticalproper

4、ties.AccordingtotheuniversalityconjectureforrandomSchr•odingeroper-ators,therearetwodistinctiveregimesdependingonthedisorderstrength.Inthestrongdisorderregime,theeigenfunctionsarelocalizedandthelocalspectralstatisticsisPoisson.Intheweakdisorderregime,theei

5、genfunctionsaredelocalizedandarepulsivepotentialgovernstheinteractionbetweeneigenvalues.InthelatticeapproximationoftheSchr•odingeroperator
+Visreplacedwithalargesymmetricsparsematrixwithrandomdiagonalentries.Wignerproposedtostudythestatisticsofeigenvalueso

6、flargerandommatricesasamodelfortheenergylevelsofheavynuclei.ForaWignerensemblewetakealargesymmetricorhermitianNNmatrix[hij]wherefhij:ijgareindependentidenticallydistributedrandomvariablesofmeanzeroandvarianceN1.ThecentralquestionforWignerensembleistheuni

7、versalityconjecturewhichassertsthatthelocalstatisticsoftheeigenvaluesareindependentofthedistributionsoftheentriesasNgetslarge.ThislocalstatisticscanbecalculatedwhentheentrydistributionischosentobeGaussian.ThedensityofeigenvaluesinlargeNlimitisgivenbythecele

8、bratedWignersemicirclelawintheinterval[2;2].Jointdistributionofeigenvaluesawayfromtheedges2hasadeterminantalstructureandisobtainedfromasinekernel.ThesinekernelisreplacedwiththeAirykernelneartheedges