Long range correlations in branched polymers

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

2、atthecanonicalensemblewefindnegativelongrangepowerlikecorrelations.Wepro-posethatasimilarmechanismexplainstheshapeofrecentlymeasuredcorrelationfunctionsintheelongatedphaseof4dsimplicialgravity[1].1IntroductionRecentlysomemeasurementsofthecurvature–curvaturecorrelationfunctionsin4dsim-plicial

3、gravitywerepresented[1].Thedataseemstoindicatenegativeandpowerlikebehaviorbothatthetransitionandintheelongatedphase.Thelatterseemssurprisingasonewouldexpecttheexponentialdecayofcorrelationsoutsidethecriticalregion.Thoseissuesarehardtostudynumericallyandevenharderanalytically.Togainsomeinsig

4、htintopossiblemechanismofthosecorrelationsweinvestigatehereasimplegeometricalarXiv:hep-lat/9511024v122Nov1995model:branchedpolymers(BP).Besidesitssimplicitytherearesomeothermotivationsforusingthismodel.TheBPdescribethec→∞limitof2dsimplicialgravity[2,3].ItisbelievedthatBPalsodescribetheelong

5、atedphaseof4dsimplicialgravity[4].Howevertheexactcorrespondencebe-tweenthequantitiesmeasuredinref.[1]andBPisnotknown.NeverthelesswecanhopethattheBPmodelcapturesthegeneralfeaturesoftheelongatedphaseof4dsimplicialgravity.2ThemodelWeconsiderheretheensembleofplanarrootedplantedtrees.Atreeisagra

6、phwithouttheclosedloops.Arootedtreeisatreewithonemarkedvertex(root)andplantedtreeisatreewhoseroot’sdegree(numberofbranches)isone.Twotreesareconsideredasdistinctiftheycannotbemappedoneachotherbyacontinuousdeformationoftheplane(seefig.1).Wedenotebynthetotalnumberofnon-rootverticesinthetreeandb

7、yni∗permanentaddress:InstituteofComp.Science,JagellonianUniversity,ul.Nawojki11,30-072Krak´ow,Poland1Figure1:Exampleofdistinctplantedrootedtrees=++...Figure2:Partitionfunctionthenumberofnon-rootverticesofthedegreei.Thenthe(grand)partitionfunctionisdefined