Introduction to the Random Matrix Theory Gaussian

arXiv:math-ph/0412017v17Dec2004IntroductiontotheRandomMatrixTheory:GaussianUnitaryEnsembleandBeyondYanV.FyodorovDepartmentofMathematicalSciences,BrunelUniversity,Uxbridge,UB83PH,UnitedKingdom.AbstractTheselecturesprovideaninformalintroductionintothenotionsandtoolsusedtoanalyzestatisticalpropertiesofeigenvaluesoflargerandomHermitianmatrices.Afterdevelopingthegeneralmachineryoforthogonalpolynomialmethod,westudyinmostdetailGaussianUnitaryEnsemble(GUE)asaparadigmaticexample.Inparticular,wediscussPlancherel-RotachasymptoticsofHermitepolynomialsinvariousregimesandemployitinspectralanalysisoftheGUE.Inthelastpartofthecoursewediscussgeneralrelationsbetweenorthogonalpolynomialsandcharacteristicpolynomialsofrandommatriceswhichisanactiveareaofcurrentresearch.1PrefaceGaussianEnsemblesofrandomHermitianorrealsymmetricmatricesalwaysplayedaprominentroleinthedevelopmentandapplicationsofRandomMatrixTheory.GaussianEnsemblesareuniquelysingledoutbythefactthattheybelongbothtothefamilyofinvariantensembles,andtothefamilyofensembleswithindependent,identicallydistributed(i.i.d)entries.Ingeneral,mathematicalmethodsusedtotreatthosetwofamiliesareverydifferent.Infact,allrandommatrixtechniquesandideascanbemostclearlyandconsistentlyintroducedusingGaussiancaseasaparadigmaticexample.Inthepresentsetoflectureswemainlyconcentrateonconsequencesoftheinvarianceofthecorrespondingprobabilitydensityfunction,leavingasidemethodsofexploitingstatisticalindependenceofmatrixentries.Underthesecircumstancesthemethodoforthogonalpolynomialsisthemostadequateone,andfortheGaussiancasetherelevantpolynomialsareHermitepolynomials.BeingmostlyinterestedinthelimitoflargematrixsizeswewillspendaconsiderableamountoftimeinvestigatingvariousasymptoticregimesofHermitepolynomials,sincethelatteraremainbuildingblocksofvariouscorrelationfunctionsofinterest.InthelastpartofourlecturecoursewewilldiscusswhystatisticsofcharacteristicpolynomialsofrandomHermitianmatricesturnsouttobeinterestingandinformativetoinvestigate,andwillmakeacontactwithrecentresultsinthedomain.ThepresentationisquiteinformalinthesensethatIwillnottrytoprovevariousstatementsinfullrigororgenerality.Iratherattemptoutliningthemainconcepts,ideasandtechniquespreferringagoodilluminatingexampletoageneralproof.Amuchmorerigorousanddetailedexpositioncanbefoundinthecitedliterature.Iwillalsofrequentlyemploythesymbol∝.Inthepresentsetoflecturesitalwaysmeansthattheexpressionfollowing∝containsamultiplicativeconstantwhichisofsecondaryimportanceforourgoalsandcanberestoredwhennecessary.2IntroductionIntheselecturesweusethesymbolTtodenotematrixorvectortranspositionandtheasterisk∗todenoteHermitianconjugation.Inthepresentsectionthebarzdenotescomplexconjugation.1LetusstartwithasquarecomplexmatrixˆZofdimensionsN×N,withcomplexentrieszij=xij+iyij,1≤i,j≤N.Everysuchmatrixcanbeconvenientlylookedatasapointina2N2-dimensionalEuclideanspacewithrealCartesiancoordinatesxij,yij,andthelengthelementinthisspaceisdefinedinastandardwayas:(ds)2=TrdˆZdˆZ∗=Xijdzijdzij=Xij(dx)2ij+(dy)2ij.(1)Asiswell-known(seee.g.[1])anysurfaceembeddedinanEuclideanspaceinheritsanaturalRiemannianmetricfromtheunderlyingEuclideanstructure.Namely,letthecoordinatesinan−dimensionalEuclideanspacebe(x1,...,xn),andletak−dimensionalsurfaceembeddedinthisspacebeparameterizedintermsofcoordinates(q1,...,qk),k≤nasxi=xi(q1,...,qk),i=1,...n.ThentheRiemannianmetricgml=glmonthesurfaceisdefinedfromtheEuclideanlengthelementaccordingto(ds)2=nXi=1(dxi)2=nXi=1kXm=1∂xi∂qmdqm!2=kXm,l=1gmndqmdql.(2)Moreover,suchaRiemannianmetricinducesthecorrespondingintegrationmeasureonthesurface,withthevolumeelementgivenbydμ=p|g|dq1...dqk,g=det(gml)kl,m=1.(3)Fork=nthesearejustthefamiliarformulaeforthelengthsandvolumeassociatedwithchangeofcoordinatesinanEuclideanspace.Forexample,forn=2wecanpassfromCartesiancoordinates−∞x,y∞topolarcoordinatesr0,0≤θ2πbyx=rcosθ,y=rsinθ,sothatdx=drcosθ−rsinθdθ,dy=drsinθ+rcosθdθ,andtheRiemannianmetricisdefinedby(ds)2=(dx)2+(dy)2=(dr)2+r2(dθ)2.Wefindthatg11=1,g12=g21=0,g22=r2,andthevolumeelementoftheintegrationmeasureinthenewcoordinatesisdμ=rdrdθ;asitshouldbe.Asthesimplestexampleofa“surface”withkn=2embeddedinsuchatwo-dimensionalspaceweconsideracircler=R=const.Weimmediatelyseethatthelengthelement(ds)2restrictedtothis“surface”is(ds)2=R2(dθ)2,sothatg11=R2,andtheintegrationmeasureinducedonthesurfaceiscorrespondinglydμ=Rdθ.The“surface”integrationthengivesthetotal“volume”oftheembeddedsurface(i.e.circlelength2πR).zyxFigure1:Thesphericalcoordinatesforatwodimensionalsphereinthethree-dimensionalEuclideanspace.2Similarly,wecanconsideratwo-dimensional(k=2)sphereR2=x2+y2+z2embeddedinathree-dimensionalEuclideanspace(n=3)withcoordinatesx,y,zandlengthelement(ds)2=(dx)2+(dy)2+(dz)2.Anaturalparameterizationofthepointsonthesphereispossibleintermsofthesphericalcoordinatesφ,θ(seeFig.1)x=Rsinθcosφ,y=Rsinθsinφ,z=Rcosθ;0≤θ≤π,0≤φ2π,whichresultsin(ds)2=R2(dθ)2+R2sin2θ(dφ)2.Hencethematrixelementsofthemetricareg11=R2,g12=g21=0,g22=R2sin2θ,andthecorresponding“volumeelement”onthesphereisthefamiliarelementaryareadμ=