Long range correlations in branched polymers

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

2、nicalensemble,whenlookingatthecanonicalensemblewefindnegativelongrangepowerlikecorrelations.Wepro-posethatasimilarmechanismexplainstheshapeofrecentlymeasuredcorrelationfunctionsintheelongatedphaseof4dsimplicialgravity[1].1IntroductionRecentlysomemeasurementsofthecurv

3、ature–curvaturecorrelationfunctionsin4dsim-plicialgravitywerepresented[1].Thedataseemstoindicatenegativeandpowerlikebehaviorbothatthetransitionandintheelongatedphase.Thelatterseemssurprisingasonewouldexpecttheexponentialdecayofcorrelationsoutsidethecriticalregion.Th

4、oseissuesarehardtostudynumericallyandevenharderanalytically.TogainsomeinsightintopossiblemechanismofthosecorrelationsweinvestigatehereasimplegeometricalarXiv:hep-lat/9511024v122Nov1995model:branchedpolymers(BP).Besidesitssimplicitytherearesomeothermotivationsforusin

5、gthismodel.TheBPdescribethec→∞limitof2dsimplicialgravity[2,3].ItisbelievedthatBPalsodescribetheelongatedphaseof4dsimplicialgravity[4].Howevertheexactcorrespondencebe-tweenthequantitiesmeasuredinref.[1]andBPisnotknown.NeverthelesswecanhopethattheBPmodelcapturesthegen

6、eralfeaturesoftheelongatedphaseof4dsimplicialgravity.2ThemodelWeconsiderheretheensembleofplanarrootedplantedtrees.Atreeisagraphwithouttheclosedloops.Arootedtreeisatreewithonemarkedvertex(root)andplantedtreeisatreewhoseroot’sdegree(numberofbranches)isone.Twotreesarec

7、onsideredasdistinctiftheycannotbemappedoneachotherbyacontinuousdeformationoftheplane(seefig.1).Wedenotebynthetotalnumberofnon-rootverticesinthetreeandbyni∗permanentaddress:InstituteofComp.Science,JagellonianUniversity,ul.Nawojki11,30-072Krak´ow,Poland1Figure1:Example

8、ofdistinctplantedrootedtrees=++...Figure2:Partitionfunctionthenumberofnon-rootverticesofthedegreei.Thenthe(grand)partitionfunctionisdefined