random electrical networks on complete graphsnew

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1、RANDOMELECTRICALNETWORKSONCOMPLETEGRAPHSII:PROOFSGeoffreyGrimmettandHarryKestenAbstra t.ThispapercontainstheproofsofTheorems2and3ofthearticleenti-tledRandomelectricalnetworksoncompletegraphs,writtenbythesameauthorsandpublishedintheJournaloftheLondonMathemati

2、calSociety,vol.30(1984),pp.171–192.Thecurrentpaperwaswrittenin1983butwasnotpublishedinajournal,althoughitsexistencewasannouncedintheLMSpaper.ThisTEXversionwascreatedon9July2001.Itincorporatesminorimprovementstoformattingandpunctuation,butnochangehasbeenmadet

3、othemathematics.WestudytheeffectiveelectricalresistanceofthecompletegraphKn+2wheneachedgeisallocatedarandomresistance.Theseresistancesareassumedindepen-dentwithdistribution−1−1(R=∞)=1−nγ(n),(R≤x)=nγ(n)F(x)for0≤x<∞,whereFisafixeddistributionfunctionandγ(n)→γ≥

4、0asn→∞.Theasymptoticeffectiveresistancebetweentwochosenverticesisidentifiedinthetwocasesγ≤1andγ>1,andthecaseγ=∞isconsidered.Theanalysisproceedsviadetailedestimatesbasedonthetheoryofbranchingprocesses.1.IntroductionInthesenoteswegivecompleteproofsofTheorems2and

5、3andafurtherindicationoftheproofofTheorem1inGrimmettandKesten(1983).Weusethesamenotationasinthatpaperandwethereforerepeatonlythebarestnecessities.Kn+2denotesthecompletegraphwithn+2vertices,whichwelabelas{0,1,...,n,∞}.(SeeBollob´as(1979)fordefinition).Eachedge

6、eisgivenarandomresistanceR(e)withdistributionγ(n)P(R(e)≤x)=F(x)for0≤x<∞n(1.1)γ(n)P(R(e)=∞)=1−,nwhereFisafixeddistributionfunctionconcentratedon[0,∞)andγ(n)asequenceofnumberssuchthat0≤γ(n)≤n.AlltheresistancesR(e),e∈Kn+2,areassumedindependent.Rndenotestheresult

7、ing(random)effectiveresistanceinKn+2betweenthevertices0and∞.Weshallprovethefollowingresult(thenumberingistakenfromGrimmettandKesten(1983)):MathematicsSubjectClassification(2000).60K35,82B43.Keywordsandphrases.Electricalnetwork,completegraph,randomgraph,branchi

8、ngprocess.12GEOFFREYGRIMMETTANDHARRYKESTENTheorem2.If(1.2)limγ(n)=γ≤1n→∞then
(1.3)limPRn=∞=1.n→∞TodescribethelimitdistributionofRnwhenγ(n)→γ>1weneeda(one-type)Bienaym´e–Galton–Watsonprocess{Zn}