Long range correlations in branched polymers

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1、Longrangecorrelationsinbranchedpolymers.PiotrBialas∗UniversiteitvanAmsterdam,InstituutvoorTheoretischeFysica,Valckenierstraat65,1018XEAmsterdam,TheNetherlandsAbstractWestudythecorrelationfunctionsinthebranchedpolymermodel.Althoughtherearenocorrelationsinthegrandcanonicale

2、nsemble,whenlookingatthecanonicalensemblewefindnegativelongrangepowerlikecorrelations.Wepro-posethatasimilarmechanismexplainstheshapeofrecentlymeasuredcorrelationfunctionsintheelongatedphaseof4dsimplicialgravity[1].1IntroductionRecentlysomemeasurementsofthecurvature–curvat

3、urecorrelationfunctionsin4dsim-plicialgravitywerepresented[1].Thedataseemstoindicatenegativeandpowerlikebehaviorbothatthetransitionandintheelongatedphase.Thelatterseemssurprisingasonewouldexpecttheexponentialdecayofcorrelationsoutsidethecriticalregion.Thoseissuesarehardto

4、studynumericallyandevenharderanalytically.TogainsomeinsightintopossiblemechanismofthosecorrelationsweinvestigatehereasimplegeometricalarXiv:hep-lat/9511024v122Nov1995model:branchedpolymers(BP).Besidesitssimplicitytherearesomeothermotivationsforusingthismodel.TheBPdescribe

5、thec→∞limitof2dsimplicialgravity[2,3].ItisbelievedthatBPalsodescribetheelongatedphaseof4dsimplicialgravity[4].Howevertheexactcorrespondencebe-tweenthequantitiesmeasuredinref.[1]andBPisnotknown.NeverthelesswecanhopethattheBPmodelcapturesthegeneralfeaturesoftheelongatedphas

6、eof4dsimplicialgravity.2ThemodelWeconsiderheretheensembleofplanarrootedplantedtrees.Atreeisagraphwithouttheclosedloops.Arootedtreeisatreewithonemarkedvertex(root)andplantedtreeisatreewhoseroot’sdegree(numberofbranches)isone.Twotreesareconsideredasdistinctiftheycannotbemap

7、pedoneachotherbyacontinuousdeformationoftheplane(seefig.1).Wedenotebynthetotalnumberofnon-rootverticesinthetreeandbyni∗permanentaddress:InstituteofComp.Science,JagellonianUniversity,ul.Nawojki11,30-072Krak´ow,Poland1Figure1:Exampleofdistinctplantedrootedtrees=++...Figure2:

8、Partitionfunctionthenumberofnon-rootverticesofthedegreei.Thenthe(grand)partitionfunctionisdefined