The number of unbounded components in the Poisson

1、arXiv:math/0610202v3[math.PR]5Nov2007ThenumberofunboundedcomponentsinthePoissonBooleanmodelofcontinuumpercolationinhyperbolicspaceJohanTykesson∗February2,2008AbstractWeconsiderthePoissonBooleanmodelofcontinuumpercolationwithballsoffixedradiusRinn-dimensionalhyperbolicspaceHn.LetλbetheintensityoftheunderlyingPoissonprocess,andletNCdenotethenumberofunboundedcomponentsinthecoveredregion.ForthemodelinanydimensionweshowthatthereareintensitiessuchthatNC=∞a.s.ifRisbigenough.InH2weshowastrongerresult:fo。

2、ranyRtherearetwointensitiesλcandλuwhere0λcλu∞,suchthatNC=0forλ∈[0,λc],NC=∞forλ∈(λc,λu)andNC=1forλ∈[λu,∞).Keywordsandphrases:continuumpercolation,phasetransitions,hyperbolicspaceSubjectclassification:82B21,82B431IntroductionWebeginbydescribingthefixedradiusversionofthesocalledPoissonBooleanmodelinRn,arguablythemoststudiedcontinuumpercolationmodel.Foradetailedstudyofthismodel,wereferto[18].LetXbeaPoissonpointprocessinRnwithsomeintensityλ.AteachpointofX,placeaclosedballofradiusR.LetCbetheunionof∗Depa。

3、rtmentofMathematicalSciences,DivisionofMathematicalStatistics,ChalmersUniver-sityofTechnology,S-41296G¨oteborg,Sweden.E-mail:[e-mail]edishNaturalScienceResearchCouncil.1allballs,andVbethecomplementofC.ThesetsVandCwillbereferredtoasthevacantandcoveredregions.WesaythatpercolationoccursinC(respectivelyinV)ifC(respectivelyV)containsunbounded(connected)components.ForthePoissonBooleanmodelinRn,itisknownthatthereisacriticalintensityλc∈(0,∞)suchthatforλλc,percolat。

4、iondoesnotoccurinC,andforλλc,percolationoccursinC.Also,thereisacriticalintensityλ∗c∈(0,∞)suchthatpercolationoccursinVifλλ∗candpercolationdoesnotoccurifλλ∗c.Furthermore,ifwedenotebyNCandNVthenumberofunboundedcomponentsofCandVrespectively,thenitisthecasethatNCandNVarebothalmostsureconstantswhichareeither0or1.InR2itisalsoknownthatλc=λ∗candthatatλc,percolationdoesnotoccurinCorV.Forn≥3,Sarkar[21]showedthatλcλ∗c,sothatthereexistsanintervalofintensitiesforwhichthereisanunboundedcomponentinbothCandV.Iti。

5、spossibletoconsiderthePoissonBooleanmodelinmoreexoticspacesthanRn,andonemightaskiftherearespacesforwhichseveralunboundedcomponentscoexistwithpositiveprobability.Themainresultsofthispaperisthatthisisindeedthecaseforn-dimensionalhyperbolicspaceHn.WeshowthatthereareintensitiesforwhichtherearealmostsurelyinfinitelymanyunboundedcomponentsinthecoveredregionifRisbigenough.InH2wealsoshowtheexistenceofthreedistinctphasesregardingthenumberofunboundedcomponents,foranyR.ItturnsoutthatthemaindifferencebetweenR。

6、nandHnwhichcausesthis,isthefactthatthereisalinearisoperimetricinequalityinHn,whichisaconsequenceoftheconstantnegativecurvatureofthespaces.InH2,thelinearisoperimetricinequalitysaysthatthecircumferenceofaboundedsimplyconnectedsetisalwaysbiggerthantheareaoftheset.ThemainresultinH2isinspiredbyatheoremduetoBenjaminiandSchramm.In[6]theyshowthatforalargeclassofnonamenableplanartransitivegraphs,thereareinfinitelymanyinfiniteclustersforsomeparametersinBernoullibondpercolation.ForH2wealsoshowthatthemodeldoe。

7、snotpercolateonλc.ThediscreteanalogueofthistheoremisduetoBenjamini,Lyons,PeresandSchrammandcanbefoundin[4].ItturnsoutthatseveraltechniquesfromtheaforementionedpapersarepossibletoadopttothecontinuoussettinginH2.ThereisalsoadiscreteanaloguetothemainresultinHn.In[17],PakandSmirnovashowthatforcertainCayleygraphs,thereisanon-uniquenessphaseforthenumberofunboundedcomponents.Inthiscase,whileitisstillpossibletoadopttheirmainideatothecontinuoussetting,itismoredifficultthanforH2.Therestofthepaperisorganized。

8、asfollows.Insection2wegiveaveryshort2reviewofuniquenessandnon-uniquenessresultsforinfiniteclustersinBernoulliper-colationongraphs(foramoreextensivereview,seethesurveypaper[14]),includingtheresultsbyBenjamini,Lyons,Peres,Schramm,PakandSmirnova.Insection3wereviewsomeelementarypropertiesofHn.Insection4weintroducethemodel,andgivesomebasicresults.Section5isdevotedtotheproofofthemainresultinH2andsection6isdevotedtotheproofofthemaintheoremforthemodelinHn.2Non-uniquenessindiscretepercolationLetG=(V,E)bea。

9、ninfiniteconnectedtransitivegraphwithvertexsetVandedgesetE.Inp-BernoullibondpercolationonG,eachedgeinEiskeptwithprobabilitypanddeletedwithprobability1−p,independentlyofallotheredges.Allverticesarekept.LetPpbetheprobabilitymeasureonthesubgraphsofGcorrespondingtop-Bernoullipercolation.(Itisalsopossibletoconsiderp-Bernoullisitepercolationinwhichitistheverticesthatarekeptordeleted,andallresultswepresentinthissectionarevalidinthiscasetoo.)Inthissection,ωwilldenotearandomsubgraphofG.Connectedcomponents。

10、ofωwillbecalledclusters.LetIbetheeventthatp-Bernoullibondpercolationcontainsinfiniteclusters.Oneofthemostbasicfactsinthetheoryofdiscretepercolationisthatthereisacriticalprobabilitypc=pc(G)∈[0,1]suchthatPp(I)=0forppc(G)andPp(I)=1forppc(G).Whathappensonpcdependsonthegraph.Abovepcitisknownthatthereis1or∞infiniteclustersfortransitivegraphs.Ifweletpu=pu(G)betheinfimumofthesetofp∈[0,1]suchthatp-Bernoullibondpercolationhasauniqueinfinitecluster,Schonmann[22]showedforalltransitivegraphs,onehasuniqu。