Arbitrary Rotation Invariant Random Matrix Ensembl

arXiv:math-ph/0606014v13Jun2006ArbitraryRotationInvariantRandomMatrixEnsemblesandSupersymmetryThomasGuhrMatematiskFysik,LTH,LundsUniversitet,Box118,22100Lund,SwedenAbstract.WegeneralizethesupersymmetrymethodinRandomMatrixTheorytoarbitraryrotationinvariantensembles.Ourexactapproachfurtherextendsapreviouscontributioninwhichweconstructedasupersymmetricrepresentationfortheclassofnorm–dependentRandomMatrixEnsembles.Here,wederiveasupersymmetricformulationunderverygeneralcircumstances.Aprojectorisidentifiedthatprovidesthemappingoftheprobabilitydensityfromordinarytosuperspace.Furthermore,itisdemonstratedthatsettingupthetheoryinFouriersuperspacehasconsiderableadvantages.Generalandexactexpressionsforthecorrelationfunctionsaregiven.Wealsoshowhowtheuseofhyperbolicsymmetrycanbecircumventedinthepresentcontextinwhichthenon–linearσmodelisnotused.WeconstructexactsupersymmetricintegralrepresentationsofthecorrelationfunctionsforarbitrarypositionsoftheimaginaryincrementsintheGreenfunctions.PACSnumbers:05.45.Mt,05.30.-d,02.30.Px1.IntroductionThesupersymmetrymethodisnowadaysindispensableforthediscussionofvariousadvancedtopicsinthetheoryofdisorderedsystems[1,2],anditbecameequallyimportantinnumerousrandommatrixapproachestocomplexsystemsingeneral[3,4,5,6].RandomMatrixTheory(RMT)asoriginallyformulatedinordinaryspacedoesnotrelyonGaussianprobabilitydensities.ItisonlyimportantthattheRandomMatrixEnsemblesareinvariantunderbasisrotations.Gaussianprobabilitydensitiesarehighlyconvenientincalculations,butotherprobabilitydensitiesarealsopossible,andsomeofthosewerealreadyconsideredintheearlydaysofRMT[7].Ontheotherhand,thesupersymmetricformulationswereconstructedforGaussianprobabilitydensities[1,2,8]bymeansofaHubbard–Stratonovichtransformation.Thus,thequestionarisesnaturallywhethertheHubbard–StratonovichtransformationrestrictstheuseofsupersymmetrytotheGaussianformoftheprobabilitydensities.Weaddressthisprobleminthepresentcontribution.Wewillshowthatthesupersymmetrymethodisnotatallrestrictedinthisway,andwewillderivesupersymmetricformulationsofRMTforarbitraryrotationinvariantRandomMatrixEnsembles.ArbitraryRotationInvariantRandomMatrixEnsemblesandSupersymmetry2Wefocusonconceptualandstructuralissues.Inparticular,wearenotaimingatasymptoticresultsintheinverselevelnumberasfollowingfromthesupersymmetricnon–linearσmodel[1,2,3].ThislatterapproachwasusedinRef.[9]toshowuniversalityforinfinitelevel–numberinthecaseofnon–Gaussianprobabilitydensities.Here,however,ourgoalisdifferent:weaddressthefullproblemtoachieveexact,i.e.non–asymptoticresults.Inapreviousstudy[10],wepresentedsupersymmetricrepresentationsfornorm–dependentensembles,wheretheprobabilitydensitiesarefunctionsofthetracedsquaredrandommatricesonly.Althoughaseriesofinterestinginsightsarerevealedalreadyinthiscase,thederivationcanbedonewithoutactuallyemployingdeepfeaturesofsupersymmetry.Thisisnotsointhepresentcontributionwhichaimsatageneralconstruction.ThemethodsneededareverydifferentfromtheonesofRef.[10].Here,wehavetoexplorethealgebraicstructureofsuperspace.Onecanalsomotivatethepresentinvestigationfromtheviewpointofapplications.Werefertheinterestedreadertothecontribution[10]andtheliteraturequotedtherein.Ourgoaltoperformaconceptualstudydoesnotpreventusfromgivinggeneralexpressionsforthecorrelationfunctions,butwerefrainfromlookingtoomuchintoapplicationsanddeferthisaspecttofuturework.Itwillnotbesurprisingforthosewhoalreadyhaveexpertiseinsupersymmetrythatageneralizationasoutlinedaboverequiresananalysisofconvergencepropertiesandthusleadsinevitablytotheissueofwhatkindofsymmetriesthetheoryinsuperspaceshouldhave.ItwasarguedinRef.[11]thathyperbolicsymmetry,i.e.groupscomprisingcompactandnon–compactdegreesoffreedom,arenecessaryifoneistosetupanon–linearσmodelinordinaryspace.Thislineofreasoningcarriesovertosuperspace[1,2],seealsotherecentreviewinRef.[8].Wejustifyaprocedureforhowtoavoidhyperbolicsymmetryintheframeworkofoursupersymmetricmodels.Thenecessitytointroducehyperbolicsymmetryisexclusivelyrootedinthenon–linearσmodel,notinsupersymmetryassuch.Ifoneaimsatexact,i.e.non–asymptoticresults,compactsupergroupssuffice.Forvariousreasons,includingsomerelatedtoconvergencequestions,wefinditadvantageoustomapthetheoryontoFouriersuperspace.Moreover,werestrictourselvestounitaryRandomMatrixEnsemblesthroughoutthewholestudy.Thepaperisorganizedasfollows.HavingposedtheprobleminSection2,wegeneralizetheHubbard–StratonovichtransformationinSection3.InSection4,wederivethesupersymmetricformulationinFouriersuperspace.ThecorrelationfunctionsareexpressedaseigenvalueintegralsinSection5.SummaryandconclusionsaregiveninSection6.2.PosingtheProblemInSection2.1,thetworelevantkindsofk–pointcorrelationfunctionsaredefinedandtherelationtothegeneratingfunctionsisgiven.Therebywealsointroduceournotationandconventions.WeclarifywhatwemeanbyarbitraryrotationinvariantensemblesinArbitraryRotationInvariantRandomMatrixEnsemblesandSupersymmetry3Section2.2.InSection2.3,weshowhowdifferenttypesofcorrelationfunctionscanberelatedtoeachotherbyproperFouriertransforms.2.1.Correlatio