Vertices from replica in a random matrix theory

arXiv:0704.2044v1[math-ph]16Apr2007LPTENS07-17VerticesfromreplicainarandommatrixtheoryE.Br´ezina)andS.Hikamib)a)LaboratoiredePhysiqueTh´eorique,EcoleNormaleSup´erieure24rueLhomond75231,ParisCedex05,France.e-mail:[e-mail]1b)DepartmentofBasicSciences,UniversityofTokyo,Meguro-ku,Komaba,Tokyo153,Japan.e-mail:[e-mail]’sworkonAirymatrixintegralshasledtoexplicitresultsfortheintersectionnumbersofthemodulispaceofcurves.InasubsequentworkOkounkovrederivedtheseresultsfromtheedgebehaviorofaGaussianmatrixintegral.InourworkweconsiderthecorrelationfunctionsofverticesinaGaussianrandommatrixtheory,withanexternalmatrixsource.WedealwithoperatorproductsoftheformQni=11NtrMki,ina1Nexpansion.Forlargevaluesofthepowerski,inanappropriatescalinglimitrelatinglargek’stolargeN,universalscalingfunctionsarederived.Furthermoreweshowthatthereplicamethodappliedtocharacteristicpolynomialsoftherandommatrices,togetherwithadualityexchangingNandthenumberofpoints,allowsonetorecoverKontsevich’sresultsontheintersectionnumbers,throughasimplesaddle-pointanalysis.1Unit´eMixtedeRecherche8549duCentreNationaldelaRechercheScientifiqueetdel’´EcoleNormaleSup´erieure.1IntroductionRandommatrixtheory(RMT)hasbeenappliedtomanyphysicalprob-lems,andalsotomathematicalsubjectssuchasthedistributionofzerosofRiemannzetafunctionorcombinatorialproblemsandithasledtoseveralmeaningfulresults[1].Italsoplaysanessentialroleinthetheoryofrandomsurfacesandforstringtheory.Severalkindsofcorrelationfunctionsinran-dommatrixtheoryhavebeenstudied.Inpreviouspapers,wehavestudiedthecorrelationfunctionoftheeigenvalues[4],andthecorrelationsofthecharacteristicpolynomials[7,9],forwhichwehavederivedexplicitintegralrepresentations.Inthisarticle,weconsiderthecorrelationfunctionsofverticesonthebasisofpreviouslyderivedintegralrepresentations.Thediagrammaticrep-resentationofthevertexhtrMki,whereMisarandommatrix,isobtainedthroughWick’stheorem,bythepairingsofk-legs,eachlegcarryingthetwoindices(i,j)ofthematrixelementMij.ForN×Nmatrices,thetwoindicesrunfrom1toN:i,j=1,...,N.WerestrictourselvesinthisarticletocomplexHermitianrandommatri-ces.ThedistributionfunctionforMisGaussianwithanexternalmatrixsourceA.PA(M)=1ZAe−N2trM2−NtrMA(1)WhenonesetsA=0,itreducestotheusualGaussianunitaryensemble(GUE).ThecorrelationfunctionsfortheverticesV(k1,...,kn)aredefinedasV(k1,...,kn)=1NntrMk1trMk2···trMkn(2)Thenormalizationischosensothattheyhaveafinitelarge-Nlimit.ThesefunctionsarecloselyrelatedtotheFouriertransformofthecorrelationfunc-tionsoftheeigenvalues,U(t1,...,tn)=Z∞−∞eiPtiλiRn(λ1,...λn)NY1dλi(3)wherethecorrelationfunctionoftheeigenvaluesisRn(λ1,...,λn)=nYi=11Ntrδ(λi−M)(4)IndeedU(t1,...,tn)=1NnnYi=1treitiM(5)1aregeneratingfunctionsoftheV(k)sinceU(t1,...,tn)=∞Xki=0trMk1trMk2···trMkn(it1)k1···(itn)knk1!k2!···kn!Nn(6)WhenthedistributionoftherandommatrixisGaussian,theaverageoftheverticesgivesthenumbersofpairwisegluingofthelegsofthevertexoperators.Thedualcellsoftheseverticesarepolygons,whoseedgesarepair-wiseglued.Wetherebygenerateorientablesurfaces,whicharediscretizedRiemannsurfacesofgivengenus.OkounkovandPandharipande[16,17]haveshownthattheintersectionnumbers,computedbyKontsevich[15],maybeobtainedbytakingasimul-taneouslargeNandlargekilimit.Furthermorethecorrelationfunctionsoftheseverticesareinteresting,sincetheygiveuniversalnumbersinthelargeNlimit.WehaveinvestigatedinanearlierworktheF.T.ofthen-pointcorrela-tionfunctionU(t1,...tn)fortheGUE,andfoundasimplecontourintegralrepresentationvalidevenforfiniteN[3,4].Inthisarticle,weextendthisintegralrepresentationtothevertexcorre-lationsV(k1,..kn),andexaminethescalingregionforlargekiandlargeN.Inthisintegralrepresentation,theasymptoticevaluationbythesaddle-pointmethodrequiresacarefulexaminationtodealwithpoleterms.Thisleadstoapracticalwaytocomputeintersectionnumberswhichwediscussinde-tail.WealsoshowthattheF.T.ofthecorrelationfunctions(C.F.)ofGUEneartheedgepointofthesupportoftheasymptoticspectrum,isequiva-lenttoKontsevich’sAirymatrixmodel;theidentificationisbasedonthereplicamethodandoveradualityforcomputingaveragesofcharacteristicpolynomials.Thearticleisorganizedasfollows.Insection2,weconsidertheF.T.oftheonepointcorrelationfunctionatabulkgenericpointinthelargeNlimit.Thisisdonebyacontourintegralrepresentation,andweobtainthebehavioroftrM2kwhenNandkarelarge.Weshowthatinthislimit,onerecoversthebehavioroftheonepointfunctionneartheedgepointofthespectrum.Insection3,weconsiderthecorrelationfunctionoftwovertices.Insection4,weinvestigatethecorrelationsofthen-vertices.Insection5,weintroduceareplicamethod,relyingonaveragesofchar-acteristicpolynomials.This,togetherwithaduality,allowsustomakecon-nexionwiththeKontsevichmodel,recoveringtherebygeneratingfunctionsfortheintersectionnumbers.Insection6,wepresentashortsummary.22OnepointcorrelationfunctionThecorrelationfunctionRn(λ1,...,λn)definedbyRn(λ1,...,λn)=nYi=11Ntr(δ(λi−M)).(7)isthusequaltoRn(λ1,...,λn)=nYi=11Ntr(Z∞−∞dt2πe−iti(λi−M)).(8)TheFouriertransformU(t1,...,tn)ofRn(λ1,...,λn)isthusgivenbyU(t