A percolation process on the square lattice where large finite clusters are frozen
aa r X i v : . [ m a t h . P R ] J un A percolation process on the square latticewhere large finite clusters are frozen
Jacob van den Berg ∗ , Bernardo N.B. de Lima † and Pierre Nolin ‡ Abstract
In [1], Aldous constructed a growth process for the binary treewhere clusters freeze as soon as they become infinite. It was pointedout by Benjamini and Schramm that such a process does not exist forthe square lattice.This motivated us to investigate the modified process on the squarelattice, where clusters freeze as soon as they have diameter larger thanor equal to N , the parameter of the model. The non-existence result,mentioned above, raises the question if the N − parameter model showssome ‘anomalous’ behaviour as N → ∞ . For instance, if one looks atthe cluster of a given vertex, does, as N → ∞ , the probability thatit eventually freezes go to 1? Does this probability go to 0? Moregenerally, what can be said about the size of a final cluster? We givea partial answer to some of such questions. Key words and phrases: percolation, frozen cluster.
AMS 2000 subject classifications.
Primary: 60K35; Secondary: 82B43.
Let S denote the square lattice. The vertices of this lattice are the elementsof Z , and each vertex v has an edge to each of the four vertices v + ( i, j ), ∗ CWI and VU University, Amsterdam; [email protected] † Universidade Federal de Minas Gerais, Belo Horizonte; [email protected] ‡ Courant Institute, NYU, New York; [email protected] i | + | j | = 1. Let E denote the set of edges of S . The norm | v | of a vertex v = ( v , v ) is defined as max( | v | , | v | ), and the distance between two vertices v and w is defined as | v − w | . The diameter of a set W ⊂ Z is defined assup {| v − w | : v, w ∈ W } . By the diameter of a subgraph G of the squarelattice, we mean the diameter of the set of vertices of G .To each edge e ∈ E we assign a value τ e , where the τ e , e ∈ E are inde-pendent random variables, uniformly distributed on the interval (0 , e became open at time τ e (andremained open after that time), the configuration of open and closed edgesat time t would simply be a typical configuration for an ordinary percolationmodel with parameter t . In particular, the open cluster of a given vertex,say 0, would initially consist of 0 only, remain finite up to some (random)time t > /
2, and eventually (at time 1) be the entire lattice.However, in the process we study, each open cluster ‘freezes’ as soon as ithas diameter larger than or equal to N , the parameter of the process. Here‘freezes’ means that the external edges of the cluster remain closed forever.In other words, in this process initially all edges are closed, and an edge e becomes open at time τ e , unless at least one endpoint of e already belongs toan open cluster with diameter ≥ N (in which case e remains closed forever).We are interested in the sizes (diameters) of the final open clusters (i.e.the open clusters at time 1), for large values of the parameter N . Note thatthe rules of the process immediately imply that the diameter of an opencluster cannot be more than 2 N −
1. (The diameter 2 N − N − ∞ -parameter frozen model.Aldous made a rigorous construction of such a process for the binary tree (andproved several interesting properties). However, Benjamini and Schramm((1999), private communication via D. Aldous) showed that such a processdoes not exist for the square lattice (see the discussion in [3], Section 3).It follows from standard arguments that for each finite N , the N -parameterfrozen percolation model on S (and, more generally, on Z d ) does exist. (SeeSections 4.1 and 4.2 in [4], where also some exact computations for d = 1 areshown). It is natural to ask if the above mentioned non-existence result forthe ∞ -parameter model on S is, in some sense, reflected in the asymptoticbehaviour of the N -parameter system as N → ∞ . In particular, the follow-2ng questions arise, where we use the notation C ( N ) for the open cluster ofthe origin at time 1 in the N -parameter model, and where a cluster is calleda giant cluster if its diameter is at least N . • (1.) Do, eventually, the giant clusters cover the entire lattice? Moreprecisely,Does P (cid:0) C ( N ) has diameter ≥ N (cid:1) → , as N → ∞ ? • (2.) Do, eventually, the giant clusters cover a neglible portion of thelattice? More precisely,Does P (cid:0) C ( N ) has diameter ≥ N (cid:1) → , as N → ∞ ? • (3.) If the answer to question (1) is negative, what can be said, forlarge N , about the diameters of the non-giant clusters?Note that if the final cluster of 0 has diameter k , there is a vertex atdistance ≤ k + 1 from 0 which belongs to a giant cluster. Hence, if theanswer to question (2) is positive, then, for every k , the probability that C ( N ) has diameter k goes to 0 as N → ∞ .Theorem 1.1, below, gives a negative answer to question (1) and a partialanswer to question (3). The proof is given in the next section. Theorem 1.1.
Let, as before, C ( N ) denote the open cluster of the origin attime for the N -parameter frozen percolation model on the square lattice.For all < a < b < , lim inf N →∞ P (cid:0) C ( N ) has diameter ∈ ( aN, bN ) (cid:1) > . Deviating somewhat from the standard percolation notation, we define (fora positive even integer k ) B ( k ) as the box [ − k/ , k/ in the square lattice.Let, as before, N denote the parameter in the frozen percolation process.Let 0 < a < b < c ∈ ( a, b ). Next, take l such that l + ( b − c ) / < < l + ( b + c ) / . (1)3y the first inequality in (1) we can choose 0 < ε < l + ( b − c ) / ε < . (2)(Later we possibly make ε even smaller to satisfy additional conditions).Let R be the rectangular box of length ( l + ( b − c ) / N and width εN ofwhich the west side is a central subsegment of the east side of B ( cN ). LetΛ be the union of B ( bN ) and R . So Λ is a bN × bN square from which a lN × εN rectangle sticks out to the right (see Figure 1). Further, let Λ ′ bethe set of all points at distance ≤ εN from Λ. So Λ ′ is the disjoint union of B (( b + 2 ε ) N ) and a rectangle of width 3 εN and length lN . Let R ′ be theleftmost part of length 4 εN of that rectangle.Finally, let L (respectively, L ) be the rectangle of which the south (resp.north) side is the rightmost segment of length εN of the north (resp. south)side of B ( cN ), and the north (resp. south) side is a part of the boundary ofΛ ′ . The first part of the proof of the theorem is a deterministic Lemma. Callan edge e t -open if τ e < t . A path, or more generally a set of edges, is called t -open if every edge of that set is t -open. The terminology t -closed is definedin a completely similar way. Lemma 2.1.
Suppose that there is a τ ∈ (0 , / such that each of thefollowing, (i) - (vi) below, holds: • (i) ∃ τ -open circuit γ in the annulus B ( bN ) \ B ( cN ) . • (ii) ∃ -closed dual circuit in the annulus B ( cN ) \ B ( aN ) . • (iii) ∃ a -closed dual circuit π in the annulus Λ ′ \ Λ . • (iv) ∃ -closed dual paths π and π in L , respectively L , ‘connecting’ γ and π . • (v) ∃ a -open path in R from the right side of R to γ . • (vi) There is no τ -open horizontal crossing of the area of R ′ boundedby the two segments of π .Then the final cluster of in the N -parameter frozen percolation processhas diameter ∈ [ aN, bN ] . R ′ aNcNbN lN L L R ǫNǫNǫNǫNǫNǫNǫN ǫN (a) The different annuli and rectangles used. γ π π π (b) Open and closed paths involved in the proof. Figure 1: Construction used for the proof of Theorem 1.1. The open pathsare represented with solid lines, and the closed paths with dotted lines.5 roof.
What are the consequences of (i) - (vi) for the N -parameter frozenpercolation process? Note that before time τ nothing in the interior of π isfrozen yet, simply because, roughly speaking, by (vi) this region is split intwo subregions that are too small to freeze. Consequently, every edge e inthe interior of π with τ e < τ is indeed open at time τ in the frozen process.In particular the circuit γ (see (i) above) is open at time τ . Because of this,and by (iv), there can be no cluster in the interior of π that does not contain γ and freezes before time 1 /
2. On the other hand, something in the interiorof π does freeze before time 1 / / π with diameter ≥ ( l +( b − c ) / c ) N ,which by (1) is larger than N . Hence the open cluster containing γ is frozenbefore time 1 /
2. Further, because of this and by (ii), it follows that attime 1 / < bN which is part of a frozen cluster. Hence the diameterof the open cluster of 0 will remain smaller than bN forever. On the otherhand, every edge in the interior of the 1 / aN and bN . This completes the proof of Lemma 2.1.To complete the proof of Theorem 1.1, we will show that there exists a τ ∈ (0 , /
2) for which the probability that there is a circuit γ in the annulus B ( bN ) \ B ( cN ) and a dual circuit π in the annulus Λ ′ \ Λ such that each of theevents (i) - (vi) in Lemma 2.1 holds is bounded away from 0 as N → ∞ . Fromnow on, E ( i ) , E ( ii ) and E ( iii ) will denote the event described in (i), respectively(ii) and (iii), in Lemma 2.1. Moreover, for a given (deterministic) circuits γ in the annulus B ( bN ) \ B ( cN ) and a given dual circuit π in the annulusΛ ′ \ Λ, we define the following events: E ( i ) ( γ ) is the event that γ is the narrowest τ -open circuit in B ( bN ) \ B ( cN ); E ( iii ) ( π ) is the event that π is the widest 1 / ′ \ Λ; E ( iv ) ( γ, π ) is the event that (iv) holds; E ( v ) ( γ ) is the event that (v) holds; E ( vi ) ( π ) is the event that (vi) holds.Further, C ( i ) , C ( ii ) etc. will denote strictly positive constants that may de-pend on a , b , c , and ε , but not on N .Now, let α (= α ( N )) denote the probability that there is a 1 / R . Note that, by the well-known RSWresults in percolation (see e.g. [5], Section 11.7), α ( N ) is bounded away from0 as N → ∞ . Let τ (= τ ( N )) be such that6 ( ∃ τ -open horizontal crossing of R ′ ) = α/ . (3)Note that for all sufficiently large N such a τ exists, is unique and smallerthan 1 /
2, because the probability that there is a t -open horizontal crossingof R ′ is obviously continuous and strictly increasing in t , is 0 for t = 0, andat least α for t = 1 / f : [0 , → [0 ,
1] (which depends only on b , c , and ε , but not on N or t ) such that f (0) = 0, f (1) = 1 and P ( ∃ t -open circuit in B ( bN ) \ B ( cN )) ≥ f ( P ( ∃ t -open horizontal crossing of R ′ )) . By this, the definition of E ( i ) , and (3), there is a positive constant C ( i ) such that for all sufficiently large NP ( E ( i ) ) ≥ C ( i ) . (4)Also by RSW, there is a positive constant C ( iii ) such that for all sufficientlylarge N P ( E ( iii ) ) ≥ C ( iii ) . (5)Now, for given γ and π , we condition on the event E ( i ) ( γ ) ∩ E ( iii ) ( π )defined above.Note that the event E ( ii ) is independent of this event and (by RSW)has, for all sufficiently large N , probability larger than some positive con-stant C ( ii ) . Also note that, again by RSW, the conditional probability of E ( iv ) ( γ, π ) is (for all sufficiently large N ) bounded from below by some posi-tive constant C ( iv ) . Further, E ( v ) ( γ ) is independent of the event we conditionon, and clearly (by the definition of α ), has conditional probability ≥ α . It isalso clear that E ( vi ) ( π ) is independent of the event we condition on, and that(using the choice of τ ), for all sufficiently large N , its complement has prob-ability at most α/
2. Finally, it is easy to see that the events E ( ii ) , E ( iv ) ( γ, π ),and E ( v ) ( γ ) ∩ E ( vi ) ( π ) are conditionally independent.Combining the above facts with Lemma 2.1 we get that the probabilitythat the final cluster of 0 in the N -parameter frozen percolation process hasdiameter ∈ [ aN, bN ] is larger than or equal to7 γ,π P ( E ( i ) ( γ ) ∩ E ( iii ) ( π )) C ( ii ) C ( iv ) P ( E ( v ) ( γ ) ∩ E ( vi ) ( π )) . (6)Since, by the choice of α and by (3), P ( E ( v ) ( γ ) ∩ E ( vi ) ( π )) ≥ P ( E ( v ) ( γ )) − P (cid:0) E c ( vi ) ( π ) (cid:1) ≥ α − α/ α/ , the summation (6) is larger than or equal to α C ( ii ) C ( iv ) X γ,π P (cid:0) E ( i ) ( γ ) ∩ E ( iii ) ( π ) (cid:1) , which by (4) and (5) is larger than or equal to α C ( ii ) C ( iv ) C ( i ) C ( iii ) . This completes the proof of Theorem 1.1. (cid:3)
Acknowledgments
B.N.B.L. is partially supported by CNPq and FAPEMIG (Programa PesquisadorMineiro). P.N.’s research was supported in part by the NSF grant OISE-07-30136. B.N.B.L. and P.N. would also like to thank CWI for its hospitalityduring multiple visits.
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