The giant component of the randombipartite graphMastersThesisinEngineeringMathematicsandComputationalScienceTONY JOHANSSONDepartmentofMathematicalSciencesChalmersUniversityofTechnologyGoteborg,Sweden2012TheGiantComponentoftheRandomBipartiteGraphTONYJOHANSSONc _TONYJOHANSSONDepartmentofMathematicsChalmersUniversityofTechnologySE-41296GoteborgSwedenTelephone+46(0)31-7721000MatematiskaVetenskaperGoteborg,Sweden2012AbstractThe randombipartite graphG(m,n; p) is the randomgraphobtainedbytakingthecompletebipartitegraphKm,nanddeletingeachedgeindependentlywithprobability1 p. Using a branching process argument, the threshold function for a giant connectedcomponentin G(m,n; p)isfoundtobe1/mn,andthenumberofverticesinthegiantcomponent is proportional tomn. An attempt is made to reduce the study of G(m,n; p)tothewell-studiedrandomgraph G(m,p)usinganoriginal argument, andthisisusedtoshowthata.a.s.thegiantof G(m,n; p)iscyclicwhenm=o(n)andpmnislargeenough. Usingacolouringargument, similarresultsareshownforFortuin-Kasteleynsrandom-clustermodel withparameter1 q 2whenm=o(n). Inparticular, whenm=o(n)thecritical inversetemperaturefortheIsingmodel onKm,nisfoundtobe12 log_1 2mn_,andthenumberofverticesinthegiantisproportionalto mn.AcknowledgementsIwouldliketothankOlleHaggstromforsupervisionthroughoutthisproject. IwouldalsoliketoprimarilyacknowledgeChalmersUniversityofTechnology,butalsoGothen-burgUniversityandtheUniversityofWaterlooastheuniversitiesthatIhaveattendedduringmyuniversityyears.TonyJohanssonGoteborg,November2012Contents1 Introduction 11.1 Basicgraphtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Asymptotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Basicprobability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Branchingprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Stochasticdomination . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Randomgraphmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 TheErdos-Renyimodel . . . . . . . . . . . . . . . . . . . . . . . . 51.4.2 IsingandPottsmodels . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Therandom-clustermodel. . . . . . . . . . . . . . . . . . . . . . . 61.5 Thegiantcomponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Thebranchingprocess 82.1 Proportionofevenandoddvertices . . . . . . . . . . . . . . . . . . . . . 92.2 Subcriticalcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Supercriticalcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Atcriticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Thenumberofoddvertices . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Theevenprojection 203.1 Theevenprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Applicationsofevenprojection . . . . . . . . . . . . . . . . . . . . . . . . 264 Therandom-clustermodel 284.1 Colouringargument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Existenceofagiantcomponent . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Sizeofthegiantcomponent . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35iCONTENTSATechnicalproofs 36ii1IntroductionSinceits initiationinthe late 1950s byErdos andRenyi, the eldof randomgraphshasbecomeanimportantandlargelyindependentpartofcombinatoricsandprobabilitytheory. Therehavebeenseveral bookspublishedintheeld,themostinuential beingBelaBollobas 1985monographRandomGraphs [2].Inspired by that book,and Svante Janson,TomaszLuczak and Andrzej Ruci nskis 2001bookbythesamename[10],thisthesisextendsclassicalresultsforthecompletegraphKmtothecompletebipartitegraphKm,n. Inparticular, itfocusesonresultsontheso-called giant component,which is historically important as one of the rst remarkableresultsonrandomgraphs.Theideabehindthisthesismaybelooselydescribedasstartingwithacertainsetof vertices, randomlyaddingedgesandstudyingthepropertiesof theresultinggraph.Itisimportanttonotethatthismaybedoneinanynumberofways,andthatrandomgraphs may be obtained in other ways, e.g. by assigning numerical values to the verticesof the graph. For the most part, this thesis studies the independent Erdos-Renyi model,namedafteritsoriginators,inwhicheachedgeisincludedwithequalprobabilityintherandom graph, independently of all other edges. Other models include the closely relatedxed-edgenumberErdos-Renyimodel,inwhichaxednumberMofedgesisincludedinthegraphandanygraphwithMedgesisequiprobable.This section sets the stage for the theorems of later sections, by introducing readers tothe eld of random graphs,and xing the notation used throughout the thesis. ReadersfamiliarwithrandomgraphscanskipSection1withnoproblem.Section 2 presents proofs of the main theorems of the thesis, imitating the techniquesof [10, Theorem5.4] closely. Theargument carries well throughthetransitionfromcompletegraphstocompletebipartitegraph,withaslightalterationinwhichonlyonepartofthebipartitegraphKm,nisconsidered.Onlyconsideringpartoftheverticesinthebranchingprocessargumentsuggestsamethodwithwhichthestudyofrandombipartitegraphmaybereducedtothealready11.1.Basicgraphtheory 1.INTRODUCTIONestablishedstudyof randomcomplete graphs. Section3investigates this reductionmethod,anddiscussesitsapplicationsandlimitations.Followingacolouringargumentfroma1996paperbyBollobas,GrimmettandJan-son [3], Section 4 extends the results of previous sections to Fortuin-Kasteleyns random-clustermodel. Inparticular,thesectionshowstheexistenceandsizeofagiantcompo-nentfortheIsingmodel,importantinstatisticalphysics.1.1 BasicgraphtheoryA graphis dened as a set (V,E) of labelled vertices Vand edges E V V . All graphswill be undirected, so that (x,y) Eif and only if (y,x) E. Throughout this thesis, Vwill be a nite set consisting of indexed letters; typically V= (a1, a2, ..., am; b1, b2, ..., bn)or V= (a1,...,am) for some positive integers m,n. An edge (x,y) between vertices x, y Vwillbedenotedbytheshorthandnotationxy.Foragivenedgexy, theverticesxandyarecalledtheendpointsof xy, andtheverticesxandyareneighbours. Similarly,twoedgese,f Eareadjacent iftheyshareanendpoint. Apathfromxtoyisasetof edgese0,...,ek Esuchthatei1andeiareadjacent for all 1 i kandxandyareendpoints of e0andek, respectively.Equivalently, apathfromxtoy maybe denedas asequence of neighbours x =x0, x1,...,xk= y.Letx yifandonlyifthereisapathfromxtoy. Thisisanequivalencerelationwhichpartitions V intosets C1,...,Ckcalledthe (connected) components of G. TheconnectedcomponentofxistheuniqueCi= C(x)suchthatx Ci.a1a2a3a4a5a6Figure1.1: The complete graph K6, as dened in Section 1.1. While complete graphs willnotbestudiedinthisthesis, theresultsforcompletebipartitegraphsareinspiredbyandatsomepointsderivedfromthecorrespondingresultsforcompletegraphs.Acomplete graphwithmvertices, denotedKm, isagraphwithvertexsetVm=(a1,...,am)andedgesetEm= (ai,aj):i ,=j. AcompletebipartitegraphisagraphwithVm,n=(a1,...,am; b1,...,bn) andEm,n=AmBn, whereAm=(a1,...,am) andBn= (b1,...,bn)makeupapartitionofVm,n. Avertexai Amwillbecalledevenand21.2.Asymptotics 1.INTRODUCTIONa1a2a3a4b1b2b3b4b5Figure1.2: ThecompletebipartitegraphK4,5, asdenedinSection1.1. Graphsofthistypearetheobjectofstudyinthecurrentthesis. Completebipartitegraphswillalwaysbedrawnwiththeevenverticesa1,...,aminthetoprow, andtheoddverticesb1,...,bninthebottomrow.avertexbj Bnodd. CompletebipartitegraphsaredenotedKm,n. Throughoutthethesis,wemaketheassumptionthatm = O(n),i.e. thatthereisaconstanta 1suchthat m an for all m,n. We regard n as a function of m, so that whenever n appears itshouldbereadasn = n(m). Theoremswillbeshowntoholdform = an,a > 0and/orm = o(n). Wenotethatm impliesn .AsubgraphH= (V,F)ofG = (V,E)isdenedasagraphwiththesamevertexsetasG, andwithedgesetF E. Inmostcasesitisconvenienttocall anedgee Eopenif e Fand closedotherwise. Each subgraph corresponds to an edge-conguration 0,1Ewhere(e) =1if eisopenand(e) =0otherwise. Wedeneapartialordering _onthespace 0,1Eby _ ifandonlyif(x) (x)forallx E.Aproperty of agraph(V,E) is denedas afamilyof subgraphs Q T(E). Asubgraph(V,F)haspropertyQifF Q. AnincreasingpropertyisapropertyQsuchthatif F1 QandF1 F2, thenF2 Q. Inotherwords, apropertyisincreasingifaddingedgestoanygraphwiththepropertywillproduceagraphwhichalsohasthatproperty.1.2 AsymptoticsAsymptoticnotationisfrequentinrandomgraphs,sinceresultsaretypicallyshownasm . Wedeneforanyfunctionfandanypositivefunctiongthefollowing:f(x) = O(g(x)) if limsupx[f(x)[g(x)< (1.1)f(x) = o(g(x)) if limxf(x)g(x)= 0 (1.2)Most results in this thesis are shown to hold asymptoticallyalmostsurely, or a.a.s. Asequence of random variables (Xk)k=1has property Q a.a.s. if an only if PXm Q 1. Furthermore,forasequenceXmofrandomvariableswedeneXm= op(am) if [Xm[ = o(am)a.a.s. (1.3)Theusageof opanda.a.s. isquiteinterchangable. Typically, asymptoticboundsarestateda.a.s.andasymptoticvaluesarestatedusingopterms.31.3.Basicprobability 1.INTRODUCTION1.3 BasicprobabilityManyresultswillrelyheavilyoninequalitiesrelatedtotheMarkovinequality. Foranynon-negativerandomvariableXwithniteexpectation,MarkovsinequalitystatesPX a E[X]a, a > 0 (1.4)Supposing further that Xhas nite second moment,Markovs inequality applied on therandomvariable [X E[X] [2impliesChebyshevsinequality:P[X E[X] [ a = P_[X E[X] [2 a2_VarXa2, a > 0 (1.5)Thiswill frequentlybeappliedwhenasequenceof randomvariablesXmisshowntosatisfyVarX= op(E[Xm]). ChebyshevsinequalityimpliesP[XmE[Xm] [ E[Xm] VarXmE[Xm]2= o(1) (1.6)sothatXm= E[Xm] (1 +op(1)).Thefollowingboundsholdforalla R,andarecommonlycalledChernobounds.PX a = P_etX eta_E_etXeta, t 0 (1.7)PX a = P_etX eta_E_etXeta, t < 0 (1.8)For a random variable X, the probability-generating function GXis dened by GX(s) =E_sXwhenthisexpectationexists. Themoment-generatingfunctionMXisdened,whenitexists, byMX(t)=E_etX=GX(et). Thepropertyof thesefunctionsthatwill be used most frequently is the following: Let Nbe a positive integer-valued randomvariable, and let Y= X1 +... +XNwhere the Xiare i.i.d. and independent of N. ThenMY (t) = MN(MX(t)) and GY (s) = GN(GX(s)) (1.9)wheneverthefunctionsexist. Seee.g. [7].Arandomvariable Xis Bernoulli distributed with parameter p[0,1], denotedX Bern(p),if PX= 1 = p and PX= 0 = 1 p. It is binomiallydistributedwithparamtersn Nandp [0,1],denotedX Bi(n,p),ifforanyinteger0 k nPX= k =_nk_pk(1 p)nk(1.10)Theprobability-generatingfunctionofX Bi(n,p)isgivenbyGX(s) = (1 p +ps)n.41.4.Randomgraphmodels 1.INTRODUCTION1.3.1 BranchingprocessesAGalton-Watsonprocess orbranchingprocess isastochasticprocess(Zk)k=0, denedasfollows. Wechooseanon-negativeinteger-valuedprobabilitydistribution /andcallit the ospring distribution. Given a deterministic starting value Z0(usually taken to be1), eachZk, k 1isdenedbyZk=

Zk1i=1Xki, wheretheXki /areindependent.ThevariablesZkareinterpretedasthenumberofindividualsinthek-thgenerationofapopulation,andeachXkiisthenumberofospringsforanindividual.The extinctionprobability for abranchingprocess is the probabilityof the eventk > 1 : Zk= 0. Let X /. A classical result states that if E[X] 1, the extinctionprobabilityis 1, andif E[X] >1the extinctionprobabilityis givenbythe uniquesolutionin(0,1)toGX(s) = s. HereGXdenotestheprobability-generatingfunctionofX.1.3.2 StochasticdominationIn Section 3 we shall need the measure-theoretic concept of stochastic domination. Giveno R, a partial ordering _ on a space and probability measures , on , we say thatstochasticallydominates,written _D,if(f) (f)forallincreasingfunctionsf: o(withrespectto _). Heuristically,thismeansthatpreferslargerelementsofthan. Inparticular, foranyincreasingeventB owehave(B) (B). See[6].The following powerful result is used to characterize stochastic domination in relationtorandomgraphs. ItisaspecialcaseofamoregeneralresultcalledHolleysTheorem,see e.g. [6]. Suppose xEandlet X(x) =1 if xEis openandX(x) =0otherwise. Wedenetheevent X=oxfor 0,1E\{x}astheevent X(y)=(y)forally E x.HolleysTheorem. Let G=(V,E)beagraphandlet 1, 2beprobabilitymeasureson 0,1E. If1(X(x) = 1 [ X= ox) 2(X(x) = 1 [ X=

ox) (1.11)forall x Eandall ,

0,1E\{x}suchthat _

,then1 _D2.1.4 RandomgraphmodelsLetG=(V,E)beagraph. Arandomgraphmodel, inall instancesofthisthesis, isaprobabilitymeasureoneitherof thespaces oVand oEforsome o R. Thissectionpresentsthedierentmodelsthatwillbeusedormentionedinthethesis.1.4.1 TheErd os-RenyimodelThemodel thatErdosandRenyi introduced, whichhasbeensubsequentlynamedtheErdos-Renyi model, comes intwocloselylinkedforms. Theonemost suitedfor this51.4.Randomgraphmodels 1.INTRODUCTIONthesis is denotedG(m,p), whichgenerates asubgraphof the complete graphKm=(Vm, Em)inwhicheachedgeisincludedindependentlyofallotherswithprobabilityp.Equivalently, it may be seen as starting with Km,n and removing each edge independentlywithprobability1 p. Inotherwords,foranysubgraphH= (Vm, F)whereF Em,PF =__m2_[F[_p|F|(1 p)(m2)|F|(1.12)ForthecompletebipartitegraphKm,n= (Am Bn, AmBn)wedenotethecorre-spondingmodel G(m,n; p). AsubgraphH= (Am Bn, F)isassignedprobabilityPF =_mn[F[_p|F|(1 p)mn|F|(1.13)1.4.2 IsingandPottsmodelsAphysicallyimportantrandomgraphmodel istheIsingmodel, inventedbyWilhelmLenz[11]andhisstudentErnstIsing[9]tomodelferromagnetisism. Mathematically,itisquitedierentfromtheErdos-Renyi model inthatitassignsprobabilitytoverticesand not to edges. Each vertex is assigned a spin, which here will be denoted by the values1or2, andthemodel will prefervertexcongurationsinwhichneighbourshaveequalspin. Formally, theIsingmodel onanitegraphG=(V,E)consistsof aprobabilitymeasureon 1,2Vwhichtoeach 1,2Vassignsprobability() =1Zexp_2

xyI{(x)=(y)}_(1.14)where 0, andx yif andonlyif x, y V areneighbours. HereZisanormal-izingconstant. Inferromagneticapplications bears theinterpretationof reciprocaltemperature,i.e. = 1/TwhereTisthetemperature.Anatural generalizationof theIsingmodel istheq-statePottsmodel [12], whichassignsspinsintheset 1,...,qforsomeintegerq 2. ThePottsprobabilitymeasureon 1,...,qVisidenticalto(1.14)exceptforanewnormalizingconstantZ,q.1.4.3 Therandom-clustermodelTherandom-clustermodel,introducedbyCeesFortuinandPieterKasteleyninaseriesof papersaround1970, seee.g. [8], isarandomgraphmodel thatuniestheErdos-Renyi, IsingandPottsmodels. Foranyq>0itassignstoanysubgraph(V,F)of agraph(V,E)probabilityproportionaltoP(F; E,p,q) = p|F|(1 p)|E||F|qc(V,F)(1.15)where c(V,F) is the number of components of (V,F). Swendsen and Wang [13],followedbyasimplerproofbyEdwardsandSokal[4],showedthattherandom-clustermodelatprobability p for any q N is equivalent to the q-state Potts model at inverse temperature= 12 log(1 p).61.5.Thegiantcomponent 1.INTRODUCTION1.5 ThegiantcomponentAround1960, ErdosandRenyi [5] provedtherstresultconcerningtheexistenceandnon-existenceof aso-calledgiantcomponentin G(m,p). Essentially, theresultstatesthatthegraph G(m,/m)a.a.s.containsonecomponentwhichismuchlargerthanallother components when > 1,while no such component exists for < 1. The followingformulationoftheresultappearsin[10]:Theorem. Letmp = ,where > 0isaconstant.(i)If 1andlet = () (0,1)betheuniquesolutionto +e= 1. ThenG(m,p)containsagiant component of m(1 + op(1))vertices. Furthermore, a.a.s. thesizeofthesecondlargestcomponentof G(m,p)isatmost16(1)2log m.Thisthesiswill mainlybeconcernedwithprovingasimilarresultforthebipartitegraph G(m,n; /mn),usingprooftechniquessimilartothosein[10].Ina1996paper[3],Bollobasetal. derivedconditionsfortheexistenceandsizeofagiantcomponentintherandom-clustermodel foranyq>0, byreducingtotheq=1caseandusingknownresults. This thesis will partlyemulatethosemethods onthebipartitegraphstoprovesimilarresults.72ThebranchingprocessSincetheresultscontainedinthisthesisareconcernedwithcomponentsizes, amethodtomeasurethesizeof acomponentisneeded. Thefollowingmethodisinspiredbytheoneusedin[10], andtheprocessitdescribeswill becalledaeven search process(or simply search process) and is closely related to a breadth-rstsearch,commonincomputergraphalgorithms. Thealgorithmndstheverticesintheconnectedcomponentofanevenvertexa,andcountsthenumberofevenandoddverticesseparately.1. Leta0=aandS= , andmarka0assaturated. All otherverticesareinitiallyunsaturated. Setk = 0.2. Findall unsaturatedneighboursb1,...,brofak, andmarkthemassaturated. LetRk+1= b1,...,brandYk+1= [Rk+1[.3. For eachbi Rk+1, ndall unsaturatedneighbours ai1, ..., aisof bi, s =s(i).LetSk+1= ri=1ai1,...,ais. Thus, Sk+1isthesetof all unsaturatedverticesatdistance2fromak. LetXk+1= [Sk+1[.4. Assign S:= S Sk+1. If Sis empty, stop the algorithm. If Sis nonempty, let ak+1beanarbitraryelementof S, removeak+1fromSandmarkak+1assaturated.Returntostep2withk + 1inplaceofk.Herethesearchforoddverticesbishouldbeconsideredanintermediatestep: weareonlyinterestedinndingtheevenverticesof thecomponentcontaininga. SoX1isthenumberof 2-neighbours, i.e. verticesatdistance2froma, andX2, X3,... arethenumberofverticesatdistance2fromverticesthathavealreadybeenfoundintheprocess. Thenumberofevenverticesinthecomponentisgivenby1 +

Kk=1Xk,whereKisthetotalnumberofiterationsofthealgorithm.Thisresemblesabranchingprocess, thedierencebeingthatabranchingprocessasdenedinSection1.3.1requireseachXitobeidenticallydistributed. However, the82.1.Proportionofevenandoddvertices 2.THEBRANCHINGPROCESSresemblance is in some sense close enough, in that we can nd actual branching processesboundingthesearchprocessfromaboveandbelow.ItisclearthatY1 Bi(n,p), sinceatthestartall verticesareunsaturated, andahas n neighbours. After this the distributions are more complicated,but we may boundX1fromabovebyX+1Bi(Y1m,p), sinceitisthesumof Y1randomvariables, eachboundedfromabovebyBi(m,p). Continuinginthisfashion, wemayboundYkfromabovebyY+k Bi(n,p)forallk,andXkfromabovebyX+k Bi(Y+km,p). Using(1.9),wehavethefollowingprobability-generatingfunctionforX+k:GX+k(s) = (1 p +p(1 p +ps)m)n(2.1)Given information about the maximal size of a component, we may bound the processfrombelowbysimilarmethods,aswillbeseenbelow.2.1 ProportionofevenandoddverticesThe even search process described above nds the number of even vertices of a componentin the graph G(m,n; p). To be able to state results about the total size of the component,we show a result that relates the number of even vertices in a component to the numberof odd vertices in the same component. Heuristically, this section shows that the numberofoddverticesinacomponentispntimesthenumberofevenvertices, ifthenumberofevenverticesinthecomponentissmallenough. Notethatthisproportionequalstheexpecteddegreeofanyevenvertex. Forthis,weneedapurelyprobabilisticlemma.Lemma1. If X Bi(n,p)wherenandp=p(n)aresuchthat E[X] =np asn ,thenX= np(1 +op(1))Proof. We have E[X] =np andVarX=np(1 p) =o(E[X]2) so that by(1.6),X= np(1 +op(1)).Lemma1isusefulinprovingthefollowing.Lemma 2. Let m0=m0(m) be suchthat pm0n andpm0 0as m .LetCbeacomponentof G(m,n; p). Conditional onChavingm0evenvertices, Chaspm0n(1 +op(1))oddvertices.Proof. In each step of the search process, a number Ykof odd vertices is identied. Theserandom variables are each bounded from above by a random variable Y+k Bi(n,p). Sincethe number of oddvertices inthecomponent, conditional onthe component havingm0evenvertices, is asumof m0suchvariables, it must beboundedfromabovebyY+Bi(m0n,p). Butbytheassumptionthatpm0n , wehaveE[Y+] andbyLemma1,Y+= pm0n(1 +op(1)).Let n+=pm0n(1 + op(1)). Thentherandomvariables Ykmust all beboundedfrombelowbyYkBi(n n+,p), since throughout the process there are at leastn n+oddverticesthatarenotyetsaturated,andY isboundedfrombelowbyY 92.2.Subcriticalcase 2.THEBRANCHINGPROCESSBi(m0(nn+),p). But pm0(nn+) = pm0n(1+o(1)) since n+= op(n), so E[Y] andwehaveY= pm0n(1 +op(1)).Thus, conditional onthecomponenthavingm0evenvertices, wemusthaveY =pm0n(1 +op(1)).Corollary3. Let>0andp=mn. Suppose(a)m=o(n)or(b)m=an, a>0,andm0=o(m). Let Cbeacomponent ofG(m,n; p), andm0besuchthat m0 asm . Conditional onChavingm0evenvertices, ithasm0_nm(1 + op(1))oddvertices.Proof. (a)Wehavepm0n = m0_nm andpm0= m0mn _mn 0asm ,andtheresultfollowsfromLemma2.(b)Wehavepm0n=am0 andpm0 m0/m=o(1)sotheresultfollowsfromLemma2.Corollary3leavesonlythecasem0= m,m = anforsome (0,1]anda > 0. Inthis case, the number of even and odd vertices will both need to be calculated explicitly.2.2 SubcriticalcaseThis sectionis concernedwithprovingthenon-existenceof agiant component when 0forall (0,1),andtheresultfollows.Theorem6. If < 1andm n,thenthereisa.a.s.nocomponentin G(m,n; p)withmorethan7(1 )3log mevenvertices.Proof. Weboundthesearchprocessbyabranchingprocesswithospringdistributionhavingmoment-generatingfunctionMX(t)=_1 p +p_1 p +pet_m_n, asdiscussedatthebeginningofSection2. Considerasearchprocessinitiatedattheevenvertexa.Theprobabilitythatthetotalnumberofevenverticesfoundintheprocessisatleastkisboundedasfollows:Pabelongstoacomponentofsizeatleastk P_k

i=1Xi k 1_(2.8)WecanboundthequantitybythefollowingChernobound:P_k

i=1Xi k 1_E[exp(t(X1 +... +Xk))]exp(t(k 1))=E_etX1ket(k1), t > 0 (2.9)wherewehaveusedthefact that theXiarei.i.d. toseparatetheexpectation. Theinequalityholdsforanyt > 0,andinparticularwemaychooset = log_1_toobtainP_k

i=1Xi k 1_ k1E_X1k=1_E_X1_k(2.10)UsingLemma5, andthem nassumption, wehaveE_X1 exp_16(1 )3_.112.3.Supercriticalcase 2.THEBRANCHINGPROCESSSolettingk = 7(1 )3log m,wehavePcomponentwithatleastkevenverticesm

i=1Paibelongstoacomponentofsizeatleastk mP_k

i=1Xi k 1_mexp_k6(1 )3_=m m7/6= o(1) (2.11)2.3 SupercriticalcaseThis sectionis concernedwithprovingthe existence, uniqueness andsize of agiantcomponent when>1usingtheevensearchprocess. Hereweweakenthem nassumption, sincewewill needtocoverall casesinwhichm n. Insteadweassumethereisana 1suchthatm an.Lemma 7. Suppose there is ana 1 such that m an. Let k=3 log2m(1)2andk+=3mlog m(1). If > 1,thenthereisa.a.s.nocomponentwithkevenvertices,wherek k k+.Remark. Note, inparticular, that k+>2kfor large m. This means that twocomponents with less than keven vertices can never be joined by a single edge to formacomponentwithatleastk+evenvertices.Proof. Startingatanevenvertexa,weshowthatthesearchprocesseitherterminatesafterfewerthanksteps,oratthek-thstepthereareatleast( 1)kverticesinthecomponentcontainingathathavebeengeneratedintheprocessbutwhicharenotyetsaturated, foranyksatisfyingk k k+. Indeed, if ( 1)kunsaturatedverticeshavebeenfound,thentheprocesscontinuessince( 1)k ( 1)k 1formlargeenough.Sinceweconsideronlythepartoftheprocesswherek k+, weneedonlyidentifyatmostk+ +( 1)k+= k+evenverticesintheprocess,andwemayboundXifrombelowbyXiBi(Y (m k+), p)whereYBi(n,p). Theprobabilitythatafterkstepstherearefewerthan( 1)knotyetsaturatedvertices, orthattheprocessdiesoutaftertherstksteps,issmallerthanP_k

i=1Xi k 1 + ( 1)k_= P_k

i=1Xi k 1_(2.12)122.3.Supercriticalcase 2.THEBRANCHINGPROCESSAsinthesubcriticalsetting,weemployChernoboundstoboundthis. First,wenotethatforanyXithemoment-generatingfunctionisgivenbyM(t) =_1 p +p_1 p +pet_mk+_n(2.13)Puttingt =1k log_1_< 0,wehavethefollowingChernobound:P_k

i=1Xi k 1_M_1k log_1__kexp_1k log_1_(k 1)_= 11k_M_1k log_1___k(2.14)ThuswewishtoboundM(t),whichexplicitlyreads_1 p +p_1 p +p1k_mk+_n(2.15)wherep = /mn. Usingtheinequality(1 +x/n)n ex,whichholdsforallx Randalln > 0,wehave_1 p +p1k_mk+=_1 11kmn_mn

mnk+mn

exp__ 11k___mn k+mn__ 1 1a_ 11k___mn k+mn_(2.16)Wherea 1issuchthatmnaforall m. Thesecondinequalityholdssinceex1 x/aforx a 1. Indeed,wehave_ 11k___mn k+mn_ a 11ka a 1 (2.17)sotheinequalityisapplicable. Putting(2.16)into(2.15)yields_1 p +p_1 p +p1k_mk+_n_1 +mn_1 1a_ 11k___mn k+mn_1__n exp_1a_ 11k___mn k+mn__nm_= exp_1a_ 11k__1 k+m__(2.18)132.3.Supercriticalcase 2.THEBRANCHINGPROCESSSo,(2.14)becomesP_k

i=1Xi k 1_ 11kexp_ka_ 11k__1 k+m__(2.19) exp_1a_ 11k_kk+mka_ 11k__Notingthatk=k2+log mm, theprobabilitythatthereisacomponentwithk k k+evenverticesisthusboundedasfollows.mk+

k=kP_k

i=1Xi k 1 + ( 1)k_ mk+exp__ 11k_k2+ma ka_ 11k__= mk+exp__ 11k_k2+ma log mk2+ma_ 11k__= mk+exp_k2+ma(1 log m)_ 11k__= c0m3/2_log(m3) exp_log(m3)a( 1)2(log m1)_ 11k__= c0_log(m3)m32c1(log m1)= o(1) (2.20)wherec0, c1aresomepositiveconstantsthatdependona, only.Lemma8. If >1, thena.a.s. thereisatmostonecomponentwithmorethank+=mlog(m3)(1)evenvertices.Proof. We argue again as in [10]. Suppose there exists at least one component of size atleastk+(call suchacomponentlarge), andletaiandaj, i ,=jbetwoevenverticeswhoeachbelongtolarge, possiblyequal, componentsCiandCj, respectively. Weruntwosearchprocessesstartingatthesetwovertices. Afterk+steps, eachprocessmusthaveproduced( 1)k+notyetsaturatedevenvertices. Theprobabilitythattheredoesnotexistanopenpathoflength2betweenasyetunsaturatedverticesofCiand142.3.Supercriticalcase 2.THEBRANCHINGPROCESSCjisboundedfromabovebyPl = 1,...,n : atleastoneofaibl, ajblclosed[(1)k+]2=_1 p2_n[(1)k+]2=_1 2mn_n[(1)k+]2 exp_2m[( 1)k+]2_= exp___2m( 1)2__mlog(m3)( 1)_2___= exp_log(m3)_= o(m2) (2.21)Sincetherearelessthanm2waysof choosingai, aj, thisimpliesthattheprobabilitythatanypairofaiandajbelongtodierentlargecomponentstendsto0asm ,andtherecannotbemorethanonecomponentofsizelargerthank+.ThismeansthatwemaypartitiontheevenverticesofG(m,n; p)intolarge andsmall vertices, theformerbeingtheverticesthatbelongtotheuniquecomponentofsize at least k+, if such a component exists. To see that the component indeed does existandnditssize,wecountthenumberofsmallvertices.Theorem9. If >1, thenthereisacomponent withmm(1 + op(1))evenvertices,wherem= m() (0,1)istheuniquesolutiontom + exp_a (exp ma 1)_= 1 if m = an, a > 0m +e2m= 1 if m = o(n)(2.22)Furthermore,thesecondlargestcomponenta.a.s.hasatmost3 log2m(1)2evenvertices.Proof. ThesecondstatementfollowsdirectlyfromLemmas7and8.Fortherstpart, wecalculatetheprobabilitythatagivenevenvertexaissmall.Call this probability (m,n,p). As in [10], the probability is bounded from above by +=+(m,n,p), theextinctionprobabilityforthebranchingprocessinwhichtheospringdistributionisgivenbyBi(Y (m k),p)=Bi(Y (m o(m)),p), whereYBi(n,p)asusual. Theprobabilityis alsoboundedfrombelowby+ o(1), whereis theextinctionprobabilityforthebranchingprocesswithospringdistributionBi(Y m,p).Theo(1)termboundstheprobabilitythattheprocessdiesaftermorethanksteps.The extinction probability +is the unique positive solution to GX(+) = +, where152.3.Supercriticalcase 2.THEBRANCHINGPROCESSGX(x) =_1 p +p (1 p +px)mo(m)_n. Ifm = o(n),wehave(1 p +px)mo(m)=_1 +(x 1)mn_mo(m)=mo(m)

k=0_mo(m)k__(x 1)mn_k= 1 +(x 1)_mo(m)n+o__mn_(2.23)sothat(1 p +p(1 p +px)mo(m))n=_1 +mn_1 + 1 +(x 1)_mo(m)n+o__mn___n=_1 +2(x 1)(1 +o(1))n_n= exp_2(x 1)(1 +o(1))_(2.24)so as m , we have += 1 m, where e2m= 1 m, and replacing mo(m) bym we get = 1m+o(1). Thus = 1m+o(1), and the expected number of smallevenverticesisgivenby(1 m)m(1 + o(1)). Inotherwords, theexpectednumberofevenverticesinthelargestcomponentismm(1 +o(1)).Similarly,ifm = an,a > 0,thenlimn_1 an+an_1 an+anx_ano(n)_n= exp_a exp_(x 1)a(1 +o(1))_a_(2.25)andthe expectednumber of evenvertices inthe giant component is againgivenbymm(1 +o(1)).Lastly,weshowthatY= E[Y ] (1 +op(1)),whereYisthenumberofsmallvertices.From this follows that the giant component has size mm(1+op(1)). Let ibe 1 if vertex162.4.Atcriticality 2.THEBRANCHINGPROCESSaiissmalland0otherwise. ThenE[iY ] = E[iY [ i= 1] Pi= 1 +E[iY [ i= 0] Pi= 0= E[Y [ i= 1] (m,p)=k

k=1E[Y [ [C(ai)[ = k] P[C(ai)[ = k (m,p)= (m,p)k

k=1(k +E[Y k [ Y k]) P[C(ai)[ = k (m,p) (k + (mO(k))(mO(k),p)) (m,p)(k +m(mO(k),p)) (2.26)sowehaveE_Y2=m

i=1E[iY ] m(m,p)(k +m(mO(k),p))= m2(m,p)(o(1) +(mO(k),p))= m2(m,p)2(1 +op(1))= E[Y ]2(1 +op(1)) (2.27)Thus,VarY= E_Y2E[Y ]2= op(E[Y ]2)andby(1.6),Y= E[Y ] (1 +op(1)).Figure2.1showsmasafunctionof [0,2]inthecasem = o(n).2.4 AtcriticalityWe conclude this sectionwithaquicklookintothe case =1. Here, the theorybecomesmuchmoreinvolvedandthetoolsweusedabovewill notbeenoughtoproveanydetailedresults. Weconneourselvestoshowingthatthelargestcomponenthasop(m)evenvertices. Thisisdonebyshowingthatincreasingpropertiesaremorelikelywhenpincreases;seeSection1.3.2.Lemma10. Let0 p1 p2 1,andlet1, 2bethemeasureson 0,1Em,ninwhicheachedgeinKm,nisopenwithprobabilityp1,p2respectively, independentlyof all otheredges. Then1 _D2.Proof. Theproof is asimpleapplicationof Holleys theorem. Let x Em,nandlet1, 2 0,1Em,n\{x}besuchthat1 _ 2. Bytheindependenceofedgesunder1, 2,wehaveiX(x) = 1 [ X= iox = iX(x) = 1 = pi, i = 1,2 (2.28)sop1 p2implies1X(x) = 1 [ X= 1ox 2X(x) = 1 [ X= 2ox (2.29)172.5.Thenumberofoddvertices 2.THEBRANCHINGPROCESS0 0.5 1 1.5 200.20.40.60.81m as a function of , 0 2m m = o(n)m = nm = 10nFigure2.1: The proportion of even vertices belonging to the giant component for [0,2]whenm = o(n),nor10nasm . ThevaluesofmaregivenbyTheorem9.andbyHolleystheorem,1 _D2.Theorem11. Let > 0. When = 1,thegraph G(m,n; /mn)a.a.s.hasnocompo-nentwithmorethanmevenvertices. Inotherwords,thelargestcomponenthasop(m)evenvertices.Proof. Let > 0,andlet > 1besuchthatm= m() < withm()asinTheorem9. Whenm=o(n), suchaexistsbythecontinuityof (, m) e2m+ m 1,andsincem=0istheonlysolutionof em+ m 1=0. Whenm n, asimilarargumentisusedtoensuretheexistenceof.LetQbetheincreasingpropertyof G(m,n; p)havingacomponentwithatleastmevenvertices. Withnotationasintheproof of Lemma10, letp1=1/mnandp2=/mn. Then1 _D2andinparticular1(Q) 2(Q). ButbyTheorem9,wehave2(Q) = o(1). Thus,P_G(m,n; 1/mn) Q_= 1(Q) = o(1) (2.30)andthelargestcomponentof G(m,n; 1/mn)hasop(m)evenvertices.2.5 ThenumberofoddverticesNow that results have been shown for the even vertices of G(m,n; p),Section 2.1 is usedto deduce the total size of the largest component. The results can be summarized in thefollowingtheorem.182.6.Concludingremarks 2.THEBRANCHINGPROCESSTheorem12.(a) (Subcriticality) If 1, thenthere is a component with mm(1+op(1))evenvertices, where mis givenby (2.22). If m=o(n), the total number of ver-tices is mmn(1+op(1)). If m=an, a >0, the total number of vertices is(mm+nn)(1+op(1)), where mis given by (2.22) and nis the unique positive solutionton + exp_a_exp_na_1__ = 1. Ineachcase,thesecondlargestcomponenta.a.s.hasatmost3 log2m(1)2_nmverticesintotal.Proof. The statements about even vertices are Theorems 6, 9 and 11. Statements aboutoddverticesin(a)and(b)followfromCorollary3. Thesizeofninthecasem=anisfoundbyconsideringtheevenverticesform =1an.The following lemma about the relative number of even and odd vertices in the giantshallbeneededinSection4. TheproofislefttoAppendixA.Lemma13. Supposea>0, >1, andlet m, nbetheuniquesolutionsin(0,1)tom + exp_a (expam 1)_= 1andn + exp_a_exp_an_1__= 1respectively. Thenn min(1,a)m.2.6 ConcludingremarksIn the complete random graph setting, authors (e.g. [10]) frequently mention the follow-ing heuristic: A giant component appears when the average vertex degree becomes greaterthanone. Led by this heuristic, I initially looked for a threshold function of either of theforms/ minm,n,/(m+n)or/ maxm,n,orsimilar. However,Icouldnotmakeanyoftheformulastintothebranchingprocessargument,soinsteadstartedtoworkmywaythroughthemethodsusedin[2]. Thesecombinatorial proofsquicklyshowedthatmandntendtoappearasaproduct, whichsuggestedthesomewhatsurprisinguseofageometricmean. Itisworthnotingthatthethresholdfunction1/mnisnotconsistentwiththeheuristicaboveasitstands,butinsteadwehaveseeninthissectionthat thefollowingwouldbethecorrespondingheuristic: Agiant component appearswhentheaveragenumberofpathsoflength2fromavertexbecomesgreaterthanone.193TheevenprojectionThebranchingprocess argument of theprevious sectionwas accomplishedbyconsideringonlytheevenverticesofKm,n. Theusual branchingprocesswasreplacedbyadouble-jumpprocesswhichinsteadofsearchingforedges, con-sideredpathsof length2betweenevenvertices. Thissuggestsamethodinwhich a subgraph G Kmis obtained from a subgraph H Km,nby removing the oddverticesofHandassigninganedgetoGifandonlyifthecorrespondingverticesinHare connected by a path of length 2. See Figure 3.1. We will call G the even projection ofH. ByusingknownresultsaboutrandomsubgraphsonKm,onemaydrawconclusionsaboutpropertiesofrandomsubgraphsofKm,n.Therearetwomajorproblemswiththismethod. Firstly,theedgesoftheevenpro-jectionarehighlydependent, limitingthecompatibilitywiththeusual randomgraphtheorywhichrequiresindependentedges. Secondly,somepropertiesoftheevenprojec-tionarediculttotranslatebacktothebipartitesetting. Wewillseethatthismethodisgoodenoughtoprovetheexistence, butnotuniqueness, ofagiantcomponentwhen > 1,and with help of Section 2 it is shown that for large enough,the giant is cyclic.Itshouldalsobementionedthattherearenosignicantresultsinthissectionforthecasem n, sinceHolleystheorem, whichtheproofsrelyheavilyupon, cannotbeappliedinthiscasetotheauthorsknowledge.3.1 TheevenprojectionGivenasubgraphG=(Am Bn,E)ofKm,n,wedenetheevenprojectionofGtobethe graph with vertex set Amand in which any edge aiajis included if and only if thereisa1 k nsuchthataibkandajbkarebothinE. Further,denetherandomevenprojection, denoted Gpn(m), as the random graph obtained by taking the even projectionof G(m,n; p).Equivalently, given 0,1Em,n, we dene the evenprojectionof througha203.1.Theevenprojection 3.THEEVENPROJECTIONmapping : 0,1Em,n0,1Emby() = ,whereforalli,j()(aiaj) = (aiaj) =max1kn(aibk)(ajbk). (3.1)We dene pnto be the measure on 0,1Emassociated with even projections, i.e. themeasure which to any 0,1Emassigns probability pn() = (1()), where is thei.i.d.measureon 0,1Em,nwhichassignsprobabilityptoeachedge.Thepismovedoutofitsusualpositioninthenotationtoavoidmisunderstanding:asweshall see, Gpn(m)doesnothaveedge-probabilityp. Thenotationwill bekeptforthei.i.d.measureonKm,n. Weshallnotneedthefollowingresult, butitservesasawarmupforwhatistocome,andmotivatesthecomparisonofpntoani.i.d.measurewithapproximateedge-probability2/m 1 (1 p2)n.Proposition14. Theprobabilitythatanedgein Gpn(m)isopenisgivenbypnaiajopen = 1 (1 p2)n(3.2)forall 1 i,j m.Proof. Hereweintroducetheevent a1b1a2open = a1b1open a2b1openandleta1b1a2closedbethecomplementaryevent a1b1closed a2b1closedThis andsimilar results will beprovedbypassingtotherandombipartitegraphG(m,n; p). Fromthedenitionof Gpn(m),wehavepnaiajopen = k : aibkajopen= 1 k : aibkajclosed= 1 aib1ajclosedn= 1 (1 aib1, ajb1open)n= 1 (1 p2)n(3.3)We dene as the measure on 0,1Emfor which each edge is open with probability(2 )/m, independentlyofall otheredges. Thegoal ofthefollowingtwolemmasistoshowthat _DpnforlargemusingHolleysTheorem. SeeSection1.3.2.a1a2a3a4b1b2b3b4b5a1a2a3a4Figure3.1: Left: ExamplesubgraphHofK4,5. Right: TheevenprojectionofH.213.1.Theevenprojection 3.THEEVENPROJECTIONa1a4a2a3a5a6a7a8Figure3.2: ExamplecongurationonK8. Wesetx = a1a2sothatd1= 2,d2= 3andd = 1.Lemma15. Letx Em. Denotetheendpointsofxbya1,a2andlet 0,1Em\{x}beanedge-congurationonthecompletegraphinwhicha1, a2haved1,d2neighboursrespectively,andtherearedverticeswhichareneighbourstobotha1anda2. Thenpn(X(x) = 1 [ X= ox) 1 (1 p)2d_1 p2(1 +p)2md1d24_nd1d2(3.4)withequalityifd = 0. Inparticular,wehavepn(X(x) = 1 [ X= 0ox) pn(X(x) = 1 [ X= ox) (3.5)forall 0,1Em\{x}andall x Em.Proof. Letxbetheedgea1a2andx 0,1Em\{x}. Forsimplicity, wewill maketheassumptionthat(aiaj)=0wheneveraiajisnotadjacenttoa1a2. Thiscanbedone without loss of generality, since aiajis independent of a1a2when the edges are notadjacent.Suppose 0,1Em,nis such that the even projection of is . Then must containd1 doddverticesbksuchthata1bkaiisopenfori=i1, i2, ..., id1d, andd2 doddverticesbksuchthata2bkajisopenforj =j1,j2,...,jd2d, wherei1,...,id1d,j1,...,jd2dareall distinct andgreater than2. Thebkmust all bedistinct, sinceotherwisetheprojectionwouldcontainanopenedgeaiajwhichisnotadjacenttoa1a2.Furthermore, foreachvertexaisuchthata1aianda2aiarebothopenin, theremust be odd vertices bk1,bk2, possibly equal, such that a1bk1ai and a2bk2ai are both open.Note that if k1= k2, then a1bk1a2will be open, and X(x) = 1 follows immediately. Thusweassumethatwhenevera1aianda2aiarebothopenin, therearedistinctbk1,bk2denedas above. Since therearedvertices aisatisfyingthis, werequire dpairs ofverticesbk1, bk2intobesuchthata1bk1aianda2bk2aiarebothopen.Bytheargumentabove,thereisinsomesenseaminimalcongurationwhichhasevenprojectionandhasanontrivialprobabilityofhavingX(x) = 1. Withoutlossofgenerality,wecansupposethathasacanonicalformobtainedasfollows.1. Leta1bkak+2beopenfork = 1,...,d1.223.1.Theevenprojection 3.THEEVENPROJECTIONa1a2a3a4a5a6a7a8b1b2b3b4b5b6b7b8b9Figure 3.3: Conguration on K8,9 obtained from in gure 3.2 by the algorithm describedin the proof of Lemma 15. The set of edges in this gure is denoted A. No edges can be addedto b1, b3 or b4 without changing the projection of , whence R = 2,5,6,7,8,9. Opening eitheroneof the2ddashededgeswill makea1a2openintheprojectionwithoutaectingotheredges. Thiscanotherwiseonlybeachievedby,forsomek> 5,openingthetwoedgesa1bkanda2bk.2. Leta2bkak+2beopenfork = d1 + 1,...,d1 +d2d.3. Leta2bkak+2d2beopenfork = d1 +d2d + 1,...,d1 +d2.Denotethesetof edgesopenedinthisprocessbyA. Also, letB= aibkaj:1 k n, (ij) = 0. ThenpnX(x) = 1 [ X= ox r : a1bra2open [ Aopen, Bclosed (3.6)where A open should be read as open for all A, and Bclosed means closedforall B. Heretheinequalitycomesfromhowweassignedoddverticestothedcommon neighbours. If d = 0, there is no such freedom of choice and equality must hold.Note that for r = 1,...,d1d, we have a2brclosed since a2brar+2 Bbut a1brar+2 A. Similarly, a1brar+2 Bwhilea2brar+2 Aforr=d1 + 1,...,d1 + d2 dsoa1brisclosed. LetR = 1,...,n (1,...,d1d d1 + 1,...,d1 +d2d). Then r : a1bra2open [ Aopen, Bclosed= r R : a1bra2open [ Aopen, Bclosed= 1 r R : a1bra2closed [ Aopen, Bclosed (3.7)LetR

1= d1d + 1,...,d1,R

1= d2d + 1,...,d2,andR2= d2 + 1,...,n. ThenR = R

1R

1 R2,andthesetsaredistinct. Intherstcase,sayr = r0 R

1,wehave,sincea1br0a2dependsonlyoneventsinAandBcontaininga1br0ora2br0, a1br0a2closed [ Aopen, Bclosed= a1br0a2closed [ a1br0ar0+2open, a1br0ajclosed, j> d1 + 2= a2br0closed [ ar0+2br0open, ajbr0closed, j> d1 + 2= 1 p (3.8)Thesamewillholdforr0 R

1. Nowletr0 R2. Wehave a1br0a2closed [ Aopen, Bclosed= 1 a1br0a2open [ Aopen, Bclosed= 1 a1br0open [ Aopen, Bclosed a2br0open [ Aopen, Bclosed (3.9)233.1.Theevenprojection 3.THEEVENPROJECTIONThetwofactorsaresimilar. Wehave a1br0open [ Aopen, Bclosed= a1br0open [ a1br0ajclosed, j> d1 + 2= a1br0open, a1br0ajclosed, j> d1 + 2 a1br0ajclosed, j> d1 + 2= a1br0open, ajbr0closed, j> d1 + 2 a1br0ajclosed, j> d1 + 2=p(1 p)md12(1 p2)md12=p(1 +p)md12(3.10)andthesameformulawithd2holdsfora2br0. Thus1 a1br0open [ Aopen, Bclosed a2br0open [ Aopen, Bclosed= 1 _p(1 +p)md12__p(1 +p)md22_= 1 p2(1 +p)2md1d24(3.11)Returningto(3.7),weobtainpnX(x) = 1 [ X= ox 1 r R : a1bra2closed [ Aopen, Bclosed= 1 _r R

1: a1bra2closed [ Aopen, Bclosed__r R

1: a1bra2closed [ Aopen, Bclosed_ r R2: a1bra2closed [ Aopen, Bclosed= 1 _a1br

1a2closed [ Aopen, Bclosed_|R

1|_a1br

1a2closed [ Aopen, Bclosed_|R

1| a1br2a2closed [ Aopen, Bclosed|R2|= 1 (1 p)2d_1 p2(1 +p)2md1d24_nd1d2(3.12)whichisthedesiredresult. Inparticular,pnX(x) = 1 [ X= 0ox = 1 _1 p2(1 +p)2m4_n(3.13)anditisnothardtoseethatpnX(x) = 1 [ X= 0ox pnX(x) = 1 [ X= ox (3.14)243.1.Theevenprojection 3.THEEVENPROJECTIONThefollowinglemmaisaconsequenceofLemma15.Lemma16. Letm = o(n)and > 0. Letbethemeasureon 0,1Emwhichassignsprobability2mtoeachedgeindependently. Thenformlargeenough,X(x) = 1 [ X= 0ox pnX(x) = 1 [ X= 0ox , x Em(3.15)Proof. ByLemma15andtheindependenceofedgesunder,thisisequivalentto2m 1 _1 _p(1 +p)m2_2_n(3.16)Wehave(1 +p)m= 1 +_mn (1 +o(1)),so1 _1 _p(1 +p)m2_2_n= 1 __1 _p(1 +p)21 +_mn (1 +o(1))_2__n= 1 ___1 __mn_1 + 2mn+2mn_1 +_mn (1 +o(1))__2___n= 1 _1 _(mn + 2mn +2)mnmn +m2n(1 +o(1))_2_n= 1 _1 _mn(1 +o(1))_2_n= 1 _1 2mn(1 +o(1))_n=2m+o_1m_(3.17)andsoformlargeenough,theresultfollows.Remark. Theinequality(3.16)infactdoesnotholdforanysequencem,nsuchthatm > n. Thisisthereasonthatwestudyonlytheevenvertices,justasinSection2.Theorem17. Letm = o(n). Forany > 0weeventuallyhave _Dpn.Proof. Usingtheedge-independenceoffollowedbyLemmas16and15, wehaveforlargemX(x) = 1 [ X= ox = X(x) = 1 [ X= 0ox pnX(x) = 1 [ X= 0ox pn_X(x) = 1 [ X=

ox_(3.18)forany,

0,1Em\{x}. Thus, _DpnfollowsfromHolleystheorem.253.2.Applicationsofevenprojection 3.THEEVENPROJECTION3.2 ApplicationsofevenprojectionTheorem 17 means that any increasing property of the i.i.d. graph with edge-probability(2 )/mwill eventuallybesatisedbytheevenprojection. Propertiesof G(m,n; p)maythenbeconcludedfrompropertiesofitsevenprojection. Inthissectionwestatecorollaries of Theorem 17. First,we reprove the existence of a giant component withoutusinganyresultsfromSection2.Corollary18. Letm=o(n). If >1, thenthereisa.a.s. acomponentin G(m,n; p)withat least mevenvertices, where m=m() (0,1) is the unique solutiontom +e2m= 1.Proof. Let > 0beconstantandsuchthat2 > 1. Form,nlargeenoughtheevenprojectionmeasuresatisespn _Danditfollowsthatif wedene (0,1)tobetheuniquesolutionto +e(2)= 1,pnComponentofsize m(1 +o(1)) Componentofsize m(1 +o(1))= 1 o(1) (3.19)byTheorem9. Bythecontinuityof ,wemusthavepnComponentofsize mm(1 +o(1)) 1 (3.20)Thus the even projection has a component with at least mm(1+op(1)) even vertices,whichmeansthattherandombipartitegraphmusthaveatleastmm(1 + op(1))evenverticeswhichbelongtothesamecomponent.Note that without using results from Section 2, this corollary only shows the existenceof a large component, and it is possible that there is more than one component with (m)evenvertices. However,resultsobtainedfromTheorem17typicallyshowpropertiesforanylargecomponent, andusingresultsfromSection2will thenimplythattheresultholdsfortheuniquegiantcomponent.While revertingfromthe projectedgraphtoG(m,n; p) without losingtoomuchinformationwas possible inCorollary18, the statement of the followingtheoremissignicantlyweakenedcomparedtothecorrespondingstatementforcompletegraphs.Thereasonforthisis,looselyspeaking,thatthepropertyofhavingagiantcomponentis a property relating to connectivity of vertices, while the property of containing a cyclerelatestoadjacencyofedges.Theorem19. Letm = o(n). Thenthereexistsa0suchthatif > 0,thena.a.s.thegiantcomponentin G(m,n; p)containsacycle.Proof. Let , > 0. According to [1], there is a 0> 1 such that the giant component ofG(m, (2)/m)containsacycleoflengthatleastmforall > 0. Since _Dpn263.2.Applicationsofevenprojection 3.THEEVENPROJECTIONforlargem,theremustbealargecomponentwithacycleoflengthatleastmintheevenprojectionof G(m,n; p).Supposethereis nocycleinthegiant component of G(m,n; p). Then, sincetheevenprojectionhas acycleof lengthm, theremust beanoddvertexb0suchthatdeg b0 m, asthisistheonlywaytoobtaincyclesintheevenprojectionfromanacyclicconiguraftion. Butwehavedeg b0 Bi(m,p),so k : deg bk m = 1 k : deg bk< m= 1 deg b1< mn= 1 __1 m

j=m_mj_pj(1 p)mj__n= 1 (1 o(1/n))n= o(1) (3.21)so there must a.a.s. be a cycle in the giant component of G(m,n; p), if we can show thatthesumisindeedo(1/n). Butnm

j=m_mj_pj(1 p)mj maxjmmn_mj_pj(1 p)mj maxjmnmjjj!mnj maxjmj+1nm(m)!mnj maxjmj+1nm(m)memmnjm3mem(m)m= o(1) (3.22)andtheresultfollows.274Therandom-clustermodelThissectionprovestheexistenceof agiantcomponentfortherandom-clustermodel G(m,n; p,q)withparameterq 1andp=/mn. When1 q 2andm=o(n), asharpthresholdvalueof isfoundfortheexistence, whileforothercasespartialresultsarefound. Mostimportantly,asharpthresholdvalueisfoundfortheIsingmodelwhenm=o(n). Theprooftechniquesimitatethoseof [3] closely, andthissectionshowsthatsomeof thereductionargumentsusedthereapplydirectlytothebipartitesetting.Werecall fromSection1.4.3thattherandom-clustermodel withparameterq>0assignsprobabilityproportionaltoP(F; E,p,q) = p|F|(1 p)|E||F|qc(V,F)(4.1)toanysubgraph(V,F)of(V,E)whichhasc(V,F)connectedcomponents.4.1 ColouringargumentThe proof techniques in [3] depend strongly on a colouring argument, which reduces thestudy of G(m,n; p,q) to that of G(M,N; p,1) for some random numbers M,N. Fix 0 r 1. Givenarandomgraph G(m,n; p,q),eachcomponentiscolouredredwithprobabilityrandgreenwithprobability1 r. Thecomponentsarecolouredindependentlyofeachother. Lettheunionofthered(green)componentsbethered(green)subgraph,andletRdenotethevertexsetoftheredsubgraph. Thefollowinglemmarelates G(m,n; p,q)toarandom-clustergraphwithparameterrq.Lemma20. LetV1beasubsetofVm,nwithm1evenandn1oddvertices. ConditionalonR = V1,theredsubraphof G(m,n; p,q)isdistributedas G(m1,n1; p,rq)andthegreensubgraphas G(mm1,nn1; p,(1 r)q);furthermore,theredsubgraphisconditionallyindependentofthegreensubgraph.284.2.Existenceofagiantcomponent 4.THERANDOM-CLUSTERMODELProof. SetV2= V V1,m2= mm1,n2= n n1,andletE1, E2 Em,nbesuchthatedges in Ei have endpoints in Vi only, i = 1,2. Write c(V,E) for the number of componentsofagraphwithvertexsetV andedgesetE. Thenc(V, E1 E2) = c(V1,E1) +c(V2,E2)sinceE1andE2denedisjointcomponents,andtheprobabilitythattheredsubgraphis(V1,E1)andthegreensubgraphis(V2,E2)satisesPG(m,n; p,q) = (V,E1 E2)andR = (V1,E1)=_p|E1E2|(1 p)mn|E1E2|qc(V,E1E2)Z(m,n; p,q)_rc(V1,E1)(1 r)c(V2,E2)= C(m,n,p,q,m1,n1)p|E1|(1 p)m1n1|E1|(qr)c(V1,E1)p|E2|(1 p)m2n2|E2|(q(1 r))c(V2,E2)= C(m,n,p,q,m1,n1)PE1; V1,p,rq PE2; V2,p,(1 r)q (4.2)forsomepositiverealC(m,n,p,q,m1,n1)dependingonlyonm,n,p,q,m1andn1. Hence,conditional onR=V1andthegreensubgraphbeing(V2,E2), theprobabilitythattheredsubgraphis(V1,E1)ispreciselyPE1; V1,p,rq.Writing(M,N)=(m1,n1)forthenumberof redvertices, RisthusdistributedasarandomgraphG(M,N; p,rq) onarandomnumber of vertices. Inparticular, it isdistributedas G(M,N,p,1)ifq 1andr=q1. Byusingdistributional propertiesofM,Nandusingresultsfromearliersections,onemaydeducepropertiesof G(m,n; p,q).4.2 ExistenceofagiantcomponentImitating [3] further, we start the search for a giant component by proving the followinglemma. Weassumethroughoutthissectionthatthereisana > 0suchthatm an.Lemma 21. Let q 1. For any ,=1, a.a.s. G(m,n; /mn,q) has at most onecomponentsuchthatatleastoneofthefollowinghappens: Thecomponenthasatleast(mn)1/3vertices Thecomponenthasatleastm3/4evenvertices Thecomponenthasatleastn3/4oddverticesProof. Let Lm,n,p,qbe the number of components of G(m,n; p,q) having at least (mn)1/3vertices, oratleastm3/4evenvertices. SupposeLm,n,p,q 2, andpicktwooftheseinsome arbitrary way. With probability r2both of these are coloured red. Setting r = q1,wendthatr2PLm,n,p,q 2

k,lkl(mn)1/3orkm3/4PLk,l,p,1 2 P[R[ = (k,l) maxk,lkl(mn)1/3orkm3/4PLk,l,p,1 2 0 asm (4.3)294.2.Existenceofagiantcomponent 4.THERANDOM-CLUSTERMODELfromwhichfollowsthata.a.s.Lm,n,p,q 1,usingknownresultsaboutLk,l,p,1.Thelaststatementfollowsfrom(mn)1/3n3/4forlargem,n, whichfollowsfromtheassumptionthatm anforsomea > 0.Let m(0,1] be the maximal number suchthat acomponent has mmevenvertices, andletn (0,1] bethemaximal numbersuchthatacomponenthasnnoddvertices. Notethat weneednot (apriori) haveacomponent withmm + nnvertices, becausetwodierentcomponentsmightdenem,n. ThisissueisresolvedinTheorem28.Lemma22. Letq 1andr= q1. Withprobabilityr, G(m,n; p,q)hasmm+ r(1 m)m+op(m)evenredvertices,ofwhichmmbelongtothecomponentwiththemostevenvertices. Withprobability1 r, therearer(1 m)m + op(m)evenverticesandnocomponentwithmorethanm3/4evenvertices.Thesamestatementswithmreplacedbynholdfortheoddcase.Proof. Suppose G(m,n; p,q)hascomponents, andsupposethatthecomponentshavemm, v2, v3,...,vevenvertices. Callanyevenvertexnotinthecomponentwithmmeven vertices small. By the lemma above, we have max vi m3/4. Conditional on m,theexpectednumberofsmallevenredverticesis

i=2vir = r(1 m)m (4.4)andthevarianceisgivenby

i=2v2ir(1 r)

i=2v2i mmaxi2vi m7/4(4.5)Hencetherearer(1 m)m + op(m)smallredevenverticesby(1.6). Thecomponentwhichhasmmevenverticesisredwithprobabilityr,andtheresultfollows.Thesameargumentappliestotheoddvertices,withmax vi n3/4.Lemma23. Ifm = o(n),thena.a.s.there isnocomponentwithatleastnoddverticesforany> 0. Inotherwords,wehavenP 0andN= rn(1 +op(1)).Proof. Let>0andletKm,n,p,qbethenumberof componentsin G(m,n; p,q)whichhaveatleastnoddvertices. Notethata.a.s.M=o(N)byLemma22, andweneedonlyconsider G(k,l; p,1)fork,lsuchthatk = o(l). Thus,rPKm,n,p,q 1

k=o(l)PKk,l,p,1 1 P[R[ = (k,l) maxk=o(l)PKk,l,p,1 1 0 (4.6)whichfollowsfromearliersections.304.2.Existenceofagiantcomponent 4.THERANDOM-CLUSTERMODELLemma24.(a)Letm = o(n). If > q 1,thenthereexistsa0> 0suchthata.a.s.m 0for G(m,n; /mn,q).(b)Let m=an, a>0. If >q 1, thenthereexistsa0>0suchthat a.a.s.m 0andn 0for G(m,n; /mn,q).Proof. (a)Forq=1, theassertionwasshowninTheorem9. Henceassumeq>1andthusr = q1< 1.Let0= 1 _+q2_2,m= Pm< 0,and > 0. Byconsideringtheeventthatthe component with mm even vertices is coloured green we see that, with probability atleast (1r)m+o(1), the number Mof red even vertices satises M r(10)mm,andtherearenoredcomponentswithatleastm3/4evenvertices. Whenthishappens,MNp _r(1 0)mm_rn +op(n)mn= _r2(1 0) r +op(1)=_ 2q_2+_12_2+2q r +op(1)> 1 (4.7)for mlargeenoughwithanappropriatechoiceof >0. Thus theredsubgraphisasupercritical Erdos-Renyi graph, andbyTheorem9hasacomponentwithatleastM (r(1 0) )mevenverticesforsome> 0. Butthisiseventuallylargerthatm3/4,sowemusthave(1 r)m 0asm 0,andinparticularPm< 0 0.(b)Thecasem=anissimilarlyhandled, butmoreworkneedstobedonesinceLemma23cannotbeapplied. Let1= 1 +q2,m= Pm< 1,n= Pn< 1and>0. Arguingasinthepreviousproof, withprobability(1 r)2mn + o(1)wehaveM r(1 1)mmandN r(1 1)n nandnoredcomponentwithmorethanm3/4evenverticesormorethann3/4oddvertices. Whenthishappens,MNp _r(1 1)mm_r(1 1)n nmn= (r(1 1) )=2q+12 > 1 (4.8)formlargeenough. Asabove,itfollowsthatmn 0,i.e. m,norbothgotozero.Supposem 0. Theothercaseishandledidentically. Wewill showthatn min(a, q1)m. DeneMastheproportionof redevenverticesthatbelongtotheredcomponentwiththemostevenvertices, anddeneNsimilarly. ThesenumbersmustsatisfyMM=mmandNN=nn. ByLemma13wea.a.s. haveNmin(A,1)M,whereA = M/N. Wehave314.3.Sizeofthegiantcomponent 4.THERANDOM-CLUSTERMODELn=NNnNnmin(A,1)mmM(4.9)butnotingthatN/M= 1/Aandm/n = a,wegetn min_a,aA_m. (4.10)Wea.a.s.haveM mandN rn = n/q,soA qaandn min(a, q1)m(4.11)Soletting0= min(a,q1)1,wea.a.s.havem 0andn 0.4.3 SizeofthegiantcomponentGiven > 0anda > 0,wedenethefunctions: R Rand,a: R2R2by() = e2q1 1 + (q 1)(4.12)and,a(1,2) =__exp_aq(1 + (q 1)2)_exp_a1_1__111+(q1)1exp_qa(1 + (q 1)1) (exp a2 1)_121+(q1)2__(4.13)Lemma25.(a)Letm = o(n). Ifq 1,thenforanysequence = mwehave(m)P 0.(b) Let m=an, a >0. If q 1, then for any sequence =mwe have,a(m,n)P (0,0).Proof. (a)For G(m,n; p,1), wehaveshown(Theorems6, 9, 11)thatforconstant [0,)e2m+ m1P 0 (4.14)Denem= 1if = ,sothattheconvergenceholdsforallsequencesinthecompactset[0,]. Letmbeonesuchsequence. Bylookingdownconvergentsubsequencesofm,wehavee2mm+ m1P 0 (4.15)Ifweapplythistotheredsubgraph,distributedas G(M,N; p,1),conditionalonthecomponentwithmmevenverticesbeingredwehaveforanysequencepMep2MMMN+ M 1P 0 (4.16)324.3.Sizeofthegiantcomponent 4.THERANDOM-CLUSTERMODELSincem = o(n),wehaveN= rn(1 +op(1))byLemma23andexp_2mqm_1 m1 + (q 1)m= exp_2mrnnm_+qm1 + (q 1)m1= exp_2mN(1 +op(1))nm_+mm(r + (1 r)m)m 1 (4.17)Letpmbesuchthatm= pmmn. Thenthisequalsexp_p2mmmN(1 +op(1))_+mmM1= exp_p2MMMN_+ M 1P 0 (4.18)wherepM= pm(1 +op(1))issomesequence,andtheresultfollows.(b)Likewise,let m = an,a > 0. For q= 1,we have showninTheorem9 thatforallsequencesM= pMMN,theredsubgraph G(M,N; M/MN,1)mustsatisfyM+ exp_M_NM_exp_M_MN M_1__1P 0 (4.19)Note that N=1q(1+(q1)n)n. For any sequence m= pmmn = pmm/a = pmna,wehaveexp_mqa(1 + (q 1)n)_exp_mam_1__1 m1 + (q 1)m= exp pmN (exp pmmm 1) 1 m1 + (q 1)m=mm(r + (1 r)m)m+ exp pmN (exp pmMM 1) 1=mmM+ exp_pmMN_NM_exp_pmMN_MN M_1__1P 0 (4.20)which follows from (4.19) by letting M= pmMN. The other part of the statement isprovedsimilarly.Becauseof therelativesimplicityof , wemaymakemoredetailedclaimsaboutthegiantcomponentinthe casem = o(n). Form = anhowever,the complexityof,apreventsusfromprovingmoreresultsatthismoment.Tostudy, wedenef()=_q(log(1 + (q 1)) log(1 ))_1/2for0 0. Thus,theonlypossibilityismP (). Thisis(b).When m = o(n), Theorem 26 gives a sharp value of the threshold c(q) for 1 q 2,whileforq> 2allweknowismin c(q) q. Gettingmoredetailedthanthiswhenq> 2islikelytobemuchmoreinvolvedthantheproofspresentedhere,judgingbythecomplexityoftheproofin[3].Wehaveseenabovethatif m=o(n), thennP0. Thefollowingresultslookscloseratthisconvergenceinthesupercriticalregime.Theorem27. Letm = o(n),andsupposeq 1and > q. Thenn=()q_mn (1 +op(1))where()istheuniquepositivesolutiontoe2/q=11+(q1).Proof. WehaveseeninTheorem26thattherea.a.s. isacomponentwithmmevenvertices, wheremP. Withprobabilityr, thiscomponentiscolouredred, andbyLemma 22 the red subgraph is distributed as G((r +(1 r))m,rn,/mn,1). To makethistintoCorollary3wemakeachangeofvariables.Put m

= (r+(1r))m, n

= rn. Then we have a component with (r+(1r))1m

evenverties inthegraphG(m

,n

,_r + (1 r)r/m

n

,1). ByCorollary3, thiscomponenthas_r + (1 r)rm

n

m

r + (1 r)n

(1 +op(1))= rm

r + (1 r)_n

r(1 +op(1))=qmn(1 +op(1)) (4.21)oddvertices.Usingtheproofsinthissection, thefollowingtheoremresolvestheissuedescribedbefore Lemma 22. We show that the unique component containing a positive fraction oftheevenverticesmustbethecomponentwiththemostoddvertices.344.4.Conclusion 4.THERANDOM-CLUSTERMODELTheorem28. Supposem> 0. Thenthereisa.a.s.oneuniquecomponentcontainingmmevenverticesandnnoddvertices.Proof. Supposem=o(n), andletCbeanycomponentbuttheonecontainingmmevenvertices. ByLemma21, thiscomponenthasop(m)evenvertices. Goingthroughthe proof of Theorem 27 for this component, it is seen that it has op(mn) odd vertices,whichisclearlylessthannn.Supposem=an, a>0. ByLemma25, thereisa>0suchthata.a.s., m>and n> . Suppose C1is a component with mm even vertices and C2is a componentwithnnoddvertices. Sincem=an, bothof thesecomponentsa.a.s. haveatleast(mn)1/3vertices,andbyLemma21wemusthaveC1= C2.4.4 ConclusionWeconcludethissectionbystatingthecollectiveresultsinatheorem.Theorem29. Supposeq 1.(i)Letm = o(n).(a)(Subcritical)If 1 q 2and2and q,then G(m,n; /mn,q)hasagiantcompo-nentwithqmn(1 +op(1))vertices,ofwhichmareeven,whereistheuniquepositivesolutiontoe2/q=11+(q1).(ii) Let m = an,a > 0. Then a.a.s. the largest component of G(m,n; p,q) has mm+nnvertices, of which mm are even and nn odd, for some solution (m, n) to ,a(m, n) =0,with,aasin(4.13). If > q,thenm> 0andn> 0.Inparticular,wehavethefollowingimportantcorollaryconcerningtheIsingmodel.Notethatnoinformationisgivenin(b)forthecase < 2.Corollary30. (a) If m=o(n), thenunder the Ising model at inverse temperature= 12 log_1 mn_thereisa.a.s.agiantcomponentifandonlyif > 2. Thegiantcomponenthas2mn(1 +op(1))vertices,ofwhichm(1 +op(1))areeven,whereistheuniquepositivesolutiontoe2/q=11+.(b)Ifm = an,a > 0,theIsingmodel atinversetemperature= 12 log_1 mn_a.a.s. hasagiant component if >2. Thegiant component hasm(1 + op(1)) evenverticesandnn(1 +op(1))oddvertices,where(m,n)issomepairofpositivenumberssatisfying,a(m, n) = 0.Proof. ThisfollowsimmediatelyfromTheorem29andthecorrespondencebetweentherandom-clustermodelandtheIsingmodelexplainedinSection1.4.3.35ATechnicalproofsThisappendixisintendedtospelloutthefullproofsofsometechnicallemmas,whichwereleftunprovedsoastonotclogupthemaintext. Thelemmaswillberestatedandproved.Lemma4. Let < 1,m1 m2 nandp1= /m1n,p2= /m2n. Then_1 p1 +p1_1 p1 +p1_m1_n_1 p2 +p2_1 p2 +p2_m2_n(2.2)Proof. Settingb = nandx = m,theresultfollowsifweshowthatf(x) = xb+xb_1 +1 xb_x(A.1)isincreasingforintegersxsatisfying1 x b2. Usingthebinomial expansionrule,thisequalsf(x) =xb_x

k=1_1 xb_k_xk__=x

k=1(1 )k(xb)k+1(x)kk!(A.2)Here(x)k=x(x 1)...(x k + 1). Thisisclearlyincreasingifeachofthetermsisincreasing,i.e. if(x)k/(xk+1)isincreasing. Sincex N,thisisequivalentto(x)kxk+1 (x + 1)kx + 1k+1, forallx k 1 (A.3)orinotherwords,usingthedenitionof(x)k= x(x 1)...(x k + 1),x k + 1xk+1x + 1x + 1k+1, x k 1 (A.4)36A.TECHNICALPROOFSTheaboveisequivalentto_1 k2_log(x + 1) +_k + 12_log(x) log(x k + 1) 0, x k 1 (A.5)Denotetheleft-handexpressionbygk(x). Theresultwillfollowbyshowingthatgk(x)isdecreasingandhaslimit0asx . Wehaveg

k(x) =1x + 1_1 k2_+1x_k + 12_1x k + 1=x(x k + 1)(1 k) + (x + 1)(x k + 1)(k + 1) 2x(x + 1)2x(x + 1)(x k + 1)=x kx k2+ 12x(x + 1)(x k + 1)= x(k 1) + (k21)2x(x + 1)(x k + 1) 0, x k 1 (A.6)Wealsohavegk(x) = log_xk+12(x + 1)k12(x k + 1)_= log_1(1 +x1)1k2(1 (k 1)x1)_(A.7)sothatlimxgk(x) = 0. Theresultfollows.Lemma13. Supposea>0, >1, andlet m, nbetheuniquesolutionsin(0,1)tom + exp_a (expam 1)_= 1andn + exp_a_exp_an_1__= 1respectively. Thenn min(1,a)m.Proof. Suppose a 1, and let f(x) = x1+exp_a (exp ax 1)_for 0 < x < 1.This function is convex and satises f(0) = f(m) = 0. Thus, we may show that n mbyshowingthatf(n) 0. Wehave,bythedenitionofn,f(n) = exp_a_exp_an_1__+n1 (A.8)= exp_a_exp_an_1__exp_a_exp_an_1__Sinceexisastrictlyincreasingfunction,f(n) 0isequivalentto37A.TECHNICALPROOFSa_exp_an_1_a_exp_an_1_ 0 (A.9)orexp_an_1 a exp_an_+a 0 (A.10)Toshowthat this indeedholds whenever a 1, set y =nandz = a, andconsiderthefunctiong(z) = eyz1 z2ey/z+z2. Thissatisesg(1) = 0andg

(z) = y_eyz eyz_+ 2z_1 eyz_ 0. (A.11)Thusg(z) 0andn mfollows.Wearguesimilarlytoshown amwhena 1. Theconvexfunctionh(x) =xa 1 + exp_a_exp_ax_1__satisesh(am) = 0,soweshowthath(n) 0.Bythedenitionofn,wehave_exp_an_1_=1a log(1 n),soh(n) =na1 + exp_1a log(1 n)_=na1 + (1 n)1/a(A.12)Lettingk(x) =(1 x)1/a+ x/a 1for 0 x 1, wehavek(0) =0andk

(x) =a1_1 (1 x)1/a1_ 0. Soh(n) 0andn am.Fromthisfollowsthatn min(1,a)m.38Bibliography[1] Bollobas, B., Fenner, T. I., Frieze, A. M. (1984) LongCycles inSparseRandomGraphs,GraphTheoryandCombinatorics,AcademicPress,London,U.K.[2] Bollobas,B.(1985)RandomGraphs,CambridgeUniversityPress,Cambridge,U.K.[3] Bollobas,B.,Grimmett,G.andJanson,S.(1996)Therandom-clustermodelonthecompletegraph,Probab.TheoryRelat.Fields104,283-317.[4] Edwards, R.G., andSokal, A.D. (1988) Generalizationof the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm, Phys.Rev. D, 38, 2009-2012[5] Erdos, P. andRenyi, A. (1960)OntheEvolutionof RandomGraphs, PublicationsoftheMathematical InstituteoftheHungarianAcademyofSciences,5,17-61.[6] Georgii, H-O., Haggstrom, O., Maes, C. (2001) The Random Geometry of EqulibirumPhases, Phasetransitionsandcritical phenomena, 18, AcademicPress, SanDiego,CA,1-142.[7] Grimmett, G., Stirzaker, D. (2001) Probability and Random Processes, Third Edition,OxfordUniversityPress,Oxford,U.K.[8] Grimmett,G.(2006)TheRandom-ClusterModel,Springer[9] Ising,E.(1925)BeitragzurTheoriederFerromagnetismus,Z.Phys.31,253-258[10] Janson, S.,Luczak, T. andRucinski, A. (2000) RandomGraphs, JohnWiley&Sons,NewYork[11] Lenz, W. (1920) Beitrag zum verstandnis der magnetischen Erscheinungen in festenKorpern,Phys.Zeitschr.21,613-615[12] Potts,R.B.(1952)Somegeneralizedorder-disordertransformationsm,Proc.Cam-bridgePhil.Soc.48,106-10939BIBLIOGRAPHY[13] Swendsen, R.H. andWang, J-S. (1987)Nonuniversal Critical DynamicsinMonteCarloSimulations,Physical ReviewLetters58,86-8840