On the size of the largest cluster in 2D critical percolation

We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0<a<b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is between an^2 pi(n) and bn^2 pi(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the argument.


Introduction and main result
Consider bond percolation on Z 2 with parameter p. (See [G99] for a general introduction to percolation theory.) Let Λ n = [−n, n] 2 ∩ Z 2 and let, for v ∈ Λ n , C n (v) denote the size of the open cluster of v inside the box Λ n : C n (v) = |{w ∈ Λ n : v ↔ w, inside Λ n }|, where we use the standard notation v ↔ w for the existence of an open path from v to w, and where the addition 'inside Λ n ' means that we require the existence of such a path which is located entirely in Λ n . For a set A ⊂ Z 2 we denote by ∂A the (internal) boundary of A: ∂A = {v ∈ A : ∃w ∈ A : (v, w) is an edge }.
The remaining part of A will be called the interior of A. Let π p (n) be the probability P p (O ↔ ∂Λ n ). For simplicity we write π(n) for π 1 2 (n). We are interested in the size of 'large' open clusters in Λ n for the case where p is equal (or close) to the critical value 1/2. It is known in the literature that, informally speaking, the size of the largest open cluster is typically of order n 2 π(n): For any c > 0, there is a 'reasonable' probability that it is larger (smaller) than cn 2 π(n), and this probability goes to 0 uniformly in n as c → ∞ (c → 0). (See [BCKS99], [BCKS01]; see also [J03] Section 3.) However, the question whether for all 0 < a < b there is a 'reasonable' probability that there is an open cluster with size between an 2 π(n) and bn 2 π(n) has not been investigated in the literature.
This question, which is also natural by itself, arises e.g. in the study of finite-parameter frozen-percolation models. In these models each edge is closed at time 0 and 'tries' to become open at some random time, independently of the other edges. However, an open cluster stops growing as soon as its size has reached a certain (large) value M , the parameter of the model. (See [BLN12] where this was studied for the case where the 'size' of a cluster is defined as its diameter instead of its volume.) The investigation of such processes leads to the question how two open clusters which both have size of order M but smaller than M , merge to a cluster of size bigger than M , which in turn leads to the question at the end of the previous paragraph. To state our main result, an affirmative answer to that question, we first need a few more definitions.
For k, l ∈ N, we denote by HC(k, l) the event that there is an open horizontal crossing in the box [0, k] × [0, l]. (This is an open path from the left side to the right side of the box, of which all vertices, except the starting and end point, are in the interior of the box). Let the "characteristic length" be as defined in e.g. [N08] and [K87]: For a fixed ∈ (0, 1 2 ): and L( 1 2 ) = ∞. The precise value of is not essential. Throughout this paper we will consider it as being fixed, and therefore we omit it from our notation.
As said before, our main question concerns the existence of some open cluster in Λ n with size in some specific interval. The proof we obtained gives, with only a tiny bit of extra work, something stronger; it shows that with 'reasonable' probability the maximal open cluster has this property. Therefore we state our main result in this stronger form (and remark that we do not know an essentially simpler proof of the original weaker form): Denote by M n the size of the maximal open cluster in Λ n . More precisely, Theorem 1 Let 0 < a < b. There exist δ > 0 and N ∈ N such that, for all n ≥ N and all p with L(p) ≥ n, P p M n ∈ (an 2 π(n), bn 2 π(n)) > δ.
The proof is given in Section 3. Section 2 will list the main ingredients used in the proof. The proof involves a suitable partition of Λ n in smaller boxes and annuli. A brief and informal summary is given in the beginning of Section 3, after the description of these objects.
2 Ingredients for the proof of Theorem 1 We will make amply use of standard RSW results of the following form: For all l > 0 there exists δ(l) > 0 such that, for all k and all p with L(p) ≥ k, P p (HC(k, lk )) ≥ δ(l). For a set W of vertices defineC For k, n ∈ N, we use the notation Λ k,n for the rectangle [−k, k] × [−n, n]. We will use the following properties of π p (n) from the literature.
(iv) For all k, n ∈ N and p with L(p) ≥ k ∧ n, Proof: The inequalities in (i) are well-known. (The first follows easily from RSW arguments, and the second goes back to [BK85]; see also for example [BCKS01]). Part (ii) follows from (7) in [K86]. Part (iii) is Theorem 1 in [K87]. Part (iv), of which versions are explicitly in the literature (see e.g. [K86], [K87] and [N08]), is proved as follows (where we assume that k ≤ n): where the last inequality uses part (ii) and (iii). Define We need the following result for the distribution of Y (m), which is essentially in [BCKS99] and (for the special case p = 1/2) [K86].
It was shown in [K86] (and extended/generalized in [BCKS99] and [BCKS01]) that M n , the size of the largest open cluster in Λ n , is typically of order n 2 π(n). In particular, its expectation has an upper and a lower bound which are linear in n 2 π(n). In the proof of Theorem 1 we use the following result from [BCKS01].
Finally, to streamline the arguments in Section 3.5 at the end of the proof of Theorem 1, we state here the following fact about 'steering' the outcome of the sum of independent random variables. It is a simple observation rather than a lemma, and versions of it have without doubt been used in the probability literature in various contexts.

Proof of Theorem 1
We first give a proof for the special case p = 1/2 and therefore drop the subscript p from the notation P p and E p . At the end of Section 3.5 we point out that (due to the 'uniformity' of the ingredients stated in Section 2) the proof for the general case is essentially the same.

More definitions, and brief outline of the proof
Let s, t ∈ N with t ≤ 1 3 s. The proof involves a construction using the following boxes and annuli.
Before we go on, we give a very brief and informal summary of the proof (see Figure 1): The box Λ n in the statement of the theorem will be (roughly) partitioned in m 2 boxes B i,j defined above, where the s (and hence m) and t will be chosen appropriately, depending on n, a and b. (For elegance/symmetry we take the number m odd). We will 'construct' an open cluster of which the 'skeleton' consists of circuits in the annuli A II i,j , 'glued' together by connections in the 'corridors' V i,j and H i,j . (The other annuli defined above will be used for technical reasons in the proof). The setup is such that the contributions from the different B i,j 's to the total cluster size are roughly independent, and that these contributions can be 'steered' to get the total sum inside the desired interval. In some sense this replaces the original problem for the box Λ n by a similar problem, but now for the smaller boxes B i,j . Apart from the technicalities involving the control of local dependencies, there is a subtle aspect in the proof related to the asymptotic behaviour of π(n): Although the precise power-law behaviour of π(n) is not important, it seems to be essential for the arguments that the exponent in a power-law upper bound is strictly smaller than 1 (see the note at the end of the proof of Lemma 11)). Now we continue with the precise constructions mentioned above. First we give some more notation and definitions. Let E denote the set of edges of Z 2 and Ω = {0, 1} E . For ω ∈ Ω and F ⊂ E we will write ω F ∈ {0, 1} F for the 'restriction' (ω e , e ∈ F ) of ω to F . Let A ⊂ Ω and W ⊂ Z 2 . We write E(W ) for the set of all edges of which both endpoints are in W . Informally, we use the notation A(W ) for the set of all configurations ω ∈ Ω that belong to A or can be turned to an element of A by modifying ω outside E(W ). More precisely, We denote byÕ s,t the event that (i) -(iii) below occur (see Figure 2): The introduction of this event looks meaningless since it has probability 0. It will only be used to give a 'compact' description of the following events (which do play a key role in the proof).

Expected cluster size in a narrow annulus
For a circuit γ in Z 2 we denote by Int(γ) the bounded connected component of Z 2 \ γ, and define C γ = |{v ∈ Int(γ) : v ↔ γ}|.
Further, for all i, j, let γ i,j denote the widest open circuit in the annulus A II i,j , and define, for W ⊂ Λ n , If there is no open circuit in A II i,j , then C i,j (W ) = 0. Recall the definition ofC in (2).
Lemma 7 There exists a constant C 10 > 0 such that for all s ∈ N, t ≤ 1 3 s and all i, j: Proof: The first inequality follows immediately from the obvious fact that, on the event O s,t i,j , C i,j (A i,j ) is smaller than or equal toC(A i,j ). We prove the second inequality. Without loss of generality we take i = j = 0. We subdivide the annulus A = A 0,0 = 8 l=1 A l , where A 1 , A 2 , A 3 , A 4 are the (3t × 3t)-squares in the four corners, and A 5 , A 6 , A 7 , A 8 the remaining rectangles (see Figure 3). Note thatC(A ) ≤ 8 l=1C (A l ). Hence it is sufficient to show that there is a constant C 11 such that for each l = 1, · · · , 8, By symmetry we only have to handle the cases l = 1 and l = 5. For each l the l.h.s. of (7) is Recall the notation (4). For each v ∈ A 1 , obviously,  (8) and (9) gives For the case l = 5 a similar argument (now the 'surgery' can be done on the union of the region V 0,−1 and the rectangle Application of part (iv) of Theorem 2 to the right-hand sides of (10) and (11) gives (7).

Properties of nice circuits
Let m, s, t be as in Definition 6, and recall the Remark about the values of the indices i, j at the end of Section 3.1. Let, for each i, j, γ i,j be as in the beginning of Section 3.2, and letγ i,j be a deterministic circuit in the annulus A II i,j . Further we will denote the collection of all γ i,j 's by (γ), and the collection of alγ i,j 's by (γ).

Definition 8 We say thatγ
with C 10 as in Lemma 7. Further, the collection (γ) is called (m, s, t)-nice if each circuit in the collection is (s, t)-nice.
We define Γ s,t i,j as the event that γ i,j is (s, t)-nice, and define Recall the notation (6). We will study the open cluster C 0,0 (Q), where Lemma 9 There exist positive constants C 14 and C 15 such that, for all m, s and t, Proof: We claim that there is a constant C 16 > 0 such that for all i, j: where (with the notation (4)) To prove this claim we write We also need an event D 2 which, informally speaking, connects the structures in the definition of D 1 with those in D. More precisely, and H i,j ∩ A I i+1,j both contain a vertical crossing. The other Dĩ ,j 2 's are defined similarly. By RSW (and FKG) there is a positive constant C 16 such that P(D 2 ) > C 16 . Note that (with the notation in Section 3.1), the event D is increasing with respect to the edges outside E(K), and that D 1 is increasing with respect to the edges outside E(G). We get where we used FKG in the third line, and in the fourth line we used that D doesn't depend on the configuration on G, D 1 doesn't depend on the configuration on K, and D 2 doesn't depend on the configuration on G ∪ K. This proves the claim. By repeating the same arguments for each B i,j , we eventually get the following 'extension' of (14): Now we are ready to prove part (i): where the inequality follows from (15) and the obvious inequality P(O m,s,t ) ≤ i,j P(O s,t i,j ). This gives part (i) of the lemma because for each factor in the product of the last expression in (16) we have, by Definition 8, Markov's inequality and Lemma 7, To prove part (ii) first note that where we used (14) in the denominator. Hence where the last inequality is just the 'niceness' property (Definition 8) of (γ). To finish the proof of part (ii), note that, for each K > 0, Km 2 stπ(t) .
Applying part (iv) of Theorem 2 to the first expectation in the r.h.s. of the last expression, and (17) to each of the other expectations gives, by choosing K sufficiently large, the desired result. This completes the proof of part (ii) of Lemma 9.

Cluster-size contributions inside the circuits
In this section we write the value t (the width of the relevant annuli and 'corridors' in the construction) as εs . A suitable value for ε (depending on the values of a and b in the statement of Theorem 1) will be determined in the next section. The main result in the current section concerns the contribution from the interior of a nice circuit to the cluster of that circuit. Recall the notation (5).

Completion of the proof of Theorem 1
We are now ready to prove Theorem 1. First we still restrict to the case p = 1/2. Let 0 < a < b be given. See the brief outline in Section 3.1. The lengths of the building blocks B i,j and the widths of the annuli and 'corridors' in the partition of Λ n , will be taken proportional to n, say (roughly) xn and εxn respectively, with suitably chosen x and ε. For this purpose we will use the following lemma: Lemma 11 There exist x > 0, ε ∈ (0, 1 12 ) and N ∈ N, with 1 x an odd integer, such that for all n ≥ N the following inequalities hold: Proof: It is easy to see (a weak form of the lower bound in part (i) of Theorem 2 suffices) that if x is sufficiently (depending on a) small, then (23) holds for all sufficiently large n. It also easily follows (now from the upper bound in the same Theorem) that if x is sufficiently (depending on b−a) small, (24) holds for all sufficiently large n. Finally, for x fixed, it follows (again from the upper bound in part (i) of Theorem 2) that if ε is sufficiently (depending on b − a and x) small, then (25) holds for all sufficiently large n.
Note that for this last step it is essential that the exponent (1/2) in part (i) of Theorem 2 is strictly smaller than 1. This completes the proof of Lemma 11.
Now let x, ε and N be as in Lemma 11. Moreover we assume (which we may, because we can enlarge N if necessary) that n − 1 x xn ≤ ε xn for all n ≥ N . Denote by D n the event D n = {∃v ∈ Λ n : C n (v) ∈ (an 2 π(n), bn 2 π(n))}.
Let n ≥ N and let m = 1 x , s = xn and t = ε xn . By straightforward RSW and FKG arguments, there is a δ 4 (x, ε) > 0 such that P(O m,s,t ) > δ 4 (x, ε). Hence From Lemma 9 (i) it follows that The next step is conditioning on the widest open circuits. (28) For each (γ) we denote by C in (γ) the number of all vertices that are in the interior of a circuit in the collection (γ) and connected to that circuit, and by C out (γ) the number of vertices outside these circuits that are connected to one or more of these circuits, plus the number of vertices on these circuits. We have where the last inequality holds by (25) and Lemma 9 (ii), and because the configurations in the interiors of theγ i,j 's are obviously independent of the event conditioned on in the expression in the r.h.s. of the first inequality. Note that C in (γ) = i,j Cγ i,j . The Cγ i,j 's are independent and for each i, j we have P Cγ i,j ∈ (ax 2 n 2 π(n), 1 3 (b − a)n 2 π(n)) (23),(24) ≥ P Cγ i,j ∈ C 17 xn 2 π( xn ), C 18 xn 2 π( xn ) Lem.10(i) ≥ δ 2 , and P Cγ i,j ≤ x 2 1 3 (b − a)n 2 π(n) ≥ P Cγ i,j ≤ 4C 10 ε xn xn π( ε xn ) Lem.10(ii) ≥ δ 3 (ε).
Hence the conditions of Lemma 5 (with k = (1/x) 2 , α = an 2 π(n), β = ( 1 3 a + 2 3 b)n 2 π(n)) are satisfied. Hence, by that lemma the l.h.s. of (29) is at least 1 2 (δ 2 ∧ δ 3 (ε)) (1/x) 2 . Together with (26) -(28) this shows that P(D n ) > δ 5 , with δ 5 a positive constant which depends only on a and b. Now we will show that, by the way we 'constructed' the open cluster, a similar result holds for the maximal open cluster in Λ n . First note that the 'constructed' cluster has the property that it contains an open horizontal and an open vertical crossing of the box Λ ms . Also note that there is at most one open cluster with this property. Given the exact location of the (unique) open cluster with this property, the conditional probability that it is the maximal cluster in Λ n is, if its size is larger than an 2 π(n), clearly larger than or equal to the probability that the remaining part of Λ n contains no open cluster of size larger than an 2 π(n). By obvious monotonicity this probability is at least P(M n ≤ an 2 π(n)), which by Theorem 4 is at least some positive constant δ 2 (which depends only on a). This argument gives P M n ∈ (an 2 π(n), bn 2 π(n)) ≥ δ 2 δ 5 , which completes the proof of Theorem 1 for p = 1/2. Now let, more generally, p be such that L(p) ≥ n. It is straightforward to check that (due to the 'uniformity' in p of the results in Section 2) each step in the proof remains essentially valid. For instance, it is easy to see from the arguments used that Lemma 7 (now with P replaced by P p ) remains valid as long as L(p) ≥ s. Since we take s ≤ n (application of) this lemma (and, similarly, the other lemma's) can be carried out as before. This completes the proof of Theorem 1.