The survival probability for critical spread-out oriented percolation above 4+1 dimensions, II. Expansion Citation for published version (APA): Hofstad, van der, R. W., Hollander, den, W. T. F., & Slade, G. (2007). The survival probability for critical spread- out oriented percolation above 4+1 dimensions, II. Expansion. Annales de l'institut Henri Poincare (B): Probability and Statistics, 43(5), 509-570. https://doi.org/10.1016/j.anihpb.2006.09.002 DOI: 10.1016/j.anihpb.2006.09.002 Document status and date: Published: 01/01/2007 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 12. Mar. 2023 Ann.I.H.Poincaré–PR43(2007)509–570 www.elsevier.com/locate/anihpb The survival probability for critical spread-out oriented percolation + above 4 1 dimensions. II. Expansion Remco van der Hofstada,∗, Frank den Hollanderb,c, Gordon Sladed aDepartmentofMathematicsandComputerScience,EindhovenUniversityofTechnology,P.O.Box513,5600MBEindhoven,TheNetherlands bMathematicalInstitute,LeidenUniversity,P.O.Box9512,2300RALeiden,TheNetherlands cEURANDOM,P.O.Box513,5600MBEindhoven,TheNetherlands dDepartmentofMathematics,UniversityofBritishColumbia,Vancouver,BCV6T1Z2,Canada Received22September2005;receivedinrevisedform12July2006;accepted12September2006 Availableonline15December2006 Abstract Wederivealaceexpansionforthesurvivalprobabilityforcriticalspread-outorientedpercolationabove4+1dimensions,i.e., theprobabilityθn thattheoriginisconnectedtothehyperplaneattimen,atthecriticalthresholdpc.Ourlaceexpansionleads toanon-linearrecursionrelationforθn,withcoefficientsthatweboundviadiagrammaticestimates.Thislaceexpansionisfor point-to-planeconnectionsanddifferssubstantiallyfrompreviouslaceexpansionsforpoint-to-pointconnections.Inparticular,to beabletodeducetheasymptoticsofθnforlargen,weneedtoderivetherecursionrelationuptoquadraticorder. ThepresentpaperisPartIIinaseriesoftwopapers.InPartI,weusetherecursionrelationandthediagrammaticestimates toprovethatlimn→∞nθn=1/B∈(0,∞),andalsodeduceconsequencesofthisasymptoticsforthegeometryoflargecritical clustersandfortheincipientinfinitecluster. ©2006ElsevierMassonSAS.Allrightsreserved. Keywords:Orientedpercolation;Laceexpansion;Survivalprobability;Criticalexponent;Nonlinearrecursion 1. Introductionandresults FororientedbondpercolationonZd×Z+ withparameterp,thesurvivalprobabilityθn=θn(p)attimen∈Z+ is theprobabilitythatthereexistsan x∈Zd suchthat (0,0)isconnectedto(x,n).Intheorientedsetting,itisknown thatthereisnopercolationatthecriticalthresholdp=pc [2,4],sothatlimn→∞θn(pc)=0.Ourgoalistostudythe mannerinwhichθ (p )tendstozeroasn→∞whend>4. n c In the present paper, we derive a lace expansion for θ (p), valid in all dimensions d (cid:2)1 and for quite general n modelsoforientedpercolation.Thislaceexpansiongivesanonlinearrecursionrelationforθ (p).Iftheexpansionis n tobeuseful,thenthecoefficientsintherecursionrelationneedtobeestimated.Weproveestimatesvalidatp=p in c dimensionsd >4,forsufficiently“spread-out”orientedbondpercolation(definedbelow),withthedegreetowhich connectionsarespreadoutinspaceparameterisedbyasufficientlylargeL∈N. * Correspondingauthor. E-mailaddresses:[email protected](R.vanderHofstad),[email protected](F.denHollander),[email protected](G.Slade). 0246-0203/$–seefrontmatter ©2006ElsevierMassonSAS.Allrightsreserved. doi:10.1016/j.anihpb.2006.09.002 510 R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 In Part I [7], we have shown how these results can be used in an induction analysis for the recursion relation to concludethatthereisaconstantB=B(d,L)suchthat,asn→∞, (cid:2) (cid:3) (cid:4) (cid:5) 1 θn(pc)−θn+1(pc)= Bn2 1+O n−1logn +L−dO(δn) ford>4andLsufficientlylarge, (1.1) where ⎧ ⎨n−(d−4)/2logn (4<d<6), δn=⎩n−1log2n (d=6), (1.2) n−1logn (d>6). In other words, the critical extinction probability θn(pc)−θn+1(pc), which is the probability that the cluster of the originsurvivestotimenbutnottotimen+1,isasymptoticto1/(Bn2)asn→∞,withaccurateerrorbounds.By summingovern,weconcludethat (cid:2) (cid:3) (cid:4) (cid:5) 1 θ (p )= 1+O n−1logn +L−dO(δ ) ford>4andLsufficientlylarge, (1.3) n c n Bn which is the main conclusion of Part I. In terms of the critical exponent ρ, defined by the conjecture that θ (p ) n c behavesliken−1/ρ asn→∞,(1.3)impliesthatρ existsandisequalto1,ford>4andLsufficientlylarge. Also in Part I, interesting consequences for the geometry of large critical clusters and for the incipient infinite cluster were deduced from (1.3), using results from [8]. In particular, (1.3) implies that two constructions for the incipientinfiniteclustercoincideandthat,conditionallyonsurvivaluptotimen,thenumberofverticestowhichthe originisconnectedattimenscaleslikentimesanexponentialrandomvariable. 1.1. Themodel Thespread-outorientedpercolationmodelisdefinedasfollows.LetZ+={n∈Z: n(cid:2)0}.Considerthegraphwith vertices Zd ×Z+ andwithdirectedbonds ((x,n),(y,n+1)),for n∈Z+ and x,y∈Zd.Let D beafixedfunction D:Zd →[0,1],satisfying (cid:9) D(x)=1. (1.4) x∈Zd The function D will be assumed to be invariant under the symmetries of Zd (permutation and reflection of coor- dinates). Let p ∈[0,(cid:6)D(cid:6)−∞1], where (cid:6)·(cid:6)∞ denotes the supremum norm, so that pD(x)(cid:3)1 for all x ∈Zd. We associatetoeachdirectedbond((x,n),(y,n+1))anindependentrandomvariabletakingthevalue1withprobabil- itypD(y−x)andthevalue0withprobability1−pD(y−x).Wesaythatabondisoccupiedwhenthecorresponding randomvariableis1andvacantwhenitis0.Notethatp isnotaprobability.Rather,p istheaveragenumberofoc- cupiedbondsfromagivenvertex.ThejointprobabilitydistributionofthebondvariableswillbedenotedbyP and p thecorrespondingexpectationbyE ,withtheparameterpusuallysuppressedfromthenotation. p Forthediagrammaticestimates,weneedtomakefurtherassumptionsonD.Wewillrefertotheassumptionson D inthepreviousparagraphastheweakassumptionsonD.Wedefinethespread-outmodeloforientedpercolation to be the model in which D obeys the weak assumptions together with Assumption D in [13, Section 1.2] (whose preciseformisnotimportantforthepresentpaper),and[14,Eq.(1.2)].AssumptionDin[13,Section1.2]involvesa parameterL∈N,whichservestospreadouttheconnectionsandwhichwillbetakentobefixedandlarge.Asimple andbasicexampleis (cid:10) D(x)= (2L+1)−d if(cid:6)x(cid:6)∞(cid:3)L, (1.5) 0 otherwise. Inthisexample,thebondsaregivenby((x,n),(y,n+1))with(cid:6)x−y(cid:6)∞(cid:3)L,andabondisoccupiedwithprobability p(2L+1)−d.AssumptionDalsoallowsforcertaininfiniterangemodels.Forthespread-outmodel,wewilluse β=L−d (1.6) asasmallparameter.AssumptionDimpliesthatthereisafinitepositiveconstantC suchthat sup D(x)(cid:3)Cβ. (1.7) x∈Zd R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 511 We say that (x,n) is connected to (y,m), and write (x,n)→(y,m), if there is an oriented path from (x,n) to (y,m)consistingofoccupiedbonds.Notethatthisisonlypossiblewhenm(cid:2)n.Byconvention,(x,n)isconnected toitself.Wewrite (cid:11) (cid:12) C(x,n)= (y,m)∈Zd ×Z+: (x,n)→(y,m) (1.8) todenotetheforwardclusterof(x,n).Wealsowrite(x,n)→mtodenotetheeventthatthereisay∈Zd suchthat (x,n)→(y,m). The event {(0,0)→∞} is the event that {(0,0)→n} occurs for all n. There is a critical threshold p ∈(0,∞) c suchthattheevent{(0,0)→∞}hasprobabilityzeroforp(cid:3)p andhaspositiveprobabilityforp>p .Thepara- c c metrisationwehavechosenisconvenient,sinceforthespread-outmodelitisknownthat (cid:3) (cid:4) p =1+cL−d +O L−d−1 asL→∞, (1.9) c ford>4,withthepositiveconstantc givenexplicitlyintermsoftheGreenfunctionfortherandomwalkwithstep distributionD[10]. Thesurvivalprobabilityattimenisdefinedby (cid:3) (cid:4) θ (p)=P (0,0)→n . (1.10) n p Generalresultsof[2,4]implythat limn→∞θn(pc)=0.Forthespread-outmodelindimension d >4,with L suffi- cientlylarge,thesameconclusionwasshownin[1]tofollowfromthetrianglecondition.Thetriangleconditionwas verifiedundertheabovehypothesesin[14,16],yieldinganalternateproofthatlimn→∞θn(pc)=0ford>4,andL sufficientlylarge. 1.2. Maintheorem Forn∈Z+,x∈Zd,andp∈[0,(cid:6)D(cid:6)−∞1],wedefinethetwo-pointfunction (cid:3) (cid:4) τ (x)=P (0,0)→(x,n) . (1.11) n p Wewrite (cid:9) τ = τ (x) (1.12) n n x∈Zd for the expected number of vertices in C(0,0) at time n. The lace expansion for the two-point function [5] (see also[14])yieldsarecursionrelationforτ ,whichreads n (cid:9)n−1 τn= πmpτn−m−1+πn, (1.13) m=0 where(π )arecertainp-dependentcoefficients.Infact,(1.13)uniquelydefines(π ),butthelaceexpansionprovides m m a useful representation for (π ). In [14, Proposition 2.2], this representation was used to prove that π =1,π =0 m 0 1 andthatthereexistsafinitepositiveconstantC suchthat π C β |π |(cid:3) π (p=p , m(cid:2)2), (1.14) m (m+1)d/2 c forthespread-outmodelindimensionsd>4,withβ of(1.6)sufficientlysmall.Inaddition,underthesameassump- tions,itisshownin[14,Eq.(2.11)]that (cid:9)∞ π p =1. (1.15) m c m=0 In the present paper, we obtain a lace expansion for the survival probability θ , with good bounds valid for the n spread-outmodelindimensionsd >4atp=p .Ourmainresultisthefollowingtheorem.Initsstatement,weuse c thenotation ⎧ ⎨n−(d−4)/2logn (4<d<6), Δn=⎩n−1logn (d=6), (1.16) n−1 (d>6). 512 R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 Theorem1.1(Laceexpansionanddiagrammaticestimates). (i) Ford(cid:2)1,p∈[0,(cid:6)D(cid:6)−∞1],andn(cid:2)1,andundertheweakassumptiononD, (cid:9)n−1 (cid:7)(cid:9)n/2(cid:8) (cid:9)n θn(p)= πm(p)pθn−1−m(p)− φm1,m2(p)θn−m1(p)θn−m2(p)+en(p), (1.17) m=0 m1=1m2=m1 where(π )areasin(1.13),and(φ )and(e )aregivenbyexplicitformulas(seeSections4,5). m m1,m2 n (ii) For the spread-out model in dimensions d >4, at the critical value p=p , there are finite positive constants c C ,C ,andβ suchthat,for0<β(cid:3)β ,thecoefficients(φ )andtheerrorterms(e )satisfythefollowing φ e 0 0 m1,m2 n estimates: (cid:13) • φ (p )= 1p2 D(x)(1−D(x))= 1[1+O(β)]and,form (cid:2)m (cid:2)1suchthat(m ,m )(cid:9)=(1,1), 1,1 c 2 c x∈Zd 2 2 1 1 2 (cid:14) (cid:14) (cid:14)φ (p )(cid:14)(cid:3)C β(m +1)−(d−2)/2(m −m +1)−(d−2)/2. (1.18) m1,m2 c φ 1 2 1 • Ifθ (p )(cid:3)C (m+1)−1 for0(cid:3)m(cid:3)nandsomeC (cid:2)1,then m c θ θ (cid:14) (cid:14) (cid:2) (cid:5) (cid:14)en+1(pc)(cid:14)(cid:3)CeCθ3(n+1)−2 (n+1)−1+βΔn+1 . (1.19) Notethatthediagrammaticestimate(1.19)foren+1,whichistheerrortermin(1.17)forθn+1,assumesabound in the recursion relation for θ only for 0(cid:3)m(cid:3)n. This is precisely what opens up the possibility of the inductive m analysisemployedinPartI.Namely,inPartI,(1.1)isdeducedfromTheorem1.1byapplyinganinductionanalysis to (1.17), which makes use of the bounds in (1.14), (1.18) and (1.19) in order to moderate the coefficients of the recursion. When we derive (1.17) in Sections 2–5, we will fix an arbitrary p∈[0,(cid:6)D(cid:6)−∞1] and assume only the weak as- sumptiononD.InSections6–8,whereweprovethediagrammaticestimates(1.18),(1.19),wewillspecialisetothe spread-outmodelwithd>4,p=p ,andsmallβ. c WeexpectthatTheorem1.1hasimplicationsalsoforthecriticalcontactprocessinspatialdimensiond>4.Indeed, ithasbeenshownin[9]thatthelaceexpansionforthetwo-pointfunctioncanbeappliedtotheorientedpercolation modelresultingfromtimediscretisationofthecontactprocess.Weexpectthatpart(i)ofthetheoremcanbeapplied similarlytostudythesurvivalprobabilityforthecriticalcontactprocess,inconjunctionwithasuitablemodification ofpart(ii).Theextensionofourresultstothecontactprocesswillbetakenupin[11,12]. 1.3. TheconstantB Itwasshownin[7,Eq.(1.36)]thattheconstantB in(1.3)isgivenby (cid:13) (cid:13) ∞ ∞ φ (p ) B= 1m1+=1p (cid:13)m2∞=m1mπm1,m(p2 )c . (1.20) c m=2 m c Itfollowsfrom(1.9),(1.14)and(1.18)thatB<∞ford>4andβ sufficientlysmall,withB= 1 +O(β)asβ↓0. 2 ˆ Thesurvivalprobabilityθ ofaGalton–Watsonbranchingprocesswhoseoffspringdistributionhasmean1,vari- n anceσˆ2,andfinitethirdmoment,obeysthesimplerecursionrelation θˆn=θˆn−1− σˆ22θˆn2−1+eˆn, (1.21) where en =O(θˆn3−1). This leads to the conclusion that limn→∞nθˆn =2σˆ−2.(cid:13)We sketch the proof of these well- knownfactsinPartI.Considerthebranchingprocesswithoffspringdistribution I ,wheretheI areindependent x x x Bernoullirandomvariableswithpa(cid:13)rameterD(x).Thisoffspringdistributionhasmean1,bythenormalisationassump- tionforD,andhasvarianceσˆ2= D(x)(1−D(x))=1+O(β),asL→∞inthespread-outmodel,by(1.7).We x regardthecriticalspread-outorientedpercolationmodelindimensions d >4 asasmallperturbationofthiscritical branchingprocess—theformerallowsatmostoneparticlepervertex,whereasthelatterallowsmultipleoccupancy. Therecursionrelation(1.17)canbeviewedasaperturbationof (1.21).ThefactthatB = 1[1+O(β)] as L→∞ 2 R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 513 showsthatthesolutionto(1.17)forthespread-outmodelremainsclosetothesolutionof(1.21),toleadingorder,for Llarge. LetN denotethenumberofverticesinC(0,0)attimen,whenp=p ,anddefinetheconstantsAandV by n c (cid:2) (cid:5) 1 A= lim E [N ], V = lim E N2 . (1.22) n→∞ pc n n→∞A3n pc n Itispartoftheresultsin[14]thattheseconstantsexistwhend>4andLissufficientlylarge.Itisshownin[8]that, givennθ (p )→1/B (whichfollowsfrom(1.3)), n c AV B= . (1.23) 2 Itisshownin[14,Eqs.(2.12)and(2.49)]that (cid:9)∞ (cid:9)∞ A= p +p2(cid:13)∞1 mπ (p ), V = ψˆm1,m2(0,0), (1.24) c c m=2 m c m1=2m2=2 ˆ where(ψ )arecoefficientsarisinginthelaceexpansionforthecriticalthree-pointfunction m1,m2 (cid:3) (cid:4) τ (x ,x )=P (0,0)→(x ,n ),(0,0)→(x ,n ) . (1.25) n1,n2 1 2 pc 1 1 2 2 Itfollowsfrom(1.20)and(1.23),(1.24)that (cid:9)∞ (cid:9)∞ (cid:9)∞ (cid:9)∞ V = ψˆ (0,0)=2p φ (p ). (1.26) m1,m2 c m1,m2 c m1=2m2=2 m1=1m2=m1 This implies that the coefficients (φ ) in our lace expansion for the survival probability are related to those m1,m2 appearinginthelaceexpansionforthethree-pointfunction.However,ourapproachdoesnotrevealanexplicitrelation ˆ between ψ (0,0) and φ for fixed m ,m . In [11], an alternate expansion for the three-point function is m1,m2 m1,m2 1 2 derived,whichisquitedifferentfromtheexpansionof[14]andcloserinspirittotheexpansionderivedhereforthe survivalprobability.Theexpansionof[11]leadstoadirectproofthat (cid:9)∞ (cid:9)∞ V =2p φ (p ). (1.27) c m1,m2 c m1=1m2=m1 1.4. Organisation The remainder of the paper is devoted to the proof of Theorem 1.1. The proof is divided into two main parts: (a)thederivationoftheexpansion(1.17)forθ ,and(b)theproofofthediagrammaticestimates(1.18),(1.19)forthe n expansioncoefficients.Thebasicstepsintheproofofeachpartareasfollows. (a)Derivationofthelaceexpansion(1.17).Thestartingpointfortheexpansionisthepercolationlaceexpansionof [5]forthetwo-pointfunction.Thisexpansionwasappliedtoorientedpercolationin[14],whereaderivationof(1.13) can be found. We will extend this lace expansion for the two-point function (a point-to-point expansion) to a lace expansionforthesurvivalprobability(apoint-to-planeexpansion).Therearealternateexpansionsforthetwo-point functionof orientedpercolation,dueto[16] and[17] (see[18] for adescriptionofallthreeexpansions),butwedo notknowhowtousethesealternateexpansionstoobtainanexpansionforthesurvivalprobability. Theexpansionof[5]isbasedonafactorisationlemma,whichweisolateinSection2.InSection3,weextractthe lineartermin(1.17)usingarelativelyminorextensionofthelaceexpansionforthetwo-pointfunction.Thisproduces anequation (cid:9)n−1 θn= πmpθn−1−m+χn, (1.28) m=0 wherethetermχ involvesconfigurationswithtwoconnectionstothehyperplaneattimen.Thesetwoconnections n lead to the quadratic term in (1.17), but two further expansions are required to obtain the two factors θn−m1θn−m2 in(1.17). 514 R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 Thefirstoftheseexpansionsforχ isthemostdelicateandnovelpartofourmethod.Acrucialroleisplayedby n arandomset PA of bonds,whichisdefinedinSection4for anyfixedsubset A of Zd ×Z+.Using PA,weextract afactorθn−m1 fromχn inSection4,completingthefirstexpansionforχn.Then,inSection5,weperformasecond expansion for χn to extract an additional factor θn−m2. Our treatment of this second expansion is different in spirit than the expansion methods used in [6,14], and is simpler due to a careful use of independence that is due to the orientation. Thispartoftheargumentappliesforgeneralpandd,andmakesonlytheweakassumptiononD. (b)Thediagrammaticestimates(1.18),(1.19).Asisusualinlaceexpansionanalyses,wewillprove(1.18),(1.19) byboundingφm1,m2 anden+1 bydiagramsofthesamecharacterastheFeynmandiagramsofphysics,i.e.,bysums of products of two-point functions and survival probabilities. The two-point functions are bounded using estimates provedin[14],andthesurvivalprobabilitiesareboundedusingtheassumptiononθ (p )givenabove(1.19). m c The first step in this procedure is carried out in Section 6, where we generalise the bound on π of [14], stated m abovein (1.14), and proverelated bounds on χn. The bounds on φm1,m2 and en+1 are in terms of diagramsthat are builtfromthediagramsencounteredinSection6usingcertaindiagrammaticconstructions.Usingthese,inSection7, we complete the proof of the bound (1.18) on φ , and in Section 8, we complete the proof of the bound (1.19) m1,m2 onen+1. This part of the argument is for the spread-out model. It relies on d >4 and small β, and the bounds we obtain applyatp=p . c 2. TheFactorisationLemma Thissectioncontainssomepreliminariesthatwillbecrucialintheexpansionforthesurvivalprobability.Themain resultistheFactorisationLemmastatedinLemma2.2below.Throughouttherestofthepaper,wewrite Λ=Zd ×Z+, (2.1) andweuseboldletterssuchasx,y,zforelementsofΛ.TobeabletostatetheFactorisationLemma,weneedsome definitions. Definition2.1. (i) Givena(deterministicorrandom)setofverticesAandabondconfigurationω,wedefineω ,therestrictionof A ωtoA,tobe (cid:10) (cid:3) (cid:4) ω({x,y}) ifx,y∈A, ω {x,y} = (2.2) A 0 otherwise, foreveryx,y suchthat{x,y}isabond.Inotherwords,ω isobtainedfromωbymakingeverybondthatdoes A nothavebothendpointsinAvacant. (ii) Given a (deterministic or random) set of vertices A and an event E, we say that E occurs in A, and write {E inA},ifω ∈E.Inotherwords,{E inA}meansthatE occursonthe(possiblymodified)configurationin A whicheverybondthatdoesnothavebothendpointsin A ismadevacant.Weadopttheconvenientconvention that{x→x inA}occursifandonlyifx∈A. (iii) Given a bond configuration and x ∈Λ, we define C(x) to be the set of vertices to which x is connected, i.e., C(x)={y ∈Λ:x →y}. Given a bond configuration and a bond b, we define C˜b(x) to be the set of vertices y∈C(x)towhichx isconnectedinthe(possiblymodified)configurationinwhichbismadevacant. Wewilloftenusethefollowingeasilyverifiedrulesforoccursin: {EinB}∩{F inB}={E∩F inB}, (2.3) {EinB}∪{F inB}={E∪F inB}, (2.4) {EinB}c={Ec inB}. (2.5) Eqs.(2.3)–(2.5)implythat“occursin”iswellbehavedundersetoperations. R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 515 The following Factorisation Lemma lies at the heart of the expansion method.1 We write I[E] for the indicator functionofaneventE.ThestatementoftheFactorisationLemmaisintermsoftwoindependentpercolationconfigu- rations.Thelawsoftheseindependentconfigurationsareindicatedbysubscripts,i.e.,E denotestheexpectationwith 0 respect to the first percolation configuration, and E denotes the expectation with respect to the second percolation 1 configuration.Wealsousethesamesubscriptsforrandomvariables,toindicatewhichlawdescribestheirdistribution. Thus,thelawofC˜(u,v)(y)isdescribedbyE . 0 0 Lemma 2.2 (Factorisation Lemma).Fix p∈[0,(cid:6)D(cid:6)−∞1], a bond (u,v), a vertex y, a positive integer n,and events E,F whichdependonlyonthestatusofbondswhoseverticeshavetimevariablesatmostn.Then (cid:3) (cid:2) (cid:5)(cid:4) (cid:3) (cid:2) (cid:5) (cid:3) (cid:2) (cid:5)(cid:4)(cid:4) E I EinC˜(u,v)(y),F inΛ\C˜(u,v)(y) =E I EinC˜(u,v)(y) E I F inΛ\C˜(u,v)(y) . (2.6) 0 0 1 0 Moreover, when E ⊆{u∈C˜(u,v)(y),v ∈/ C˜(u,v)(y)}, the event on the left-hand side of (2.6) is independent of the occupationstatusof(u,v). Proof. BecauseofourassumptionontheeventsE andF,wecanreplacethesetC˜(u,v)(y)in(2.6)byitsrestriction toverticeswhichareendpointsofbondswhoseverticeshavetimevariablesatmostn(i.e.,wesetallotherbondsto ˜(u,v) be vacant).We denotethis restrictionby C (y),andnotethat thisis a finiteset withprobability 1 by (1.4).The n proofproceedsbyconditioningon C˜(u,v)(y).Weemphasisethat C˜(u,v)(y) isasetofvertices.Thus, C˜(u,v)(y)=S n n n doesnotdeterminetheoccupationstatusofallthebondsbwithbothverticesinS.Theleft-handsideof(2.6)equals (cid:9) (cid:3) (cid:14) (cid:4) (cid:3) (cid:4) P {EinS}∩{F inΛ\S}(cid:14)C˜(u,v)(y)=S P C˜(u,v)(y)=S , (2.7) n n S wherethesumoverS isoverfinitesubsetsofΛcontainingy. ByDefinition2.1(ii),theevent {EinS} dependsonlyonbondswithbothendpointsin S,whiletheevent {F in Λ\S}dependsonlyonbondswithbothendpointsinΛ\S.Thelatterisequivalenttosayingthat{F inΛ\S}depends onlyonbondsthathavenoendpointsinS.Thus,bytheindependenceofthebondvariables,weobtainthat (cid:3) (cid:14) (cid:4) (cid:3) (cid:14) (cid:4) (cid:3) (cid:14) (cid:4) P {EinS}∩{F inΛ\S}(cid:14)C˜(u,v)(y)=S =P EinS(cid:14)C˜(u,v)(y)=S P F inΛ\S(cid:14)C˜(u,v)(y)=S . (2.8) n n n Moreover,theevent{C˜(u,v)(y)=S}dependsonlyonbondsthathaveatleastoneendpointinS.Therefore,forfixedS, n theevents{F inΛ\S}and{C˜(u,v)(y)=S}areindependent,andhence n (cid:3) (cid:14) (cid:4) P F inΛ\S(cid:14)C˜(u,v)(y)=S =P(F inΛ\S). (2.9) n Thus,weobtain (cid:3) (cid:14) (cid:4) (cid:3) (cid:14) (cid:4) (cid:3) (cid:4) P {EinS}∩{F inΛ\S}(cid:14)C˜(u,v)(y)=S =P EinS(cid:14)C˜(u,v)(y)=S P F inΛ\S , (2.10) n 0 n 1 where we have added subscripts to the probabilities on the right-hand side to distinguish the different expectations. ˜ ˜ Wesubstitute(2.10)into(2.7),performthesumoverS,andreplaceC byC,toget(2.6). n Finally, when E ⊆{u∈C˜(u,v)(y),v∈/ C˜(u,v)(y)}, the event on the left-hand side of (2.6) is independent of the occupationstatusofthebond(u,v).For{EinC˜(u,v)(y)},thisisbecausev∈/C˜(u,v)(y),and,for{F inΛ\C˜(u,v)(y)}, itisbecauseu∈C˜(u,v)(y). (cid:2) Althoughwedonotneedithere,wenotethatLemma2.2alsoapplies(bothfororientedandunorientedpercolation) toarbitraryeventsEandF,ifwereplacetheassumptionthatEandF aredeterminedbybondslyingbelownbythe assumptionthatP (|C(0)|=∞)=0. p We will refer to a bond (u,v) to which we can effectively apply Lemma 2.2 as a cutting bond. In the nested ˜(u,v) expectation on the right-hand side of (2.6), the set C (y) is random with respect to the outer expectation, but 0 ˜(u,v) deterministic with respect to the inner expectation. We have added a subscript “0” to C (y) and subscripts “0” 0 1 SomeversionsofLemma2.2publishedpreviously[5,6,14]containnon-essentialerrors.However,oneachoccasioninthesepaperswherethe FactorisationLemmahasbeenapplied,theclaimedfactorisationdoesinfacthold. 516 R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 and“1”totheexpectationsontheright-handsideof(2.6)toemphasisethisdistinction.Theinnerexpectationonthe right-hand side effectively introduces a second percolation model on a second lattice, which is coupled to the first ˜(u,v) percolationmodelviathesetC (y). 0 3. Thelinearterm In this section, we prove (1.28) by expanding the survival probability to linear order. In Section 3.1, we define pivotalbonds,andrewriteeventsdealingwithpivotalbondsusingDefinition2.1.InSection3.2,weperformafirst expansionstepmakingcrucialuseoftheFactorisationLemma,Lemma2.2.Thefirstexpansionisvirtuallyidentical totheexpansionforthetwo-pointfunctionperformedin[5]and[14].InSection3.3,weiteratethisexpansionstep indefinitelytoobtain(1.28). 3.1. Pivotalbonds Definition3.1. (i) Givenabondconfiguration,wesaythatx isdoublyconnectedtoy,writtenx⇒y,ifthereareatleasttwobond- disjointpathsfromx toy consistingofoccupiedbonds.Byconvention,wesaythatx⇒x forallx.Similarly, we say that y is doubly connected to n, and write y ⇒n, if there exist x ,x ∈Zd (possibly equal) and two 1 2 bond-disjointpathsfromy to(x ,n)and(x ,n). 1 2 (ii) Givenabondconfiguration,wesaythatabondispivotalforx→yifx→yinthe(possiblymodified)configura- tioninwhichthebondismadeoccupied,whereasxisnotconnectedtoyinthe(possiblymodified)configuration inwhichthebondismadevacant.Similarly,wesaythatabondis pivotalfor y→n if y→n inthe(possibly modified) configuration in which the bond is made occupied, whereas y is not connected to n in the (possibly modified)configurationinwhichthebondismadevacant. Thesetofpivotalbondsfor x→y or y→n isorderedintime,whichallowsustospeakaboutthefirstpivotal bondhavingacertainproperty.Wecanvisualiseaconfigurationwhere0→nasconsistingofastringofsausages, thestringsrepresentingthepivotalbonds,andthesausagesthepartsoftheclusterof0thatareseparatedbythepivotal bonds.SeeFig.1foraschematicrepresentationoftheevent0→nasastringofsausages. IntermsofDefinitions2.1and3.1,wehaveacharacterisationofapivotalbondforv→y as (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (u(cid:15),v(cid:15))pivotalforv→y = v→u(cid:15)inC˜(u(cid:15),v(cid:15))(v) ∩ v(cid:15)→y inΛ\C˜(u(cid:15),v(cid:15))(v) . (3.1) Similarly,wehaveacharacterisationofapivotalbondforv→nas (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (u(cid:15),v(cid:15))pivotalforv→n = {v→u(cid:15)}∩{v→n}c inC˜(u(cid:15),v(cid:15))(v) ∩ v(cid:15)→ninΛ\C˜(u(cid:15),v(cid:15))(v) . (3.2) Theright-handsidesof(3.1),(3.2)areconvenientforapplicationoftheFactorisationLemma2.2. 3.2. Thefirstexpansionstep Wewillsuccessivelyexpandthesurvivalprobability,usingthenotionofpivotalbondsdefinedinSection3.1,the rewrite in (3.2) and Factorisation Lemma 2.2. It turns out that in the process of deriving the expansion for θ , we n Fig.1.Schematicrepresentationoftheevent0→nasastringofsausages. R.vanderHofstadetal./Ann.I.H.Poincaré–PR43(2007)509–570 517 encounteranadaptationofthesurvivalprobabilitythatwedenotebythegeneralisedsurvivalprobability.Therefore, itismostconvenienttoimmediatelyexpandthegeneralisedsurvivalprobability.Westartbydefiningthegeneralised survivalprobability.Thisdefinitionwillbecrucialthroughouttheexpansion. Definition 3.2. Given a bond configuration and a set A⊆Λ, we say that y is connected to x through A, and write y−→A x,ifeveryoccupiedpathconnectingy tox hasatleastonebondwithanendpointinA.Byconvention,x−→A x holdsifandonlyifx∈A.Similarly,wesaythaty isconnectedtonthroughA,andwritey−→A n,ifeveryoccupied pathconnectingy toavertexinZd ×{n}hasatleastonebondwithanendpointinA,orify∈(Zd ×{n})∩A. As mentioned above it will be convenient to expand not only θ , but also the generalised survival probability n P(v−→A n)forafixedvertexvandsetofverticesA.Wenotethat,with0=(0,0),wehave (cid:3) (cid:4) P 0−{−0→} n =θ . (3.3) n ToanalyseP(v−→A n),wedefinetheevents (cid:11) (cid:12) (cid:11) (cid:12) E(cid:15)(v,x;A)= v−→A x ∩ (cid:2)pivotalbond(u(cid:15),v(cid:15))forv→x suchthatv−→A u(cid:15) , (3.4) (cid:11) (cid:12) (cid:11) (cid:12) F(cid:15)(v;A)= v−→A n ∩ (cid:2)pivotalbond(u(cid:15),v(cid:15))forv→nsuchthatv−→A u(cid:15) , (3.5) n whicharedepictedschematicallyinFig.2. Givenaconfigurationinwhichv−→A n,thecuttingbond(u(cid:15),v(cid:15))isdefinedtobethefirstoccupiedandpivotalbond forv→nsuchthatv−→A u(cid:15).Itispossiblethatnosuchbondexists.Bypartitioning{v−→A n}accordingtothelocation ofthecuttingbond(orthelackofacuttingbond),weobtainthedecompositiongiveninthefollowinglemma.Here andelsewhere,wewrite∪˙ foradisjointunion. Lemma3.3(Thepartition).Foranyv∈Λ,A⊆Λ,n(cid:2)0, (cid:11) (cid:12) (cid:15)˙ (cid:2) (cid:11) (cid:12)(cid:5) v−→A n =F(cid:15)(v;A)∪˙ E(cid:15)(v,u(cid:15);A)∩ (u(cid:15),v(cid:15))occupiedandpivotalforv→n . (3.6) n (u(cid:15),v(cid:15)) Proof. Wedecomposetheevent{v−→A n}dependingonwhetherthereisacuttingbondornot.TheeventF(cid:15)(v;A)is n (cid:15) (cid:15) (cid:15) (cid:15) thecontributionwheresuchacuttingbonddoesnotexist.Otherwise,let(u,v)bethecuttingbond.Then,(u,v)is occupiedandpivotalforv→nand{v−→A u(cid:15)}holds.Moreover,therecannotbeapreviouspivotalbondsatisfyingthe samerequirements.Thelatterisequivalenttothestatementthat,forall b thatareoccupiedandpivotalfor v→u(cid:15), theevent{v−→A b}cannothold.Therefore,E(cid:15)(v,u(cid:15);A)holds. (cid:2) Define (cid:3) (cid:4) γ(0)(v;A)=P F(cid:15)(v;A) . (3.7) n n (a) (b) Fig.2.(a)SchematicrepresentationoftheeventE(cid:15)(v,x;A).TheintersectionofAwiththefourthsausageisoptional,whiletheintersectionwith thesixthisrequired.(b)SchematicrepresentationoftheeventF(cid:15)(v;A).TheintersectionofAwiththethirdsausageisoptional,whiletheother n intersectionisrequired.