A lower bound on the two-arms exponent for critical percolation on the lattice
aa r X i v : . [ m a t h . P R ] O c t The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2015
A LOWER BOUND ON THE TWO-ARMS EXPONENT FORCRITICAL PERCOLATION ON THE LATTICE
By Rapha¨el Cerf
Universit´e Paris Sud and IUF
We consider the standard site percolation model on the d -dimensionallattice. A direct consequence of the proof of the uniqueness of the in-finite cluster of Aizenman, Kesten and Newman [ Comm. Math. Phys. (1987) 505–531] is that the two-arms exponent is larger than orequal to 1 /
2. We improve slightly this lower bound in any dimen-sion d ≥
2. Next, starting only with the hypothesis that θ ( p ) >
1. Introduction.
We consider the site percolation model on Z d . Eachsite is declared open with probability p and closed with probability 1 − p ,and the sites are independent. Little is rigorously known on the percolationmodel at the critical point p c in three dimensions. Barsky, Grimmett andNewman have proved that there is no percolation at the critical point in ahalf-space. Grimmett and Marstrand have proved that the critical points ina half-space and in the full space coincide. A full account of these results andtheir proofs can be found in Grimmett’s book [4]. Kesten’s book presents alsosome estimates valid at the critical point (see Chapter 5 of [9]). There existsone remarkable result, a rigorous lower bound on the two-arms exponent,which says that, for any d ≥ ∃ κ > , ∀ n ≥ P p c (two-arms(0 , n )) ≤ κ ln n √ n . The event “two-arms(0 , n )” is the event that two neighbors of 0 are con-nected to the boundary of the box Λ( n ) = [ − n, n ] d by two disjoint openclusters. Although some percolationists are aware of this estimate (e.g., itis explicitly used by Zhang in [14]), it does not seem to be fully written inthe literature. This estimate can be obtained as a byproduct of the proof Received July 2013; revised May 2014.
AMS 2000 subject classifications.
Primary 60K35; secondary 82B43.
Key words and phrases.
Critical percolation, arms exponent.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2015, Vol. 43, No. 5, 2458–2480. This reprint differs from the original inpagination and typographic detail. 1
R. CERF of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman[1]. This deep proof was originally written for a quite general percolationmodel. A simplified and illuminating version has been worked out by Gan-dolfi, Grimmett and Russo [3]. The two-arms estimate is obtained by taking ε = κ ln n/ √ n in the proof of [3]. Nowadays the uniqueness of the infinitecluster in percolation is proved with the help of the more robust Burton–Keane argument; see, for instance, [4] or [6]. Yet the Burton–Keane argumentrelies on translation invariance, and it does not yield any quantitative es-timate, contrary to the argument of Aizenman, Kesten and Newman. Thefirst main result of this paper is a slightly improved lower bound on thetwo-arms exponent. Theorem 1.1.
Let d ≥ and let p c be the critical probability of the sitepercolation model in d dimensions. We have lim sup n →∞ n ln P p c ( two-arms (0 , n )) ≤ − d + 3 d − d + 5 d − . In two dimensions, our two-arms event corresponds to a four-arms eventwith alternating colors. The corresponding exponent is rigorously known tobe equal to 5 / /
21. In high dimensions, the exponent associated to
WO-ARMS EXPONENT FOR CRITICAL PERCOLATION two arms is rigorously proven to be equal to four [10], and our lower boundconverges to 1 / ∀ γ < , ∃ c > , ∀ n ≥ P p c (two-arms(0 , n )) ≤ cn γ . To prove Theorem 1.1, we rework the proof of [3] in order to obtain aninequality of the form P p c (two-arms(0 , n )) ≤ d ln n p | Λ( n ) | E ( p |C| ) + negligible term , where C is the collection of the clusters joining Λ( n ) to the boundary ofΛ(2 n ). From this inequality, we obtain the previously known estimate on thetwo-arms event by bounding the number of clusters in C by 2 d (2 n + 1) d − .We next try to enhance the control on the number of clusters. It turns outthat the expectation of this number can be bounded with the help of theprobability of the two-arms event. Our strategy consists in controlling thetwo-arms event associated to a box. This is the purpose of our second mainresult. The event “two-arms(Λ( n ) , n α )” is the event that two sites of the boxΛ( n ) = [ − n, n ] d are connected to the boundary of the box Λ( n + n α ) by twodisjoint open clusters. Theorem 1.2.
Let d ≥ and let p c be the critical probability of the sitepercolation model in d dimensions. Let α be such that α > d + 2 d − d + 3 d − d + 5 d − . We have lim n →∞ P p c ( two-arms (Λ( n ) , n α )) = 0 . For d = 3, this giveslim n →∞ P p c (two-arms(Λ( n ) , n )) = 0 . Next, we cover the boundary of the box Λ( n ) = [ − n, n ] d by a collection ofboxes of side length n β , with β small. Theorem 1.2 yields an estimate onthe number of small boxes joined to the boundary of Λ(2 n ) by at most onecluster, from which we obtain an upper bound on the mean number of openclusters joining Λ( n ) to the internal vertex boundary of Λ(2 n ). This gives anupper bound on E ( |C| ) in terms of the two-arms event. This way we obtainan inequality of the form P (two-arms(0 , n )) ≤ c ′ ln n √ n (cid:18) k d − + k d +2 d − P (two-arms(0 , n )) (cid:19) / . R. CERF
Iterating this inequality with an adequate choice of k ≤ n , we progressivelyimprove the exponent 1 /
2. We obtain a sequence of exponents converginggeometrically toward the limiting value presented in Theorem 1.1. The finalimprovement is quite disappointing and the value is probably quite far fromthe correct one.Our third main result is a little minor step for the establishment of long-range order in a finite box. This is a central question, which if correctlyanswered, should lead to a proof that θ ( p c ) = 0. For Λ a box and x, y in Λ,we denote by { x ←→ y in Λ } the event that x, y are connected by an openpath inside Λ. Theorem 1.3.
Let d ≥ and let p be such that θ ( p ) > . Let α be suchthat α > d + 5 d − d + 3 d − d − . We have inf n ≥ inf { P p ( x ←→ y in Λ( n α )) : x, y ∈ Λ( n ) } > . For d = 3, this gives the following estimate: ∃ ρ > , ∀ n ≥ , ∀ x, y ∈ Λ( n ) P p ( x ←→ y in Λ( n )) ≥ ρ. WO-ARMS EXPONENT FOR CRITICAL PERCOLATION One of the most important problems in percolation is to prove that, inthree dimensions, there is no infinite cluster at the critical point. The mostpromising strategy so far seems to perform a renormalization argument [4,12]. The missing ingredient is a suitable construction helping to define agood block, starting solely with the hypothesis that θ ( p ) >
0. For instance,it would be enough to have the above estimate within a box of side lengthproportional to n . Moreover, if the famous conjecture θ ( p c ) = 0 was true,such an estimate would indeed hold. Here again, we are still far from thedesired result. Our technique to prove Theorem 1.3 is to inject the hypothesis θ ( p ) >
2. Basic notation.
Two sites x, y of the lattice Z d are said to be con-nected if they are nearest neighbors, that is, if | x − y | = 1. Let A be a subsetof Z d . We define its internal boundary ∂ in A and its external boundary ∂ out A by ∂ in A = { x ∈ A : ∃ y ∈ A c , | x − y | = 1 } ,∂ out A = { x ∈ A c : ∃ y ∈ A, | x − y | = 1 } . For x ∈ Z d , we denote by C ( x ) the open cluster containing x , that is, theconnected component of the set of the open sites containing x . If x is closed,then C ( x ) is empty. For n ∈ N , we denote by Λ( n ) the cubic boxΛ( n ) = [ − n, n ] d . Let n, ℓ be two integers. We consider the open clusters of the percolationconfiguration restricted to Λ( n + ℓ ). These open clusters are the connectedcomponents of the graph having for vertices the sites of Λ( n + ℓ ) which areopen, endowed with edges between nearest neighbors. We denote by C thecollection of the open clusters in Λ( n + ℓ ) which intersect both Λ( n ) and ∂ in Λ( n + ℓ ), that is, C = { C open cluster in Λ( n + ℓ ) : C ∩ Λ( n ) = ∅ , C ∩ ∂ in Λ( n + ℓ ) = ∅ } .
3. The proof of Gandolfi, Grimmett and Russo.
We reproduce here theinitial step of the argument of Gandolfi, Grimmett and Russo to prove theuniqueness of the infinite cluster [3]. This argument was obtained from themore complex work of Aizenman, Kesten and Newman [1]. The only differ-ence is that we introduce an additional parameter ℓ . We will use specificvalues for ℓ later on. We define the following three subsets of Λ( n ): F = [ C ∈C C ∩ Λ( n ) , G = [ C ∈C ∂ out C ∩ Λ( n ) ,H = [ C ,C ∈C ,C = C ( ∂ out C ∩ ∂ out C ∩ Λ( n )) . R. CERF
A site of Λ( n ) belongs to F if it is connected to ∂ in Λ( n + ℓ ) by an open path.A site of Λ( n ) belongs to G if it is closed and it has a neighbor which isconnected to ∂ in Λ( n + ℓ ) by an open path. A site of Λ( n ) belongs to F ∪ G if it has a neighbor which is connected to ∂ in Λ( n + ℓ ) by an open path. Yet,for any x ∈ Λ( n ), the event { a neighbor of x is connected to ∂ in Λ( n + ℓ ) by an open path } is independent of the status of the site x itself, therefore, P ( x ∈ F | x ∈ F ∪ G ) = P ( x is open) = p,P ( x ∈ G | x ∈ F ∪ G ) = P ( x is closed) = 1 − p. Summing over x ∈ Λ( n ), we obtain E ( | F | ) = E (cid:18) X x ∈ Λ( n ) x ∈ F (cid:19) = X x ∈ Λ( n ) P ( x ∈ F ) = X x ∈ Λ( n ) P ( x ∈ F | x ∈ F ∪ G ) P ( x ∈ F ∪ G )= X x ∈ Λ( n ) pP ( x ∈ F ∪ G ) = pE ( | F ∪ G | ) . Similarly, we have E ( | G | ) = (1 − p ) E ( | F ∪ G | ) . We wish to estimate the cardinality of H . To this end, we write | H | = (cid:12)(cid:12)(cid:12)(cid:12) [ C ,C ∈C ( ∂ out C ∩ ∂ out C ∩ Λ( n )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X C ∈C | ∂ out C ∩ Λ( n ) | − (cid:12)(cid:12)(cid:12)(cid:12) [ C ∈C ∂ out C ∩ Λ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X C ∈C | ∂ out C ∩ Λ( n ) | − | G | . Taking the expectation in this inequality, we obtain E ( | H | ) ≤ E (cid:18)X C ∈C | ∂ out C ∩ Λ( n ) | (cid:19) − E ( | G | )= E (cid:18)X C ∈C | ∂ out C ∩ Λ( n ) | (cid:19) − − pp E ( | F | )= (1 − p ) E (cid:18)X C ∈C (cid:18) − p | ∂ out C ∩ Λ( n ) | − p | C ∩ Λ( n ) | (cid:19)(cid:19) . WO-ARMS EXPONENT FOR CRITICAL PERCOLATION For A a subset of Z d , we define h ( A ) = 11 − p |{ x ∈ A : x is closed }| − p |{ x ∈ A : x is open }| . For C an open cluster, we define C = C ∪ ∂ out C. With these definitions, we can rewrite the previous inequality as E ( | H | ) ≤ (1 − p ) E (cid:18)X C ∈C h ( C ∩ Λ( n )) (cid:19) . Our next goal is to control the expectation on the right-hand side. We firstnotice that, for x in the box Λ( n ), the expected value of h ( C ( x ) ∩ Λ( n )) iszero. Lemma 3.1.
For any x ∈ Λ( n ) , we have E ( h ( C ( x ) ∩ Λ( n ))) = 0 . Proof.
Let x ∈ Λ( n ). For any lattice animal A containing x and in-cluded in Λ( n ), we have P ( C ( x ) ∩ Λ( n ) = A ) = p | A | (1 − p ) | ∂ out A ∩ Λ( n ) | . Summing over all such lattice animals A , we get1 = X A p | A | (1 − p ) | ∂ out A ∩ Λ( n ) | . Differentiating with respect to p , we obtain0 = X A (cid:18) | A | p − | ∂ out A ∩ Λ( n ) | − p (cid:19) p | A | (1 − p ) | ∂ out A ∩ Λ( n ) | and we notice that this last sum is equal to E ( h ( C ( x ) ∩ Λ( n ))). (cid:3) It turns out that, for large clusters, the value h ( C ∩ Λ( n )) is close to 0with high probability. This is quantified by the next proposition.
4. The large deviation estimate.
The basic inequality leading to the con-trol of the two-arms event relies on the following large deviation estimate.This estimate is a variant of the one stated in [1, 3]. We have introduced anadditional parameter ℓ and we use Hoeffding’s inequality. Proposition 4.1.
For any p in ]0 , , any n ≥ , ℓ ≥ , we have ∀ x ∈ Λ( n + ℓ ) , ∀ k ≥ , ∀ t ≥ P ( | h ( C ( x ) ∩ Λ( n )) | ≥ t, | C ( x ) ∩ Λ( n ) | = k ) ≤ exp (cid:18) − p (1 − p ) t k (cid:19) R. CERF
Proof.
Let x ∈ Λ( n + ℓ ). In order to estimate the above probabil-ity, we build C ( x ) ∩ Λ( n ) in two steps. First, we explore all the sites ofΛ( n + ℓ ) \ Λ( n ). Second, we use a standard growth algorithm in Λ( n ) to findthe sites belonging to C ( x ) ∩ Λ( n ). This algorithm is driven by a sequenceof i.i.d. Bernoulli random variables ( X m ) m ≥ with parameter p . Let us de-scribe precisely this strategy. The first step amounts to condition on thepercolation configuration in Λ( n + ℓ ) \ Λ( n ). We denote this configurationby ω | Λ( n + ℓ ) \ Λ( n ) and we write P ( | h ( C ( x ) ∩ Λ( n )) | ≥ t, | C ( x ) ∩ Λ( n ) | = k )= X η P ( | h ( C ( x ) ∩ Λ( n )) | ≥ t, | C ( x ) ∩ Λ( n ) | = k, ω | Λ( n + ℓ ) \ Λ( n ) = η )= X η P ( | h ( C ( x ) ∩ Λ( n )) | ≥ t, | C ( x ) ∩ Λ( n ) | = k | ω | Λ( n + ℓ ) \ Λ( n ) = η ) × P ( ω | Λ( n + ℓ ) \ Λ( n ) = η ) . The summation runs over all the percolation configurations η in Λ( n + ℓ ) \ Λ( n ). Let us fix one such configuration η . The second step corresponds tothe growth algorithm. At each iteration, the algorithm updates three sets ofsites: • The set A k : these are the active sites, which are to be explored. • The set O k : these are open sites, which belong to C ( x ) ∩ Λ( n ). • The set C k : these are closed sites, which have been visited by the algo-rithm.All the sites of the sets A k , O k , C k are in Λ( n ). Initially, we set O = C = ∅ and A is the set of the sites of Λ( n ) which are connected to x by an openpath in η . Recall that a path is a sequence of sites such that each site is aneighbor of its predecessor. Thus a site y belongs to A if and only if ∃ z , . . . , z r ∈ Λ( n + ℓ ) \ Λ( n ) z , . . . , z r are open in η,z = x, z , . . . , z r , y is a path . Suppose that the sets A k , O k , C k are built and let us explain how to buildthe sets A k +1 , O k +1 , C k +1 . If A k = ∅ , the algorithm terminates and C ( x ) ∩ Λ( n ) = O k ∪ C k . If A k is not empty, we pick an element x k of A k . The site x k has not beenexplored previously, and its status will be decided by the random variable X k . We consider two cases, according to the value of X k . • X k = 0. The site x k is declared closed, and we set A k +1 = A k \ { x k } , O k +1 = O k , C k +1 = C k ∪ { x k } . WO-ARMS EXPONENT FOR CRITICAL PERCOLATION • X k = 1. The site x k is declared open, and we set O k +1 = O k ∪ { x k } , C k +1 = C k ,A k +1 = A k ∪ V k \ ( { x k } ∪ O k ∪ C k ) , where V k is the set of the sites of Λ( n ) which are neighbors of x k or whichare connected to x k by an open path in Λ( n + ℓ ) \ Λ( n ). More precisely, asite y of Λ( n ) belongs to V k if and only if it is a neighbor of x k or ∃ z , . . . , z r ∈ Λ( n + ℓ ) \ Λ( n ) z , . . . , z r are open in η,x k , z , . . . , z r , y is a path . Since O k ∪ C k ∪ A k is included in Λ( n ) and the sequence of sets O k ∪ C k , k ≥ A k is empty after at most | Λ( n ) | steps and the al-gorithm terminates. Suppose | C ( x ) ∩ Λ( n ) | = k . This means that the growthalgorithm stops after having explored k sites in Λ( n ). The status of these k sites is given by the first k variables of the sequence ( X m ) m ≥ , so that | C ( x ) ∩ Λ( n ) | = X + · · · + X k , | ∂ out C ( x ) ∩ Λ( n ) | = k − ( X + · · · + X k )and h ( C ( x ) ∩ Λ( n )) = 11 − p | ∂ out C ( x ) ∩ Λ( n ) | − p | C ( x ) ∩ Λ( n ) | = 11 − p ( k − ( X + · · · + X k )) − p ( X + · · · + X k )= pk − ( X + · · · + X k ) p (1 − p ) . Therefore, we can write P ( | h ( C ( x ) ∩ Λ( n )) | ≥ t, | C ( x ) ∩ Λ( n ) | = k | ω | Λ( n + ℓ ) \ Λ( n ) = η )= P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) pk − ( X + · · · + X k ) p (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ t, | C ( x ) ∩ Λ( n ) | = k | ω | Λ( n + ℓ ) \ Λ( n ) = η (cid:19) ≤ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) pk − ( X + · · · + X k ) p (1 − p ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ t | ω | Λ( n + ℓ ) \ Λ( n ) = η (cid:19) = P ( | X + · · · + X k − pk | ≥ tp (1 − p )) ≤ (cid:18) − k t p (1 − p ) (cid:19) . For the last step, we have applied Hoeffding’s inequality [8] (one could alsouse the earlier inequality due to Bernstein [2]). The above inequality is uni-form with respect to the configuration η . Plugging this bound in the initialsummation, we obtain the desired estimate. (cid:3) R. CERF
5. The central inequality.
We will now put together the previous esti-mates in order to obtain an inequality between the probability of the two-arms event and the number of clusters in the collection C . Our goal is tobound the expectation E (cid:18)X C ∈C h ( C ∩ Λ( n )) (cid:19) . Let E be the event E = {∀ C ∈ C , | h ( C ∩ Λ( n )) | < (ln n ) | C ∩ Λ( n ) | / } . On the event E , we bound the sum as follows: X C ∈C | h ( C ∩ Λ( n )) | ≤ X C ∈C (ln n ) | C ∩ Λ( n ) | / ≤ (ln n ) p |C| (cid:18)X C ∈C | C ∩ Λ( n ) | (cid:19) / . A site x belongs to at most 2 d sets of the collection { C : C ∈ C} , therefore, X C ∈C | C ∩ Λ( n ) | ≤ d | Λ( n ) | . If E does not occur, then we use the inequality ∀ C ∈ C | h ( C ∩ Λ( n )) | ≤ p (1 − p ) | C ∩ Λ( n ) | and we bound the sum as follows: X C ∈C | h ( C ∩ Λ( n )) | ≤ p (1 − p ) X C ∈C | C ∩ Λ( n ) | ≤ dp (1 − p ) | Λ( n ) | . We bound the probability of the complement of E with the help of Proposi-tion 4.1: P ( E c ) = P ( ∃ C ∈ C , | h ( C ∩ Λ( n )) | ≥ (ln n ) | C ∩ Λ( n ) | / ) ≤ P ( ∃ x ∈ Λ( n ) , | h ( C ( x ) ∩ Λ( n )) | ≥ (ln n ) | C ( x ) ∩ Λ( n ) | / ) ≤ X x ∈ Λ( n ) | Λ( n ) | X k =1 P ( | C ( x ) ∩ Λ( n ) | = k, | h ( C ( x ) ∩ Λ( n )) | ≥ (ln n ) | C ( x ) ∩ Λ( n ) | / ) ≤ | Λ( n ) | − n ) p (1 − p ) ) . WO-ARMS EXPONENT FOR CRITICAL PERCOLATION Putting together the previous inequalities, we obtain E ( | H | ) ≤ d (ln n ) q | Λ( n ) | E ( p |C| )+ 4 dp (1 − p ) | Λ( n ) | exp( − n ) p (1 − p ) ) . Definition 5.1.
For x ∈ Z d and n ≥
1, we define the event two-arms( x, n )by:two-arms( x, n ) = in the configuration restricted to x + Λ( n )two neighbors of x are connected to the boundaryof the box x + Λ( n ) by two disjoint open clusters . If x belongs to Λ( n ) and the event two-arms( x, n + ℓ ) occurs, then x belongs to H as well. Thus, | H | ≥ X x ∈ Λ( n ) two - arms( x, n + ℓ ) and taking expectation, we obtain the following central inequality. Lemma 5.2.
For any p in ]0 , , any n ≥ , ℓ ≥ , we have the inequality P ( two-arms (0 , n + ℓ )) ≤ d ln n p | Λ( n ) | E ( p |C| ) + 4 dp (1 − p ) | Λ( n ) | exp( − n ) p (1 − p ) ) . In order to obtain the initial estimate on the two-arms event stated inthe Introduction, we remark that the cardinality of C is bounded by thecardinality of ∂ in Λ( n ), because different clusters of C intersect ∂ in Λ( n ) atdifferent sites. Taking ℓ = 0 in the inequality, we have P (two-arms(0 , n )) ≤ d (ln n ) (cid:18) | ∂ in Λ( n ) || Λ( n ) | (cid:19) / + 4 dp (1 − p ) | Λ( n ) | exp( − n ) p (1 − p ) ) . This inequality readily implies the initial estimate stated in the Introduction.
Proposition 5.3.
Let d ≥ and let p ∈ ]0 , . There exists a constant κ depending on d and p only such that ∀ n ≥ P p ( two-arms (0 , n )) ≤ κ ln n √ n . In order to improve this estimate on the two-arms exponent, we will tryto improve the estimate on the cardinality of C . R. CERF
6. Lower bound for the connection probability.
For x, y two sites be-longing to a box Λ, we define the event { x ←→ y in Λ } = { the sites x and y are joined by an open path of sites inside Λ } . The next lemma gives a polynomial lower bound for the probability of con-nection of two sites of Λ( n ) if one allows the path to be in Λ(2 n ). At crit-icality, the expected behavior is indeed a power of n , but with a differentexponent. In Lemma 1.1 of [11], Kozma and Nachmias derive a smaller lowerbound, however, only paths staying inside Λ( n ) are allowed. Lemma 6.1.
There exists a positive constant c which depends only onthe dimension d such that, for n ≥ , ∀ x, y ∈ Λ( n ) P p c ( x ←→ y in Λ(2 n )) ≥ cn d − d . Proof.
The basic ingredient to prove Lemma 6.1 is the following lowerbound. For any box Λ centered at 0, we have X x ∈ ∂ in Λ P p c (0 ←→ x in Λ) ≥ . This lower bound is proved in Lemma 3.1 of [11], or in the proof of The-orem 5.3 of [5]. The reason is that, by an argument due to Hammersley[7], if the converse inequality holds, then this implies that the probability oflong connections decays exponentially fast with the distance, and the systemwould be in the subcritical regime. Applying the above inequality to the boxΛ( n ), we conclude that there exists x ∗ in ∂ in Λ( n ) such that P p c (0 ←→ x ∗ in Λ( n )) ≥ | ∂ in Λ( n ) | ≥ d )(2 n + 1) d − . Without loss of generality, we can suppose that x ∗ belongs to { n } × Z d − .Let us set e = (1 , , . . . , P p c (0 ←→ ne in Λ( n ) ∪ (2 ne + Λ( n ))) ≥ P p c (0 ←→ x ∗ in Λ( n ) ∪ (2 ne + Λ( n )) ,x ∗ ←→ ne in Λ( n ) ∪ (2 ne + Λ( n ))) ≥ P p c (0 ←→ x ∗ in Λ( n ) ∪ (2 ne + Λ( n ))) × P p c ( x ∗ ←→ ne in Λ( n ) ∪ (2 ne + Λ( n ))) ≥ P p c (0 ←→ x ∗ in Λ( n )) P p c ( x ∗ ←→ ne in 2 ne + Λ( n )) ≥ (cid:18) d )(2 n + 1) d − (cid:19) . WO-ARMS EXPONENT FOR CRITICAL PERCOLATION By symmetry, the same inequality holds for the other axis directions. Letnow x, y be two sites in Λ( n ) with coordinates x = ( x , . . . , x d ) , y = ( y , . . . , y d ) . We suppose first that y i − x i is even, for 1 ≤ i ≤ d , and we set z = x, z = ( y , x , . . . , x d ) , . . . , z d − = ( y , . . . , y d − , x d ) , z d = y. Again by the FKG inequality, we have P p c ( x ←→ y in Λ(2 n )) ≥ P p c ( ∀ i ∈ { , . . . , d − } , z i ←→ z i +1 in Λ(2 n )) ≥ Y ≤ i ≤ d − P p c ( z i ←→ z i +1 in Λ(2 n )) . Let i ∈ { , . . . , d − } and let n i = ( y i − x i ) /
2. We have n i ≤ n and( z i + Λ( n i )) ∪ ( z i +1 + Λ( n i )) ⊂ Λ(2 n ) , whence P p c ( z i ←→ z i +1 in Λ(2 n )) ≥ P p c ( z i ←→ z i +1 in ( z i + Λ( n i )) ∪ ( z i +1 + Λ( n i ))) ≥ (cid:18) d )(2 n i + 1) d − (cid:19) . Coming back to the previous inequality, we obtain P p c ( x ←→ y in Λ(2 n )) ≥ Y ≤ i ≤ d − (cid:18) d )(2 n i + 1) d − (cid:19) ≥ cn d − d , where the last inequality holds for some positive constant c . In the generalcase, if x = y and if x i − y i is not even for some 1 ≤ i ≤ d , we can find z inΛ( n ) such that | z − x | ≤ | y − x | and ∀ i ∈ { , . . . , d } | z i − y i | ≤ , z i − x i is even . We then use the FKG inequality to write P p c ( x ←→ y in Λ(2 n )) ≥ P p c ( x ←→ z in Λ(2 n )) P p c ( z ←→ y in Λ(2 n )) . The probability of connection between x and z is controlled with the helpof the previous case, while the probability of connection between z and y islarger than ( p c ) d . (cid:3) R. CERF
7. Two-arms for distant sites.
We derive here an estimate for the two-arms event associated to two distant sites, which we define next.
Definition 7.1.
For n, ℓ ≥ a, b belonging to Λ( n ), wedefine the event two-arms(Λ( n ) , a, b, ℓ ) as follows:two-arms(Λ( n ) , a, b, ℓ ) = (cid:26) the open clusters of a and b in Λ( n + ℓ )are disjoint and they intersect ∂ in Λ( n + ℓ ) (cid:27) . We will establish an inequality linking the two-arms event for distant sitesto the two-arms event for neighboring sites.
Lemma 7.2.
Let p ∈ ]0 , . For any n, ℓ ≥ and any a, b ∈ Λ( n ) , we have ∀ k ≤ ℓ P ( two-arms (Λ( n ) , a, b, ℓ )) ≤ d p ( n + k ) d P ( two-arms (0 , ℓ − k )) P ( a ←→ b in Λ( n + k )) . Proof.
Let n, ℓ ≥
1, let k ≤ ℓ and let a, b ∈ Λ( n ). We denote by C ( a )and C ( b ) the open clusters of a and b in Λ( n + ℓ ). We write P (two-arms(Λ( n ) , a, b, ℓ )) = X A,B P ( C ( a ) = A, C ( b ) = B ) , where the sum runs over the pairs A, B of connected subsets of Λ( n + ℓ )such that A ∩ B = ∅ , a ∈ A, WO-ARMS EXPONENT FOR CRITICAL PERCOLATION A ∩ ∂ in Λ( n + ℓ ) = ∅ , b ∈ B,B ∩ ∂ in Λ( n + ℓ ) = ∅ . For E a finite subset of Z d , we define E = E ∪ ∂ out E, ∆ E = ∂ in (( E ) c ) . Equivalently, we have∆ E = { z / ∈ E ∪ ∂ out E : z is the neighbor of a point in ∂ out E } . Let a, b ∈ Λ( n ) and let A, B be two connected subsets of Λ( n + ℓ ) as above.Suppose that the open clusters of a and b in Λ( n + ℓ ) are exactly A and B , that is, we have C ( a ) = A and C ( b ) = B . Suppose that ∂ out A ∩ ∂ out B ∩ Λ( n ) = ∅ . Then the event two-arms( z, ℓ ) occurs, where z is any point in theprevious intersection. Suppose next that ∂ out A ∩ ∂ out B ∩ Λ( n ) = ∅ . We will transform the configuration in Λ( n ) in order to create a two-armsevent. The idea is that, for k ≤ ℓ , the sets ∆ A and ∆ B are rather likely tobe connected by an open path inside Λ( n + k ) \ ( A ∪ B ). By modifying thestatus of one site in ∂ out B , we can then create a connection between ∆ A and ∂ in Λ( n + ℓ ), which does not use the sites of A . Let us make this strategymore precise. Any open path joining a to b in Λ( n + k ) has to go throughboth ∆ A and ∆ B , thus P ( a ←→ b in Λ( n + k )) ≤ P (∆ A ∩ Λ( n + k ) ←→ ∆ B ∩ Λ( n + k ) in Λ( n + k ) \ ( A ∪ B )) . The event { C ( a ) = A, C ( b ) = B } depends only on the sites in A ∪ B , henceit is independent from the event above, therefore, P ( C ( a ) = A, C ( b ) = B, ∆ A ∩ Λ( n + k ) ←→ ∆ B ∩ Λ( n + k ) in Λ( n + k ) \ ( A ∪ B )) ≥ P ( C ( a ) = A, C ( b ) = B ) × P ( a ←→ b in Λ( n + k )) . Let E be the event E = { C ( a ) ∩ ∂ in Λ( n + ℓ ) = ∅ , C ( b ) ∩ ∂ in Λ( n + ℓ ) = ∅ , C ( a ) ∩ C ( b ) = ∅ } . Summing the previous inequality over
A, B , we get (cid:18)X
A,B P ( C ( a ) = A, C ( b ) = B ) (cid:19) P ( a ←→ b in Λ( n + k )) ≤ X A,B P (cid:18) C ( a ) = A, C ( b ) = B ∆ A ∩ Λ( n + k ) ←→ ∆ B ∩ Λ( n + k ) in Λ( n + k ) \ ( A ∪ B ) (cid:19) R. CERF ≤ P (cid:18) E , ∆ C ( a ) ∩ Λ( n + k ) ←→ ∆ C ( b ) ∩ Λ( n + k )in Λ( n + k ) \ ( C ( a ) ∪ C ( b )) (cid:19) ≤ P (cid:18) E , ∃ u ∈ ∆ C ( a ) ∩ Λ( n + k ) , ∃ v ∈ ∆ C ( b ) ∩ Λ( n + k ) u ←→ v in Λ( n + k ) \ ( C ( a ) ∪ C ( b )) (cid:19) ≤ X u,v ∈ Λ( n + k ) P (cid:18) E , u ∈ ∆ C ( a ) , v ∈ ∆ C ( b ) u ←→ v in Λ( n + k ) \ ( C ( a ) ∪ C ( b )) (cid:19) . Let us consider the event inside the probability appearing in this sum. Let z (resp., w ) be a neighbor of u (resp., v ) belonging to ∂ out C ( a ) [resp., ∂ out C ( b )]. Suppose that we change the status of w to open. The site u is connected to v by an open path, and v is now connected to w and C ( b ),hence to ∂ in Λ( n + ℓ ), and this connection does not use any site of C ( a ).Thus, the site z , which is closed, will admit two neighbors which are con-nected to ∂ in Λ( n + ℓ ): the site u and another one belonging to C ( a ), andthese two neighbors do not belong to the same cluster in Λ( n + ℓ ). Therefore,the event two-arms( z, ℓ − k ) occurs, and we conclude that P (cid:18) C ( a ) ∩ ∂ in Λ( n + ℓ ) = ∅ , C ( b ) ∩ ∂ in Λ( n + ℓ ) = ∅ , C ( a ) ∩ C ( b ) = ∅ u ∈ ∆ C ( a ) , v ∈ ∆ C ( b ) , u ←→ v in Λ( n + k ) \ ( C ( a ) ∪ C ( b )) (cid:19) ≤ d p P (two-arms(0 , ℓ − k )) . Plugging this inequality in the previous sum, we obtain P (two-arms(Λ( n ) , a, b, ℓ )) ≤ X u,v ∈ Λ( n + k ) d p P (two-arms(0 , ℓ − k )) P ( a ←→ b in Λ( n + k )) ≤ | Λ( n + k ) | d p P (two-arms(0 , ℓ − k )) P ( a ←→ b in Λ( n + k )) ≤ d p ( n + k ) d P (two-arms(0 , ℓ − k )) P ( a ←→ b in Λ( n + k )) . This is the inequality we wanted to prove. (cid:3)
We derive next an estimate for the two-arms event associated to a box.For n, ℓ ≥
1, we define the event two-arms(Λ( n ) , ℓ ) as follows:two-arms(Λ( n ) , ℓ ) = (cid:26) there exist two distinct open clustersin Λ( n + ℓ ) joining Λ( n ) to ∂ in Λ( n + ℓ ) (cid:27) . WO-ARMS EXPONENT FOR CRITICAL PERCOLATION Corollary 7.3.
For any n ≥ , ℓ ≥ n , we have P ( two-arms (Λ( n ) , ℓ )) ≤ d p n d − P ( two-arms (0 , ℓ − n ))inf { P ( a ←→ b in Λ(2 n )) : a, b ∈ ∂ in Λ( n ) } . Proof.
From the definition of the two-arms event, we havetwo-arms(Λ( n ) , ℓ ) = [ a,b ∈ ∂ in Λ( n ) two-arms(Λ( n ) , a, b, ℓ ) . Therefore, applying the inequality of Lemma 7.2 with k = n , we obtain P (two-arms(Λ( n ) , ℓ )) ≤ X a,b ∈ ∂ in Λ( n ) P (two-arms(Λ( n ) , a, b, ℓ )) ≤ X a,b ∈ ∂ in Λ( n ) d p (2 n ) d P (two-arms(0 , ℓ − n )) P ( a ←→ b in Λ(2 n )) ≤ d p d (2 n + 1) d − (2 n ) d P (two-arms(0 , ℓ − n ))inf { P ( a ←→ b in Λ(2 n )) : a, b ∈ ∂ in Λ( n ) } . This yields the desired inequality. (cid:3)
Corollary 7.4.
We have lim n →∞ P ( two-arms (Λ( n ) , n d +4 d − )) = 0 . Proof.
We apply the inequality given in Corollary 7.3. We use Propo-sition 5.3 to control the probability of the two-arms event and Lemma 6.1to control from below the connection probability. We obtain P (two-arms(Λ( n ) , ℓ )) ≤ d pc n d +2 d − κ ln( ℓ − n ) √ ℓ − n . We take ℓ = n d +4 d − in this inequality and we send n to ∞ . (cid:3) For d = 3, this yields the exponent 4 d + 4 d −
8. Control on the number of arms.
We try next to improve the previousestimates. The idea is the following. With the help of Corollary 7.4, we willimprove slightly the control on the number of clusters in the collection C [these are the clusters intersecting both Λ( n ) and ∂ in Λ( n + ℓ )]. Thanks to thecentral inequality stated in Lemma 5.2, this will permit to improve the boundon the two-arms event for a site, and subsequently the bound on the two-arms event for a box. This leads to a better exponent in Corollary 7.4. We can R. CERF then iterate this scheme to improve further the exponents. Unfortunately,the sequence of exponents converges geometrically and the final result is stillquite weak.Let n, ℓ, k be three integers, with k ≤ n ≤ ℓ . Let Λ i , i ∈ I , be a collection ofboxes which are translates of Λ( k ) = [ − k, k ] d , which are included in Λ( n ) andwhich covers the inner boundary ∂ in Λ( n ). Such a covering can be realizedwith disjoint boxes if 2 n + 1 is a multiple of 2 k + 1, otherwise we do notrequire that the boxes are disjoint. In any case, there exists such a coveringΛ i , i ∈ I , whose cardinality | I | satisfies | I | ≤ d (cid:18) nk (cid:19) d − . Let us fix such a covering. Given a percolation configuration in Λ( n + ℓ ), abox Λ i of the covering is said to be good if the event two-arms(Λ i , ℓ ) doesnot occur. Let us compute the expected number of bad boxes: E (cid:18) number of bad boxes inthe collection Λ i , i ∈ I (cid:19) = E (cid:18)X i ∈ I the box Λ i is bad (cid:19) = | I | P (two-arms(Λ( k ) , ℓ )) . The clusters of the collection C intersect ∂ in Λ( n ), hence they have to go intoone box of the collection Λ i , i ∈ I . If two clusters of C intersect the samebox Λ i , this box has to be bad, because these two clusters go all the wayuntil ∂ in Λ( n + ℓ ), hence they realize the event two-arms(Λ i , ℓ ). Thus, a goodbox of the collection Λ i , i ∈ I , meets at most one cluster of C . Moreover, abad box of the collection Λ i , i ∈ I , meets at most | ∂ in Λ( k ) | clusters of C . Weconclude that |C| ≤ (cid:18) number of good boxes inthe collection Λ i , i ∈ I (cid:19) + | ∂ in Λ( k ) | × (cid:18) number of bad boxes inthe collection Λ i , i ∈ I (cid:19) . We bound the number of good boxes by | I | and we take the expectation inthis inequality. We obtain E ( |C| ) ≤ | I | + | ∂ in Λ( k ) | × | I | × P (two-arms(Λ( k ) , ℓ )) ≤ d d (cid:18) nk (cid:19) d − (1 + 2 d (2 k + 1) d − P (two-arms(Λ( k ) , ℓ ))) ≤ c (cid:18) nk (cid:19) d − + cn d − P (two-arms(Λ( k ) , ℓ )) , where c is a constant depending on d and p . Plugging the inequality ofCorollary 7.3 in the previous inequality, we get, with some larger constant WO-ARMS EXPONENT FOR CRITICAL PERCOLATION c , E ( |C| ) ≤ c (cid:18) nk (cid:19) d − + cn d − k d − P (two-arms(0 , ℓ − k ))inf { P ( a ←→ b in Λ(2 k )) : a, b ∈ ∂ in Λ( k ) } . Noticing that E ( p |C| ) ≤ E ( |C| ) / , we deduce from the central inequalitystated in Lemma 5.2 and the previous inequality that P (two-arms(0 , n + ℓ )) ≤ d ln n p | Λ( n ) | (cid:18) c (cid:18) nk (cid:19) d − + cn d − k d − P (two-arms(0 , ℓ − k ))inf { P ( a ←→ b in Λ(2 k )) : a, b ∈ ∂ in Λ( k ) } (cid:19) / + 4 dp (1 − p ) | Λ( n ) | exp( − n ) p (1 − p ) ) . We choose ℓ = n , and we conclude that, for some constant c , we have P (two-arms(0 , n )) ≤ c ln n √ n (cid:18) k d − + k d − P (two-arms(0 , n − k ))inf { P ( a ←→ b in Λ(2 k )) : a, b ∈ ∂ in Λ( k ) } (cid:19) / . We shall next iterate this inequality in order to enhance the lower bound onthe two-arms exponent.
9. Iterating at p c . In this section, we work at p = p c and we completethe proofs of Theorems 1.1 and 1.2. Lemma 6.1 yields that ∀ k ≥ { P ( a ←→ b in Λ(2 k )) : a, b ∈ ∂ in Λ( k ) } ≥ ck d − d . From the last two inequalities, we deduce the following lemma.
Lemma 9.1.
There exists c > such that, for ≤ k ≤ n , P ( two-arms (0 , n )) ≤ c ln n √ n (cid:18) k d − + k d +2 d − P ( two-arms (0 , n − k )) (cid:19) / . We shall next use iteratively the inequality of the lemma to improve pro-gressively the lower bound on the two arms exponent. Suppose that for somepositive constants c ′ , β, γ , with γ <
1, we have ∀ n ≥ P (two-arms(0 , n )) ≤ c ′ (ln n ) β n γ . Choosing k = n δ with δ = γ d + 3 d − , R. CERF we obtain that ∀ n ≥ P (two-arms(0 , n )) ≤ c √ c ′ (ln n ) β/ n γ ′ , where γ ′ = 12 + d − d + 6 d − γ. By monotonicity, ∀ n ≥ P (two-arms(0 , n )) ≤ P (two-arms(0 , ⌊ n/ ⌋ )) , therefore, there exists also a constant c ′′ such that ∀ n ≥ P (two-arms(0 , n )) ≤ c ′′ (ln n ) β +1 n γ ′ . The initial estimate stated in Proposition 5.3 yields that ∀ n ≥ P (two-arms(0 , n )) ≤ κ ln n √ n . We define a sequence of exponents ( γ i ) i ≥ by setting γ = 1 / ∀ i ≥ γ i +1 = 12 + d − d + 6 d − γ i . Iterating the previous argument, we conclude that, for any i ≥
1, there existsa constant α i such that ∀ n ≥ P (two-arms(0 , n )) ≤ α i (ln n ) i +1 n γ i . It follows that ∀ i ≥ n →∞ n ln P (two-arms(0 , n )) ≤ γ i . The sequence ( γ i ) i ≥ converges geometrically toward γ ∞ = 2 d + 3 d − d + 5 d − . Letting i go to ∞ in the previous inequality, we obtain the result statedin Theorem 1.1. Theorem 1.1 and the inequality of Corollary 7.3 readilyimply Theorem 1.2. To prove Theorem 1.2, we proceed as in the proof ofCorollary 7.4, but instead of the initial estimate of Proposition 5.3, we usethe enhanced estimate provided by Theorem 1.1. WO-ARMS EXPONENT FOR CRITICAL PERCOLATION
10. Proof of Theorem 1.3.
Throughout this section, we work with a pa-rameter p such that θ ( p ) >
0. We will use the hypothesis θ ( p ) > Lemma 10.1.
Let n, ℓ ≥ . For any x, y ∈ Λ( n ) , we have P ( x ←→ y in Λ( n + ℓ )) ≥ θ ( p ) − P ( two-arms (Λ( n ) , x, y, ℓ )) . Proof.
We write P ( x ←→ y in Λ( n + ℓ )) ≥ P x ←→ ∂ in Λ( n + ℓ ) y ←→ ∂ in Λ( n + ℓ ) x ←→ y in Λ( n + ℓ ) ≥ P (cid:18) x ←→ ∂ in Λ( n + ℓ ) y ←→ ∂ in Λ( n + ℓ ) (cid:19) − P x ←→ ∂ in Λ( n + ℓ ) y ←→ ∂ in Λ( n + ℓ ) x y in Λ( n + ℓ ) . By the FKG inequality, we have P (cid:18) x ←→ ∂ in Λ( n + ℓ ) y ←→ ∂ in Λ( n + ℓ ) (cid:19) ≥ P (cid:18) x ←→ ∞ y ←→ ∞ (cid:19) ≥ θ ( p ) . Moreover, P x ←→ ∂ in Λ( n + ℓ ) y ←→ ∂ in Λ( n + ℓ ) x y in Λ( n + ℓ ) ≤ P (two-arms(Λ( n ) , x, y, ℓ )) . The last two inequalities imply the inequality stated in the lemma. (cid:3)
Since θ ( p ) ≤ P (0 ←→ ∂ in Λ( n )) ≤ X x ∈ ∂ in Λ( n ) P (0 ←→ x in Λ( n )) , then there exists x n in ∂ in Λ( n ) such that P (0 ←→ x n in Λ( n )) ≥ θ ( p ) | ∂ in Λ( n ) | ≥ θ ( p )2 d (2 n + 1) d − . We apply the inequality of Lemma 7.2 to 0 and x n with k = 0: P (two-arms(Λ( n ) , , x n , ℓ )) ≤ d p n d P (two-arms(0 , ℓ )) P (0 ←→ x n in Λ( n )) . R. CERF
Combining the two previous inequalities, we conclude that P (two-arms(Λ( n ) , , x n , ℓ )) ≤ d pθ ( p ) n d − P (two-arms(0 , ℓ )) . We apply the inequality of Lemma 10.1 to 0 and x n , and, together with theprevious inequality, we obtain P (0 ←→ x n in Λ( n + ℓ )) ≥ θ ( p ) − d pθ ( p ) n d − P (two-arms(0 , ℓ )) . Let α be such that α > d + 5 d − d + 3 d − d − . We take ℓ = n α . By Theorem 1.1, for n large enough, P (0 ←→ x n in Λ( n + n α )) ≥ θ ( p ) . Suppose, for instance, that x n belongs to { n } × Z d − . Let e = (1 , , . . . , n large enough, P (0 ←→ ne in Λ(4 n + n α )) ≥ P (cid:18) ←→ x n in Λ( n + n α ) x n ←→ ne in 2 ne + Λ( n + n α ) (cid:19) ≥ P (0 ←→ x n in Λ( n + n α )) P ( x n ←→ ne in 2 ne + Λ( n + n α )) ≥ P (0 ←→ x n in Λ( n + n α )) ≥ θ ( p ) . Thus, there exists N ≥ ∀ n ≥ N P (0 ←→ ne in Λ(4 n + n α )) ≥ θ ( p ) . Let n ≥ N and let k ∈ { N, . . . , n } . We have P (0 ←→ ke in Λ(4 n + n α )) ≥ P (0 ←→ ke in Λ(4 k + k α )) ≥ θ ( p ) . This implies further that ∀ k ∈ { N, . . . , n } P (0 ←→ ke in Λ(4 n + n α )) ≥ p θ ( p ) . Since N is independent of n , we conclude that there exists ρ > ∀ n ≥ N, ∀ k ∈ { , . . . , n } P (0 ←→ ke in Λ(4 n + n α )) ≥ ρ. Since N is fixed, this lower bound can be extended to every n ≥ ρ . By symmetry, we have the same lower bounds for theprobabilities of connections along the other axis directions. Using the FKGinequality, we conclude that ∀ n ≥ , ∀ x ∈ Λ(2 n ) P (0 ←→ x in Λ(6 n + n α )) ≥ ρ d . This completes the proof of Theorem 1.3.
WO-ARMS EXPONENT FOR CRITICAL PERCOLATION Acknowledgements.
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