INCIPIENT INFINITE PERCOLATION CLUSTERS IN 2D. BY ANTAL A. JÁRAI 1 University of British Columbia

Transcription

1 The Aals of Probability 2003, Vol. 31, No. 1, INCIPIENT INFINITE PERCOLATION CLUSTERS IN 2D BY ANTAL A. JÁRAI 1 Uiversity of British Columbia We study several kids of large critical percolatio clusters i two dimesios. We show that from the microscopic lattice scale perspective these clusters ca be described by Keste s icipiet ifiite cluster IIC, as was cojectured by Aizema. More specifically, we establish this for icipiet spaig clusters, large clusters i a fiite box ad the ihomogeeous model of Chayes, Chayes ad Durrett. Our results prove the equivalece of several atural defiitios of the IIC. We also show that for ay k 1 the differece i size betwee the kth ad k + 1st largest critical clusters i a fiite box goes to ifiity i probability as the size of the box goes to ifiity. I additio, the distributio of the Chayes Chayes Durrett cluster is show to be sigular with respect to the IIC. 1. Itroductio. The term icipiet ifiite cluster has bee used, mostly i the physics literature, i referece to the large coected clusters that ca be see i umerical simulatios of critical percolatio. The ame reflects the idea that as the bod or site desity is raised above threshold, some of these large clusters coect ad give birth to the ifiite cluster; see Borgs, Chayes, Keste ad Specer A mathematically rigorous defiitio of what oe might call the iifite cluster at criticality was give by Keste The defiitio, which we review i Sectio 1.2, is obtaied by coditioig the critical percolatio process to have a ope path coectig the origi to the boudary of a large box, whose size icreases to ifiity. I the weak limit a ifiite cluster cotaiig the origi is obtaied, which we call the IIC. The followig alterative defiitio was proposed by Aizema 1997 ad expected to be equivalet. Cosider the set of sites i a large box that are coected at the critical desity to both the left ad right sides, ad pick oe uiformly, give the cofiguratio. Traslate this site to the origi ad let the size of the box go to ifiity. By its choice, the traslated site is coditioed to have a ope coectio to a log distace, so we may expect the IIC to arise i the weak limit. We prove this i Theorem 1. Received February 2001; revised February Supported i part by a NSF grat to Corell Uiversity, NSERC of Caada, ad the Pacific Istitute for the Mathematical Scieces. AMS 2000 subject classificatios. Primary 60K35; secodary 82B43. Key words ad phrases. Percolatio, icipiet ifiite cluster, spaig cluster, critical pheomea. 444

2 INCIPIENT INFINITE CLUSTERS 445 I this paper we discusss two more atural procedures that produce the IIC ad ote that aalogous results about ivasio percolatio were obtaied i Járai 2000 which will be published i Járai to appear. A procedure useful for umerical simulatios is to pick a radom site uiformly from the largest cluster i the box where the simulatio is performed. Agai, we ca ask whether the law of the cluster whe viewed from the radom site coverges to the IIC. This is ideed the case, ad more geerally, we show this for the kth largest cluster where k 1isfixed. The third settig we cosider is the ihomogeeous model of Chayes, Chayes ad Durrett I this model the probability that a bod at distace k from the origi is ope is take to be p c + k λ. Assumig that the correlatio legth expoet ν exists, it ca be show that for λ<1/ν there is a ifiite cluster ad this cluster was proposed i the above referece as a alterative to the IIC. Our aim i Sectio 4 below is to explore the relatioship betwee the CCD cluster ad the IIC. The precise formulatio of our mai results are give i Sectio 1.3. Before statig our results we fix some otatio ad termiology i the ext subsectio. For a approach to the icipiet ifiite cluster i high dimesios see Hara ad Slade 2000a, b. REMARK. We formulate results i the settig of bod percolatio o Z 2. This ca be geeralized to other commo two-dimesioal graphs, as log as the Russo Seymour Welsh Lemma ad Theorems 1 ad 2 of Keste 1987 hold Notatio ad termiology. We cosider bod percolatio o the square lattice Z 2. We deote by E 2 the set of earest eighbor bods i Z 2 ad each bod e E 2 is declared to be ope with probability p ad closed with probability 1 p, idepedetly. We let P p resp. E p deote the correspodig product measure resp. expectatio o the cofiguratio space, F = {0, 1} E2, F, where 0 meas closed, 1 meas ope, ad F is the usual σ -field o. Wesay that E F is a cylider evet,ife is give by coditios o the states of fiitely may edges oly. If A is a evet, I[A] deotes its idicator fuctio. If G is a subgraph of Z 2, E 2,wewriteEG for the set of edges of G ad v G meas that v is a vertex of G. Sometimes we defie G by specifyig EG E 2 ad it is uderstood that the vertex set cosists of those v Z 2 that are icidet to a edge i EG.IfA,B Z 2,wewriteA B for the evet that some vertex i A is coected by a ope path to some vertex i B. We deote by Cv the ope cluster cotaiig the vertex v, Cv ={w Z 2 : v w} ad write C = C0. For a coutable set A we write A for the cardiality of A. The percolatio probability is defied by θp= P p C =.

3 446 A. A. JÁRAI We also usethe otatio {0 }for the evet { C = }.Thecritical probability is p c = if{p 0:θp>0}. The fact that p c = 1/2 for the square lattice will ot play a role; our argumets ca be geeralized to other stadard two-dimesioal lattices. We itroduce the orm v =max{ v 1, v 2 } for v = v 1,v 2 Z 2.IfA ad B are two sets of vertices, we put dista, B = mi{ v w : v A,w B}. For a bod e = v,w E 2 we let e =max{ v, w }. The box of radius cetered at the vertex v is B,v ={w Z 2 : v w } ad we write B = B,0. We use the otatio A, m = B \ Bm ={v Z 2 : m< v } for a aulus. The boudary of a lattice graph G is defied as G = { v G : v is icidet to a edge ot belogig to EG }, thus B ={v Z 2 : v =}. Sometimes it will be coveiet to work with ={w Z 2 : w i <, i= 1, 2}. Whe we refer to Ba, a,etc.wherea is ot a iteger, we mea to replace a by its iteger part a. The symbol a will deote the smallest iteger that is greater tha or equal to a. A importat quatity for us is the poit-to-box coectivity: πp, = P p 0 B. We write π = π= πp c,. The quatity s = 2 π will ofte show up, sice it represets the order of magitude of the largest critical clusters i a box of liear size ; see Borgs, Chayes, Keste ad Specer It is easy to see that for p>0wehave 1.1 P p 0 πp,, where the otatio a b meas that there are costats 0 <C 1 <C 2 <, such that C 1 a b C 2 a. The costats implicitly preset i 1.1 are also idepedet of p, as log as p is bouded away from 0. The dual lattice of Z 2, E 2 is deoted by Z 2, E2,whereZ2 = 1/2, 1/2+Z2. For each e E 2 there is a uique edge e E 2 that itersects e.wecalle ope if ad oly if e is ope. We use the otatio f= og for lim f/g= 0, ad f= Og for the fact that f/gis bouded.

4 INCIPIENT INFINITE CLUSTERS Defiitio of the IIC. It was show by Keste 1986 that for ay cylider evet E the limit νe = lim P 1.2 p c E 0 B exists. It follows that ν has a uique extesio to a probability measure o F ad uder the measure ν the cluster C0 ={v Z 2 : v 0} is almost surely ifiite. Followig the termiology of Keste 1986 we call the distributio of the cluster C0 uder ν the icipiet ifiite cluster IIC Mai results. All costats i this paper are strictly positive ad fiite. Costats deoted by C i may have differet meaigs i differet theorems. Whe we eed to refer to a costat from a differet theorem or proof, this will be doe explicitly. The first object we study is the uio of spaig clusters i B. LetLS= { } [, ] ad RS ={} [, ] deote the left ad right sides of B ad defie SC ={v B:LS v RS iside B}. The followig procedure has bee proposed by Aizema Let p = p c,ad choose a site I uiformly at radom from SC if ot empty, ad shift that site to the origi. I other words: look at SC from the perspective of oe of its sites. We ca expect that as thelawofthetraslatedsc coverges weakly to the IIC. To formulate the statemet precisely let τ v deote traslatio by v Z 2. Traslatios act o by τ v ω u, x = ω u v,x v, ad o evets by τ v A ={τ v ω : ω A}. THEOREM 1. Let I deote a site of SC chose uiformly at radom, give the cofiguratio ω, whe SC is ot empty. The for ay cylider evet E F lim P p c τ I E SC = νe. REMARK. The followig questio was suggested to us by H. Keste: what do we see if we choose the vertex from the backboe of the spaig cluster? The backboe is the set of vertices that have disjoit coectios to the left ad right sides of B. We believe that the limit is νe = lim P 1.3 p c E 0 has two disjoit coectios to B. The existece of the limit i 1.3 ca most likely be show by the method of Keste We ote that ν is sigular with respect to ν. Due to the coditioig

5 448 A. A. JÁRAI i 1.3, there are ν-a.s. two disjoit ifite ope paths startig at 0, but ν-a.s. there is oly oe such path by the BK iequality [va de Berg ad Keste 1985]. We believe that similar limitig measures exist i other cases, for example, if we coditio the vertex to be the ed-vertex of a bod that is pivotal for a coectio betwee the left ad right sides. We have the followig variatio o Theorem 1 i which the site is oradom. THEOREM 2. Let h i such a way that h. The lim P pc τ v E v SC = νe. v h We ca also relate the large clusters see at criticality to the IIC. Let C 1, C 2,... deote the clusters iside B, ordered by size, that is by the umber of vertices each cotais. I determiig the clusters we use free boudary coditios, that is, for two vertices to belog to the same cluster they have to be coected iside B. If there are clusters of equal size we use some lexicographic orderig betwee them; it turs out that this has o importace. The size distributio of C 1, C 2,... both ear ad away from the critical poit was studied extesively by Borgs, Chayes, Keste ad Specer 1999, We prove the followig theorem. THEOREM 3. Fix k 1, ad let I deote a vertex chose uiformly at radom from C k, the kth largest cluster i B. The for ay cylider evet E, lim P p c τ I E = νe. REMARK. We eed to be a bit more precise about what happes whe there are fewer tha k clusters altogether. The probability of this goes to 0, so the statemet of the theorem is ot affected by whatever we defie I to be i this case. A importat step i the proof of Theorem 3 is to show the followig propositio. PROPOSITION 1. Let W 1 W 2 deote the sizes of the clusters i B i decreasig order. For ay k 1 W k W k+1 i -probability as. REMARK. We actually show a bit more: W k W k+1 caot be of smaller order tha s = 2 π. Results by Borgs, Chayes, Keste ad Specer 2000 show that at p c oe has W k s for ay fixed k. This suggests that the true order of the differece should be s.

6 ad defie for p>p c Lp, ε = mi{ : σ,, p 1 ε}. INCIPIENT INFINITE CLUSTERS 449 The last settig we cosider is the ihomogeeous model studied by Chayes, Chayes ad Durrett I this model, each bod e E 2 is ope with a probability qe p c. We assume that qe p c,as e. We deote by P q the correspodig probability measure. Recall that C ={v Z 2 :0 v} ad let C = C B. By results of Chayes, Chayes ad Durrett 1987, if the decay of qe to p c is sufficietly slow, the we have P q C = >0. We call the distributio of C uder P q the Chayes Chayes Durrett cluster. It ca be expected that away from the origi C looks similar to a critical percolatio cluster. The ext theorem makes this precise. THEOREM 4. Assume that qe is a fuctio of e oly, ad that it is decreasig i e. Let I deote a vertex chose uiformly at radom from C. The for ay cylider evet E, 1.4 lim P m, q τi E 0 Bm = νe. m I particular, if P q C = >0, we have lim P qτ I E 0 = νe. I the special case qe p c we obtai a result for the IIC itself, sayig that a typical vertex of the IIC looks like the origi. COROLLARY 1. Let qe p c. The lim ντ I E = νe. A problem related to Theorem 4 is whether the Chayes Chayes Durrett cluster is sigular with respect to the IIC. To discuss this assume that the limit µ q E = lim P 1.5 q E 0 B exist for cylider evets. [This is i fact true for ay choice of qe, as log as qe p c ad d = 2, by a geeralizatio of 1.2; see 4.1.] We give a sufficiet coditio for µ q ad ν to be sigular. To formulate the result, we use the fiite-size scalig correlatio legth itroduced by Chayes, Chayes ad Fröhlich 1985 ad further studied by Keste Let σ 1, 2,p= P p there is a ope horizotal crossig of [0, 1 ] [0, 2 ], REMARK. I the defiitio of Keste 1987 the meaig of [0,] [0,], ad hece of Lp, ε is slightly differet. However, his results remai valid i our settig.

7 450 A. A. JÁRAI We fix ε = ε 0,adletLp = Lp, ε 0,whereε 0 is a small positive costat. We will ot be cocered with how exactly ε 0 is chose, oly that Lp satisfies certai properties see the begiig of Sectio 4.2. THEOREM 5. Assume that qe p c is a fuctio of e oly ad that it is decreasig i e. Suppose that Lqe lim = 0. e e [This holds wheever P q 0 >0.] Let µ q be defied by 1.5. The µ q ad ν are sigular with respect to each other. The orgaizatio of the rest of the paper is the followig. We summarize some prelimiary results i Sectio 1.4. The we prove Theorem 1 i Sectio 2. We start the proofs with the case of the spaig clusters because it is easiest to demostrate our techique with this example. Later proofs will go alog similar lies ad will ot always be spelled out completely. Our results about large clusters are proved i Sectio 3 ad the oes about the Chayes Chayes Durrett cluster i Sectio Prelimiaries. Recet breakthroughs by Smirov 2001 ad Lawler, Schramm ad Werer 2002 give very precise iformatio about certai percolatio quatities, icludig those importat for this paper. For site percolatio o the triagular lattice Lawler, Schramm ad Werer 2002 show that π = 5/48+o1 as. Smirov ad Werer 2001 show the existece of other critical expoets, o the triagular lattice, usig the results of Keste The proofs i this paper are largely model idepedet ad ca be based o properties of π that are kow for other stadard two-dimesioal lattices as well. It might be useful for the reader to simply thik of πp c, as 5/48 throughout the paper, which makes some of the techical statemets of Theorem 7 below more clear. The statemets of the followig theorem are well-kow cosequeces of the Russo Seymour Welsh Lemma [Russo 1978, Seymour ad Welsh 1978] ad the FKG iequality [see Grimmett 1999]. THEOREM 6RSW. all 1 we have 1.6 ifor ay κ 1 there is a costat C κ, such that for there is a ope horizotal crossig of [0,κ] [0,] Cκ. ii There are costats C>0, ad µ>0, such that for all >m 1 we have 1.7 there is o ope circuit i A, m C m µ. We are goig to use the followig properties of π.

8 INCIPIENT INFINITE CLUSTERS 451 THEOREM i There are costats C 1 <C 2 such that C 1 C s π C2 C s. ii There exists a costat D>0, such that, for p p c, πp,m 1.9 πp, D, m 1. m iii There exists C 3 <C 4, such that, if >N 1, the 1.10 π C 3 πn P π p c N C 4 πn. iv There exist C 5 ad C 6 such that, for L 1, p p c ad β = 0, 1, we have L 1.11 r + 1 β πp,r C 5 L β+1 πp,l r=0 ad L 1.12 r + 1πp, r 2 C 6 L 2 πp,l 2. r=0 v If N is fixed, L N 1, p p c ad β = 0, 1 the 1.13 ad 1.14 L/N r=0 L/N r=0 Nr + 1 β πp,nr πp,n C L β+1 πp,l 5 Nπp,N πp,nr Nr + 1 πp,n 2 C 6 L 2 πp,l 2 Nπp,N 2. REMARK. For the triagular lattice, the result of Lawler, Schramm ad Werer 2002 ad Theorems 1 ad 2 of Keste 1987 imply ii with D/m 5/48+ε o the right-had side. The iequalities i 1.10 correspod to the fact that N should scale as N/ 5/48. A slightly weaker form of this scalig is proved i 3.1, Lawler, Schramm ad Werer Properties iv ad v are straightforward, give a power law decay of π that is slower tha 1. PROOF OF THEOREM 7. The lower boud i 1.8 is a special case of Propositio 4.5 of Borgs, Chayes, Keste ad Specer Note that their Assumptio II is valid for d = 2 by 1.9. Likewise the upper boud is a special case of Propositio 5.2 of Borgs, Chayes, Keste ad Specer Note that their Assumptio I is well kow for d = 2.

9 452 A. A. JÁRAI The boud 1.9 is a stadard extesio of Corollary 3.15 of va de Berg ad Keste Their proof also works for ay p p c. The bouds i 1.10 follow by a stadard RSW argumet. Statemet iv is a special case of v. The proof of v goes alog the same lies as the proof of Lemma 4.4 of Borgs, Chayes, Keste ad Specer Usig 1.9 we have L/N r=0 Nr + 1 β πp,nr πp,n = πp,l L/N πp,n C 9 D = C 9 D r=0 L/N πp,l πp,n Nr + 1 β πp,nr πp,l r=0 L/N πp,l πp,n L1/2 L β+1 πp,l C 5 Nπp,N. The proof of 1.14 is quite similar. Nr + 1 β L Nr + 1 r=0 Nr + 1 β 1/2 1/2 2. Spaig clusters. We start with a rough outlie of the argumet for Theorem 1. Recall that SC deotes the uio of spaig clusters ad I is a radom vertex from SC. We ca write 2.1 τ I E SC = v B E pc I[τv E,v SC ] SC SC. Cosider a large box BN,v cetered at v,where1 N.Theevetv SC implies that v BN,v. The latter is the coditioig i Keste s theorem, so we hope to apply 1.2 iside the box BN,v. Some work is eeded to decouple from what is outside the box BN,v. We put a thick aulus BM,v \ BN,v aroud the box, where N M. Assume that there is a ope circuit i this aulus [here, ad later always, we uderstad that the circuit surrouds the smaller box BN,v]. Cosider the outermost such circuit. A well-kow observatio is that the outermost ope circuit acts as a spatial aalogue of a stoppig time: the evet that it equals a give circuit D oly depeds o the cofiguratio outside D. Therefore, if we coditio o the outermost ope circuit, we ca apply Keste s theorem iside the circuit. Equatio 1.7 suggests that the ope circuit will exist with large probability if the aulus is thick eough. However, sice the ceter of the aulus is radom, we eed some iformatio about the size distributio of the set from which I is chose. The result providig this iformatio is discussed i Sectio 2.1 ad the proof of Theorem 1 is give i Sectio 2.2.

10 INCIPIENT INFINITE CLUSTERS Tightess of SC. We prove the followig result about SC. THEOREM 8. i There are costats C 1 ad C 2 such that, for 1, 2.2 C 1 E p c SC C 2. s 2.3 ii We have lim if P p c ε< SC ε 0 1 E pc SC < 1 SC = 1. ε REMARK. It also holds that for ay t 1wehaveE pc SC t [s] t.this extesio ca be show usig the idea of either Nguye 1988 or Theorem 8 of Keste Sice we do ot eed this extesio, we omit the proof. PROOF OF THEOREM 8. i Defie π= 0 LS iside B. Itis easy to see that π π 4 π. We first show the lower boud i 2.2. For v = v 1,v 2 B/2 let Sv =[, ] [v 2 /2,v 2 ]. Defie the evets A 1 v ={there is a ope horizotal crossig i Sv}, A 2 v ={v BS iside B/2,v}, where BS deotes the bottom side of B/2,v, that is, BS =[v 1 /2, v 1 + /2] {v 2 /2}. Note that both evets are icreasig. Usig the FKG iequality ad the RSW theorem, we get Hece v SC A1 v A 2 v A 1 v A 2 v C 3 π/2 C 3 /4π/2 C 3 /4π. E pc SC v B/2 v SC C 4 2 π. For the upper boud i 2.2 we use the iclusio of evets to write 2.4 E pc SC = {LS v RS} {v Bv,} v B v B LS v RS iside B v Bv, = π C 5 s.

11 454 A. A. JÁRAI ii The proof of 2.3 ca be broke up ito two parts: showig a upper ad a lower boud o SC i terms of E pc SC. For the upper boud we ca use Markov s iequality: SC E pc SC 1 ε 2.5 SC = 1 SC P SC p c E pc SC 1 ε ε SC. By the RSW theorem SC C 6 > 0, hece the right-had side of 2.5 goes to zero uiformly i,asε 0. To give a lower boud o SC we modify the proof of the secod part of Theorem 8 of Keste Let R deote the lowest ope crossig i B. Our pla is to show that the umber of vertices above R that are coected to R is larger tha εe pc SC with large probability. We first eed to show that R is ot too close to the top side so that there is eough space for these vertices. For 0 <a<1lets a =[, ] [, a ]. We eed the followig lemma: LEMMA 1. There are costats c 1, α such that R lies i S a R exists 1 c 1 1 a α. PROOF. The proof is almost idetical to the proof of a more geeral lemma we are goig to state later, so we omit it. See Lemma 4 i Sectio 3.1. REMARK. A stroger statemet with α = 1 ca be proved by a method of Zhag preprit. We cotiue with the proof of tightess. Let δ>0 be fixed. It follows from Lemma 1 that we ca choose a close eough to 1, so that 2.6 R lies i S a R exists 1 δ 2, uiformly i. Fix such a a. We show that for ε small eough, for ay, ad for ay crossig r 0 of S a we have SC 2.7 E pc SC <ε R = r0 δ 2.

12 This is eough because 2.6 ad 2.7 imply SC E pc SC <ε SC INCIPIENT INFINITE CLUSTERS 455 = SC E pc SC <ε R exists R S a R exists + SC E pc SC <ε R Sa δ 2 + r 0 SC E pc SC <ε R = r0 R = r 0 R S a δ. The sum is over all horizotal crossigs r 0 of S a. Note that it is eough to show 2.7 for large eough. Fix the path r 0.LetG deote the ope regio i [, ] 2 that lies above r 0, whe r 0 is viewed as a curve i R 2. We defie the graph H by EH def ={e E 2 : e lies i G apart maybe from its edpoits}. Observe that the evet {R = r 0 } oly depeds o the states of the edges o ad below r 0, hece it is idepedet of the states of the edges of H. Thus, coditioed o {R = r 0 } we still have a Beroulli percolatio process o H. Let v be ay vertex o the path r 0 with maximal secod coordiate ad fix this vertex v see Figure 1. We assume that v is i the right half of B,thatis,v 1 0; the other case ca be treated aalogously. We cosider a aulus v + A7m, m, where m will later be chose to be of order. Let Am ={w Z 2 : v 1 5m w 1 <v 1 3m, v 2 <w 2 v 2 + 5m}. We suppress the depedece o v i our otatio. I Figure 1 the striped regio is the set Am. We wat to make sure that the rectagle v +[ 7m, 0] [0, 7m] is cotaied i H.Sicev 2 a, for this we eed to have 2.8 7m 1 a. Observe that sice v 1 0, 2.8 is i fact eough to esure that v +[ 7m, 0] [0, 7m] H. We are goig to estimate Ym def = { w Am : w r 0 iside [v + A7m, m] B } ad ote that o {R = r 0 } we have SC Ym. Defie a ew radom cofiguratio ω i the followig way: for e EH,that is, for edges above r 0, put ω e = ωe. O the rest of the lattice let ω beaew idepedet cofiguratio with bod desity p c.theω is Beroulli percolatio at p c. We deote its law by P p c.

13 456 A. A. JÁRAI FIG. 1. If i the modified cofiguratio there is a ope circuit i v + A7m, 5m ad a ope path from w to the left side of Bv,7m, thew is coected to r 0, ad hece w SC i the old cofiguratio. The two pictures show the two possible ways this could happe. We let F m ={there is a ope circuit i v + A7m, 5m i the cofiguratio ω }. Now suppose that w Am is coected to the left side of Bv,7m iside [v + A7m, m] B i the cofiguratio ω ad that F m occurs. See Figure 1. The, by the geometry of the costructio, w is coected to r 0 iside [v +A7m, m] B i the cofiguratio ω. This gives us that for all w Am we have w r0 iside [v + A7m, m] B P p c F m, w left side of Bv,7m iside [v + A7m, m] B. By the FKG iequality ad the RSW Theorem the right-had side is at least C 7 P p 2.9 c w left side of Bv,7m iside [v + A7m, m] B. By the same method as i part i of this theorem, we ca show that 2.9 is at least C 8 πm C 8 /4πm. This implies that 2.10 E pc Ym C 9 m 2 πm. The proof ca ow be completed by copyig the argumet of Theorem 8 i Keste We have already oted that SC Ym. As i Keste 1986, we ca use 1.11 ad 1.12 to prove the secod momet boud E pc Ym 2 C 10 m 2 πm 2. Usig the oe-sided Chebyshev iequality [Exercise I.3.6 i Durrett 1996] this implies that Ym 1 2 E pc Ym C 11 > 0.

14 INCIPIENT INFINITE CLUSTERS 457 The we repeat the costructio with differet values of m correspodig to disjoit auli [so that the Ym s are idepedet]. Namely, if k is the uique iteger for which 7 k 1 a < 7 k+1, the let m l = 7 k l, with l = 1, 2,...,j,wherej will be chose later. Let ε be such that ε 1 a2 C j+2, C 2 where C 2 is the costat appearig i part i of this theorem. For large eough 1 7 k j, so our costructio makes sese for m = m 1,...,m j.for1 l j we have 1 2 E p c Ym l 1 2 C 9m 2 l πm l 1 2 C 97 2k j π C 91 a 2 7 2j 2 2 π εc 2 2 π, where we used 2.10, the lowest possible value of m l ad the fact πm l π, the choice of k, ad fially the choice of ε. Usig 2.11 we obtai SC E pc SC ε R = r SC εc 2 2 π R = r 0 Yml E pc Ym l 1 2 for some 1 l j R = r0 Yml = E pc Ym l 1 for some 1 l j 2 j = 1 1 Ym l 1 2 E p c Ym l l=1 1 1 C 11 j, where we used idepedece i the peultimate step. Choose j so that the righthad side of 2.12 is greater tha 1 δ/2. Decreasig ε, if ecessary, we get the statemet of 2.7 for all ad this completes the proof Spaig clusters look like the IIC. We are ready to prove Theorem 1. We are goig to write A ={SC } for short. PROOF OF THEOREM 1. Recall that i 2.1 we have writte τ I E A as a sum over v B.Letτ v AM, N = v + AM, N be a aulus cetered

15 458 A. A. JÁRAI at v. We will take 1 N M. I ay case, we are goig to take N large eough so that BN cotais all edges o which E depeds. Let 2.13 F = FM,N={there is a ope circuit i AM, N}. The evet that the aulus aroud the radom vertex does ot cotai a ope circuit is τ I F c. I view of the RSW Theorem it is ituitive that if M/N is large the the probability of this evet should be small. Our first goal is to prove this. Let ε>0. Similarly to 2.1 we write 2.14 τ I F c A = = v B 1 A E pc I[v SC,τ v F c ] SC v B A I[v SC,τ v F c ] E pc ; A. SC By part ii of Theorem 8 it follows that for sufficietly small x = xε > 0we have 2.15 SC <xe pc SC,A ε 2 P p c A. Hece, otig that v B I[v SC,τ v F c ] SC we get that the right-had side of 2.14 is at most 2.16 ε A v B 1, E pc I[τv F c ]I[v SC ] xe pc SC The variable I[τ v F c ] is decreasig ad I[v SC ] is icreasig, so the FKG iequality implies that the summad i 2.16 is less tha 1 xe pc SC P p c τ v F c v SC = P p c F c xe pc SC P p c v SC. Summig over v we get that the expressio i 2.16 is less tha ε 2 + P p c F c x A ε 2 + P p c F c, C 3 x whereweusedthat A C 3 > 0 for some costat C 3.Bypartiiofthe RSW Theorem, if M/N 1 ε the F c C 3 xε/2. Hece, τ I F c A ε.

16 INCIPIENT INFINITE CLUSTERS 459 The estimate 2.17 shows that up to a additive error ε we ca write τ I E A τ I E τ I F A 2.18 = I[τv E,v SC,τ v F ] E pc A. SC v B For techical reasos we will eed to restrict the sum over v i 2.18 to sites that are ot too close to the boudary. For this we show that for a suitable fuctio fthe evet G ={I B f} occurs with large probability. We write G c A = f< v E pc I[v SC ] SC A. Similarly to the calculatio for τ I F c, we use 2.15 to show that the right-had side is less tha ε 2 + I[v SC ] E pc A xe f< v pc SC 2.19 ε 2 + f< v 1 xe pc SC ε 2 + C 4f xe pc SC. By Theorem 8 we have E pc SC C 1 2 π. Therefore, if f= oπ,the secod term o right-had side of 2.19 will be ε/2forlarge.toedthisstep we ote that the coditio o fallows f. This follows from the fact that π D /, a cosequece of 1.9. Fix f with the above properties. The for 2 ε we have 2.20 G c A ε. TheextstepistodecomposeF accordig to the outermost ope circuit. Defie 2.21 FD ={D is the outermost ope circuit i AM, N}; thus F is the disjoit uio of the evets FD where D rus over all circuits i AM, N. Likewise we have τ v F = D τ v FD. The usig the bouds 2.17 ad 2.20 we have 2.22 τ I E A 2ε + 1 PA v f 2ε + τ I E A, if ad M/N are large eough. D I[τv E,v SC,τ v FD] E pc ; A SC

17 460 A. A. JÁRAI Fix v ad D, ad assume that f > M.Otheevetτ v FD we have that {v SC } occurs if ad oly if {v τ v D} ad {τ v D SC } both occur. Therefore, the umerator iside the expectatio i 2.22 factors as 2.23 I[τ v E, v τ v D]I[τ v FD, τ v D SC ]. Let extτ v D deote the exterior of τ v D, that is the graph cosistig of the set of edges outside τ v D or o the circuit τ v D. The bulk of the deomiator SC comes from extτ v D, so we would like to group it with the secod factor i We let 2.24 W τ v D = {w extτ v D : w SC }. For 3 M, ε we have the iequalities 2.25 W τ v D SC 1 + εw τ v D for all v ad D. We ca choose 3 so that it also takes care of the previously metioed coditio f>m. Let X D,v,E = I[τ v E, v τ v D], Y D,v, = I[τ vfd, τ v D SC ]. W τ v D It is uderstood that Y = 0, whe of the form 0/0. Deotig the expectatio i 2.22 by ED,v,,E, equatio 2.25 yields 2.26 ED,v,,E E pc X D,v,E Y D,v, 1 + εed,v,,e, provided 3. It is straightforward to check that X ad Y are idepedet. By traslatio ivariace we have E pc X D,v,E = E, 0 D. A slight geeralizatio of 1.2 yields that 2.27 lim E 0 D = νe; N D surrouds BN see Remark after Theorem 3 i Keste Therefore, if N 4 ε, E the ε P p c E, 0 D νe 0 D 1 + ε E, 0 D. [For this we eed to suppose νe > 0. This is ot restrictive, sice otherwise we cosider the evet E c.] We choose the differet quatities i the proof i the followig order. Give ε>0 we choose N 4 ε so that 2.28 is satisfied. Next we choose M N 1 ε. For max{ 2 ε, 3 M, ε} we also have 2.22 ad Now replace E by the sure evet i our argumet. The, icreasig if ecessary, we

18 INCIPIENT INFINITE CLUSTERS 461 have 2.22, 2.26 ad 2.28 with E replaced by. Combiig these iequalities with the similar iequalities for E we get τ I E A 2ε + 1 ED,v,,E PA 2ε + 1 PA 2ε + for large eough ad quite similarly This completes the proof. v D v 1 + ενe PA E pc X D,v,E Y D,v, D v D E pc X D,v, Y D,v, 2ε ε2 νe ED,v,, PA v D 2ε ε 2 νe τ I E A 2ε ε 2 νe. PROOF OF THEOREM 2. The proof is very similar to that of Theorem 1, but much simpler. By the FKG iequality we have τ v F c v SC ε uiformly i v B, ifm/n is large eough. Therefore 1 τ v E v SC τv E,τ v FD, v SC. v SC D If h > M ad v h, equatio 2.23 implies that the summad is E, 0 D τv FD, τ v D SC. For the first factor, 2.28 holds if N is large eough. Therefore choosig first N large, the M large, the large, we get the claim as i Theorem Large clusters. The goal of this sectio is to show that if we pick a site uiformly at radom from C k,thekth largest cluster i B, the, agai the IIC arises i the limit as. The mai difficulty i applyig the argumet of the previous sectio is that for a fixed site v ad circuit D we eed to break up the evet {v C k } ito two parts, oe of which oly depeds o edges i extτ v D. For this we eed to be able to idetify the kth largest cluster i terms of the cofiguratio i extτ v D oly. This is oly possible if the gap betwee the sizes of C k ad C k+1 is typically large compared to C k itτ v D, where itτ v D deotes the iterior of τ v D, that is, the graph complemet of extτ v D.

19 462 A. A. JÁRAI We show that the gap is ideed large by provig Propositio 1 i Sectio 3.1. Theorem 3 is proved i Sectio 3.2 ad its proof ca be read without the proof of Propositio 1. The followig result of Borgs, Chayes, Keste ad Specer 2000 will be used throughout this sectio. It provides basic iformatio about the size distributio of large clusters ear p c ad we state it here for coveiece. For the proof see Theorem 3.1 i, Theorem 3.3 i ad Sectio 6 i the cited referece. THEOREM 9 Borgs, Chayes, Keste ad Specer. lim if P p c K 1 W i 3.1 s K 1 For all i 1 we have as K Gaps betwee cluster sizes. The proof of Propositio 1 is based o a block argumet. We briefly explai the argumet before makig the ecessary defiitios. Divide B ito boxes with liear size N. For each box determie whether it cotais a ope circuit surroudig a small box of half the size, ad if it does, cosider the outermost such circuit. Large clusters typically will cotai may of these ope circuits. Coditio o the outermost circuit i each box ad o the cofiguratio outside the circuits i.e., we coditio o the states of all bods that are ot i the iterior of ay of the circuits. The the cofiguratios i the iteriors of circuits correspodig to differet boxes are coditioally idepedet. The idea is that this creates sufficiet radomess to prevet the cluster sizes from beig too close. We spell out the proof for k = 1; the modificatios for k 2 are straightforward. We eed some otatio ad defiitios for the block argumet. Divide B ito smaller boxes cogruet to N,whereN 8willbeafixedpowerof2for the etire proof. The boxes are of the form bk = N + 2Nk, wherek Z 2. Wheever we say box durig this proof we refer to oe of these. For each k determie if the aulus ak = N \ N/2 + 2Nk cotais a ope circuit ad cosider the outermost ope circuit D k whe there is oe. Do this oly for the boxes that are cotaied i B. We defie the graph G k by EG k = Eext D k Ebk [i.e., G k is the part of bk that lies outside D k ]. It is uderstood that G k = bk,ifd k does ot exist ad G k = bk B, ifbk itersects B but it is ot etirely cotaied i B. Give the cofiguratio i G k the cofiguratio i itd k is a idepedet percolatio process. Fix a orderig of the boxes. Let M 1 be a iteger that we are goig to choose later. We build the cofiguratio i B i three steps. A For each k pick the cofiguratio i G k. This provides a cofiguratio i the graph G = k G k. B We mark certai boxes. Let E be a ope cluster i the graph G. Cosider those bk for which D k exists ad is cotaied i E, ad mark the first M

20 INCIPIENT INFINITE CLUSTERS 463 such boxes accordig to the fixed orderig of boxes. If there are ot eough boxes available, mark as may as there are. Do the above markig procedure for all ope clusters E i the graph G. Now fill i the umarked boxes, that is, pick the cofiguratio i the set Ebk \ EG k. k : bk is ot marked At this stage we have picked the states of the edges of a graph A ω, where we ca defie the graph A ω i the followig way: EA ω def = EB \ Eit D k. k : bk is marked It follows from the costructio that if G is ay oradom subgraph def of B, the the evet K G ={A = G} depeds oly o the edges i G. 1 2 C Let C, C,... deote the ope clusters i A ω ordered by size ad 1 2 j let W W deote their sizes, respectively. To each C there correspod at most M marked boxes. Use the orderig of the boxes to label the ope circuits i these marked boxes by D j 1, Dj 2,... Our fial step is to fill i the iteriors of the circuits D t j, that is, pick the cofiguratio iside these circuits. Let X t j = { w itd t j : w D t j }, t = 1,...,M; j = 1, 2,..., where we defie X t j = 0ifD t j does ot exist. Observe that the X t j are coditioally idepedet, give the cofiguratio obtaied i step B. If M is large we ca hope that the distributio of t X t j will be sufficietly spread out to give the result of the propositio. For j = 1, 2,...let 3.2 Z j = M j W + t=1 X j t. That is, Z j j is the size of the ope cluster i B that cotais C. We prove Propositio 1 via three lemmas that correspod to the followig three steps. STEP 1. For some j 0, with large probability, W 1 ad W 2 appear amog the Z j with 1 j j 0. This meas we oly have to care about a fixed umber of Z j s. STEP 2. For large eough there are at least M marked boxes for 1 j j 0, with large probability. C j,

21 464 A. A. JÁRAI STEP 3. For a give r, the probability that Z j Z j r for some 1 j <j j 0 is small, if M is large eough. The followig lemma takes care of Step 1. LEMMA 2. Cosider the evets A 1 i = { Z j W 1 }, 1 j i A 2 i = { Z j W 2 }. 1 j i For ay ε>0 there is a i = iε such that, for all M 1, there exists 0 = 0 ε,m,nsuch that, for all 0, we have 3.3 A1 i A 2 i ε. PROOF. Write P t = C0 t, wherec0 deotes the cluster of the origi. Let Ñt j = {j : t W }. We ca boud E pcñt by a idea of Borgs, Chayes, Keste ad Specer Namely, if C v ad Cv deote the coected compoet of the vertex v i B ad i Z 2, respectively, we have 1 E pcñsm = t P p c v j j C, W = t t=sm v B 1 sm 1 sm 1 sm v B v B v B v = P sm sm C 1 m 2. I the last iequality we used 1.8. Now choose C 2 = C 2 ε > 0isuchawaythat 3.4 j C j, W sm C v sm Cv sm W 2 C 2 s 1 ε 2, which is possible by 3.1. Let m be the largest iteger such that sm C 2 /2s. Note that if is large eough there exists such m by 1.9 ad if

22 INCIPIENT INFINITE CLUSTERS 465 we choose the largest oe, we have /m C 3 for some costat C 3.Leti = iε be a iteger such that The 3.5 i + 1 2C2 3 C 1. ε W i+1 C 2 2 s W i+1 sm Ñ sm i + 1 E p cñsm i + 1 C 1/m 2 i + 1 ε 2. For idices j>i, ad o the complemet of the evet o the left-had side of 3.5 we have Z j j W + M2N 2 i+1 W + M2N C 2 2 s + M2N2, where we used the trivial boud X t j 2N 2. Let be so large that M2N 2 <C 2 s/2. The o the evet o the left had side of 3.4 we have that all clusters that are etirely cotaied i the iteriors of the circuits [i.e., i B \ A ω] have size strictly less tha W 2. Also, o this evet, 3.6 implies Z j <W 2, for j>i. Therefore W 1 ad W 2 have to occur amog Z 1,...,Z i with probability at least 1 ε. This completes the proof of Lemma 2. REMARK. Note that we ca i fact let M grow with, as log as it grows slower tha s recall that N is fixed. This is importat i provig the extesio metioed i the remark after Propositio 1. Before statig the lemma that correspods to Step 2, we itroduce some termiology. If E is a coected subgraph of B, which may or may ot be a cluster, we say that a box bk is good for E, if there is a ope circuit i the aulus ak ad the outermost ope circuit is a subset of E. LEMMA 3. For ay ε>0, i<, M>0, there is a 1 = 1 ε,i,m,n such that, for 1, we have 3.7 there are at least M good boxes for C j, 1 j i 1 ε.

23 466 A. A. JÁRAI The idea of the proof is to show that crosses some rectagle of liear size δ with large probability for some small δ>0. This is achieved by showig i that its size W is of order s, hece its diameter has to be of order.thewe apply a block-versio of the argumet of Theorem 8 to show that spaig clusters have may good boxes. PROOF OF LEMMA 3. We first wat to show that there exists a C 1 such that W i C 1 s holds with large probability. To this ed we first prove that for ay j we have 3.8 j C >W j wheever W j which implies that W j M2N 2 C j W j W j exists. Start by showig the secod iequality. If we had for some j, the for all l j we would have Z l l j W W >W j, { l : Z l } >Wj j. Sice all the Z l s are cluster sizes, this is a cotradictio. Suppose ow that the first iequality i 3.8 does ot hold. The for l j we have Z l l W + M2N 2 j W + M2N 2 <W j. We also have W 1 W j >M2N 2. These two facts together imply that the oly cadidates for W 1,...,W j are Z 1,...,Z j 1, a cotradictio. Next use 3.1 to choose C 1 = C 1 i, ε, such that for large W i 2C 1s 1 ε 2. The by 3.8 there is a 2 = 2 C 1,M, such that for 2 we have W i C 1s 1 ε 2. Note that if there are fewer tha M good boxes for the the cluster E j cotaiig C has fewer tha M good boxes. Therefore, to prove 3.7 it is sufficiet to show that for large eough 3.9 ε 2. C j there is a ope cluster E i B with E C1 s, but there are fewer tha M good boxes for E As we said our pla ow is to use that a cluster of order s has diameter comparable to ad that it ecessarily crosses some rectagle whose sides are comparable to.

24 INCIPIENT INFINITE CLUSTERS 467 REMARK. The reductio to spaig clusters seems to be a detour here; however, we were uable to use the method of Theorem 3.3 of Borgs, Chayes, Keste ad Specer 2000 directly. By Remark xiii of Borgs, Chayes, Keste ad Specer 1999 we have costats c ad d such that for x>0, 1, 4/ y 1, there is a cluster E B with diame y but E xs C 2 y 2 exp[ C 3 x/y]. This implies that there is a δ = δc 1,εsuch that for large eough there is a cluster E B with E C1 s, but diame δ 3.10 ε 4. Give that diame δ we ca fid a rectagle that is crossed by E by the followig argumet. Oe ca cover B by a family U of rectagles that cosists of traslates of [0,δ/4] [0,δ/2] ad [0,δ/2] [0,δ/4] i such a way that: a Each Q U is cotaied i B. b If E is a coected subgraph of B ad diame >δ the there is a Q U that is spaed by E i the short directio. c U C 7 /δ 2. We fix such a family U. The 3.9 will follow from 3.10 ad the followig: for each Q U ad large eough there is a cluster E such that E spas Q i the short directio, but there are fewer tha M good boxes for E 3.11 ε 4 U. Let Q 0 =[0,δ/4] [0,δ/2]. The 3.11 follows if we show the boud there is a cluster E0 i Q 0 spaig Q 0 horizotally, 3.12 εδ2, but there are fewer tha M good boxes for E 0 4C 7 where we used the boud c to replace U by a costat. We ote that the evet i 3.12 depeds oly o the cofiguratio i Q 0. Sice there is a ope horizotal crossig i Q 0 1 η<1, the BK iequality, va de Berg ad Keste 1985, implies that there is a l 0 = l 0 ε, δ such that there are more tha l 0 spaig clusters i Q 0 εδ2 8C 7.

25 468 A. A. JÁRAI So we eed 3.13 E0 is the lth lowest spaig cluster εδ2, 1 l l 0. ad E 0 has fewer tha M good boxes 8C 7 l 0 We are goig to prove this alog the same lies as the lower boud i part ii of Theorem 8. The first step, agai, is to esure that there is eough space above E 0. Put = δ/4. We use the followig geeralizatio of Lemma 1. LEMMA 4. There are costats c 1,α>0, such that, for ay l 1, the lth lowest spaig cluster of Q0 exists but it is ot cotaied i [0, ] [0,1 + a c 1 1 a α. ] We defer the proof to the ed of this subsectio. By Lemma 4 we have a a = aε,δ satisfyig 1/2 <a<1, such that there are at least l spaig clusters ad 3.14 the lth oe is ot cotaied i [0, ] [0,1 + a εδ2 ] 16C 7 l 0 uiformly i. Suppose there are at least l horizotal spaig clusters. The there exist ope horizotal crossigs R i of [0, ] [0, 2 ] 1 i l, such that: i R 1 is the lowest ope horizotal crossig, ii R i is the lowest ope horizotal crossig disjoit from the cluster cotaiig R i 1 with free boudary coditios. We are goig to coditio o {R l = r 0 },wherer 0 is a horizotal crossig iside [0, ] [0,1 + a ]. This evet oly depeds o edges o ad below r 0.We defie the graph H the regio above r 0, the highest vertex v of r 0 ad the set Am as i the proof of Theorem 8. We are goig to estimate the umber of boxes iside Am that are good for R l. We assume that v is i the right half of Q 0 ; the other case is aalogous. Because a>1/2, requirig 7m 1 a esures that the rectagle v + [ 7m, 0] [0, 7m] lies i H. Defie Vm= { bk Am : D k exists ad D k R l iside v + A7m, m }, Ym= Vm. The umber of good boxes for the lth lowest spaig cluster is at least Ym. We ext estimate the momets of Ym.

26 INCIPIENT INFINITE CLUSTERS 469 Lower boud. We use the modified cofiguratio ω, just as i the proof of Theorem 8. Recall that i H, ω is the same as ω ad it is a ew idepedet cofiguratio everywhere else. For a box bk to be i Vm it is sufficiet that i the cofiguratio ω the followig three evets occur: i B 1 ={there is a ope circuit D i ak}. { } ii B 2 = N/2 + 2Nk left side of Bv,7m. iside v + A7m, m Q 0 iii F = F m ={there is a ope circuit i v + A7m, 5m}. Applyig the FKG iequality, the RSW theorem ad 1.10 we get bk Vm P pc B 1 B 2 F P p c B 1 P p c B 2 P p c F πm C 8 πn. This implies the lower boud m 2 πm E pc Ym C 9 N 2 πn. Upper boud. We ote that if bk Am ad k 1 is the largest iteger multiple of N for which k1 + 2Nk does ot touch Am,the bk Vm Ppc bk k1 + 2Nk πk 1 C 10 πn, by For a give iteger r the umber of boxes for which k 1 = rn is bouded by C 11 m/n. Therefore, a applicatio of 1.13 with β = 0yields m/n m πrn E pc Ym C 11 C 10 N πn C m 2 πm 12 N r=0 2 πn. The upper boud m E pc Y 2 2 πm 2 m C 13 N 2 πn follows similarly usig 1.13 ad As i the proof of Theorem 8 we coclude that there is a η = ηε,a,δ,n > 0 such that, if G l deotes the set of good boxes for the lth lowest spaig cluster, the uiformly i r 0 we have 3.15 G l <η s sn R l = r 0 εδ2. 16C 7 l 0 For large eough ηs/sn > M, hece 3.15 ad 3.14 imply This completes the proof of Lemma 3.

27 470 A. A. JÁRAI REMARK. Agai, the depedece of 1 o M i Lemma 3 ca be replaced by the coditio that M grows slower tha s. I establishig Step 3 we are goig to apply the Kolmogorov Rogozi iequality [see Essee 1966] i the followig form. LEMMA 5. For a radom variable Y defie its cocetratio fuctio by QY ; λ = sup Px Y x + λ, λ 0. x Let Y 1,...,Y be idepedet, S = s=1 Y s. The there is a uiversal costat C such that, for ay real umber 0 <λ L, oe has QS ; L C L { 1 QYs ; λ } 1/2. λ s=1 The followig lemma takes care of Step 3. LEMMA 6. Let X s j with distributio X j s 1 j i,1 s M be idepedet radom variables d = { v it Dj, s : v Dj, s }, where Dj, s are arbitrary oradom circuits iside N \ N/2. Also, let a 1,...,a i be arbitrary itegers. The for ay ε>0, r< ad positive iteger i there is a iteger M = Mε,i,N,r such that 3.16 a j + ε. M s=1 X j s M a j X j s r for some 1 j <j i s=1 PROOF. It is sufficiet to show that if Y s = X j s X j s,theqs M ; 2r ε/ i 2. Sice Ys is iteger valued ad ocostat, takig λ = 1/2 we have QY s ; 1/2 <1. To get a estimate which is uiform over the circuits we ote that there are fiitely may choices for the circuits Dj,s ad Dj,s ad the set of all possible circuits does ot deped o s, oly o N. Therefore, we have This gives us δ = δn = sup s sup QY s ; 1/2 = sup QY 1 ; 1/2<1. Dj,s Dj,1 Dj,s Dj,1 QS M ; 2r C4r[M1 δ] 1/2 M 0. This proves that there is a M, such that 3.16 holds, ad Lemma 6 is proved.

28 INCIPIENT INFINITE CLUSTERS 471 REMARK. The depedece of M o r ca be replaced by the requiremet that r/ M is sufficietly small. Now we are ready to assemble the proof of Propositio 1. PROOF OF PROPOSITION 1. Fix ε>0adr<. Choose i so that 3.3 i Lemma 2 is satisfied. The choose M so that 3.16 i Lemma 6 is satisfied. UsigtheotatioofLemma2let A = A c 1 i Ac 2 i Let = { W 1 = Z j ad W 2 = Z j for some 1 j,j i }. B = { for 1 j i there are at least M good boxes for j} C deote the evet i the statemet of Lemma 3, ad let C = { Z j Z j r for some 1 j <j i }. We recall that the way the graph A ω was defied implies that the evet K G ={A = G} depeds oly o the states of the edges of G ad so does B. Let σ deote a cofiguratio o G for which K G occurs. The give K G, B ad the cofiguratio σ,thevariablesx s j are coditioally idepedet ad are determied by Beroulli percolatio processes i the iteriors of the correspodig circuits.bylemma6wehave C B,K G,σ ε, so C B = C B B ε B ε. Choosig large eough we have B c ε by Lemma 3 ad A c ε by Lemma 2. Therefore, W 1 W 2 r A c + C A c + B c + B C 3ε for large eough, which proves Propositio 1. REMARK. Propositio 1. Our remarks after Lemmas 2, 3 ad 6 prove the remark after PROOF OF LEMMA 4. We assume l 2. There are oly slight chages whe l = 1. Defie iductively the ope horizotal crossigs R i of [0, ] [0, 2 ] by: i R 1 is the lowest ope crossig, ii R i is the lowest ope crossig disjoit from the cluster cotaiig R i 1.

29 472 A. A. JÁRAI Let r 0 be a path that crosses [0, ] [0, 2 ] horizotally. We coditio o the evet {R l 1 = r 0 }, ad show that uiformly i r 0 we have Rl exists, but R l [0, ] [0,1 + a ] R l 1 = r c 1 1 a α, which implies the lemma. First assume that r 0 is cotaied i S a =[0, ] [0,1 + a ].LetQ 0 ad S a deote the followig dual graphs 3.18 Q 0 = part of [ 1 2, 1 2 ] [ 1 2, ] lyig above r0, Sa = part of [ 1 2, 1 ] [ 2 1 2, 1 + a + 1 ] 2 lyig above r0. Let U deote the top side of Q 0. If R l exists but is ot cotaied i S a, the there exists a closed vertical dual crossig of Sa. We show that with large probability this vertical crossig ca be exteded to U.LetT resp. T deote the leftmost resp. rightmost vertical dual crossig of Sa.Letv resp. v deote the top vertex of T resp. T. Whe T ad T exist, at least oe of the evets {v 1 0}, {v 1 0} has to occur. Therefore we have 3.19 R l exists, R S a R l 1 = r 0 R l exists, Tad T exist R l 1 = r 0 R l exists, Texists, v 1 0 R l 1 = r 0 + R l exists, T exists,v 1 0 R l 1 = r 0. We are goig to boud the first term o the right-had side; by symmetry of the argumet the secod term will have the same boud. Whe R l exists, there ca be o closed dual path from T to U iside Q 0.This meas that the first term o the right-had side of 3.19 is at most T exists, v 1 0, Tis ot coected to U by a closed path iside Q 0. We break up the evet { T exists, v 1 0} as a disjoit uio of the evets {T = t 0 } over paths t 0 that have their top vertex i the left half of Sa ad show 3.20 T is ot coected to U iside Q 0 T = t 0 c 1 1 a α, where by coected we mea coected by a closed dual path. Let G = G t0 deote the part of Sa to the left of t 0 icludig the vertices ad edges of t 0, ad let H = H t0 = Q 0 \ G. Cosider the sets A k = Bv,3 k+1 \ Bv,3 k H

30 for values of k that satisfy 3.21 Let INCIPIENT INFINITE CLUSTERS a 3 k /6. { } there is a closed dual path r Fk= coectig t 0 to the top side of Q 0 iside A k The sets A k have bee chose i such a way that F k T = t 0 there is a closed dual circuit i B3 k+1 \ B3 k c 2 > 0. The umber of itegers k that satisfy 3.21 is at least c 3 log 3 31 a, where c 3 > 0. Sice the evets Fk are coditioally idepedet give T = t 0,the probability o the left-had side of 3.20 is at most F c k T = t 0 = 1 c 2 c 4 1 a α, k k where α = log 3 1 c 2. Equatios 3.19 ad 3.20 imply 3.17 i the case r 0 is cotaied i S a. Whe r 0 is ot cotaied i S a, we show that 3.22 R l exists R l 1 = r 0 c 1 1 a α. For this we show that with high coditioal probability r 0 is coected by a closed dual path to the top side of Q 0, hece there is o ope horizotal crossig above R l 1.Letu be a vertex of r 0 with secod coordiate larger tha 1 + a.the same way we proved 3.20 usig dual circuits i auli cetered at u this time we ca show that there is o closed dual path from r 0 to the top side of Q 0 R l 1 = r 0 c 1 1 a α. This justifies 3.22 ad completes the proof of the lemma Large clusters look like the IIC. We are ready to prove Theorem 3, the pricipal result of this sectio. The argumet goes alog similar lies as the proof of Theorem 1. PROOF OF THEOREM 3. i 2.28, such that 3.23 Let ε>0 be give. Choose N = Nε,E,as 1 + ε 1 νe P p c E, 0 D 0 D 1 + ενe

31 474 A. A. JÁRAI for ay circuit D surroudig BN. As i 2.13 ad 2.21 we defie the evets F ad FD: F = FM,N={there is a ope circuit i AM, N} = FD. D We choose M later. By 3.1 there is a x = xε such that The we have 3.24 τ I F c = v B ε 2 + W k xs 1 ε 2. E pc I[τv F c,v C k v B ε xs ε 2 + P p c F c xs W k ] I[τv F c,v C k,w k E pc v B v B W k τv F c, Cv xs Cv xs, xs] where i the last step we used the FKG iequality. To boud the sum let m be the largest iteger such that xs sm.for large eough such m exists by 1.9 ad we have /m C 1 = C 1 x. By 1.8 we have Cv xs Ppc C0 sm C2 πm. Thus the right-had side of 3.24 is bouded by ε 2 + P p c F c C πm 3.25 ε xs 2 + P p c F c C 3 x, by virtue of 1.9. The secod term i 3.25 ca be made less tha ε/2 by choosig M large. This shows that with high probability the radom vertex I is surrouded by a ope circuit. We defie G ={I B f},wheref,adf= oπ, as i the proof of Theorem 1. Usig the tightess of W k agai, it follows that G c ε for large. For a circuit D let Ĉ k D = kth largest cluster i B \ itd.

32 INCIPIENT INFINITE CLUSTERS 475 We defie a evet B o which the cluster sizes are well-behaved ad the decouplig works. Let g be a fuctio for which g i such a way that g = os.put B = { W 1 W 2 > BM,W 1 >g } if k = 1 ad { k W B = W k+1 W k 1 > BM,W k W k > BM } >g Sice M is fixed, we have B 1 by Propositio 1 ad 3.1. It follows that for large eough 3.26 τ I E 3ε + τ I E τ I F B G = 3ε + v f if k 2. I[τv E,v C k,τ v F ]I[B ] E pc W k = 3ε + I[τv E,v C k,τ v FD]I[B ] E pc v f D W k. Now fix v ad D, ad cosider the idicators iside the expectatio o the righthad side. Suppose that the evets {v C k }, τ v FD ad B occur. The we show that for large implies 3.27 C k \ itτ v D = Ĉk τ vd ad {v τ v D} ad {τ v D Ĉk } occur. This is almost obvious, except that we eed to rule out the absurd possibility Ĉ k τ vd C k \ itτ v D. For this, first ote that the iequalities W 1 W k >g> BM imply that the k 1 largest clusters all lie etirely i extτ v D,ad The iequalities Ĉ 1 τ vd = C 1,...,Ĉk 1 τ v D = C k 1 C k \ itτ v D W k BM >W k+1. Ĉk+1 τ v D ow imply the equality i 3.27 ad the rest of 3.27 follows. O the evets i the evet B defied below also occurs: { B = Ĉ1 >g BM } if k = 1