The gaps between the sizes of large clusters in 2D critical percolation

Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 \pi(n), where \pi(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order \sqrt{n^2 \pi(n)}. Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 \pi(n). Our main result is a proof that this is indeed the case.


Introduction and statement of main results
For general background on percolation we refer to [G99] and [BR06]. We consider bond percolation on the square lattice with parameter p equal to its critical value p c = 1/2. Let Λ n = [−n, n] 2 ∩ Z 2 be the 2n × 2n box centered at 0 and let ∂Λ n = Λ n \ Λ n−1 be the (inner) boundary of the box. For each vertex v ∈ Z 2 , we write Λ n (v) = Λ n + v. Further, the open cluster in Λ n of the vertex v is denoted by C n (v). More precisely, where 'u ↔ v inside Λ n ' means that there is an open path from u to v of which all vertices are in Λ n . We write π(n) for the probability P(O ↔ ∂Λ n ), the probability that there is an open path from O to ∂Λ n . Further, we write s(n) := n 2 π(n). (1) By the size of a cluster we mean the number of vertices in the cluster. Let, for i = 1, 2, · · · , C (i) n denote the i-th largest open cluster in Λ n , and let |C (i) n | denote its size. (If two clusters have the same size, we order them in some deterministic way).
In [BCKS99] it was proved that |C (1) n | is of order s(n). In the later paper [BCKS01] by the same authors it is shown that also |C (2) n |, |C (3) n | etcetera are of order s(n). They also proved an extension of this result for the case where the parameter p is not equal but close to p c .
It was shown by Járai in [J03] that for each i the difference |C | → ∞ in probability as n → ∞ . In fact he showed that this difference is at least of order s(n). He suggested that it should be of order s(n), but did not have a proof. In this paper we show that his conjecture is correct. We became interested in such problems through our investigation of frozen-percolation processes. Our main theorem is as follows.
Theorem 1 For all k ∈ N, δ > 0, there exist ε > 0, N ∈ N such that for all n ≥ N : Remarks: (i) The analog of Theorem 1 can be proved for site and bond percolation on other common two dimensional lattices, e.g. site percolation on the square or the triangular lattice. In this latter model (site percolation on the triangular lattice) one of the last steps of the proof can be made a little bit shorter (see the Remark below the proof of Proposition 11).
(ii) The proof, which is given in Section 3, follows the main line of Járai's proof of the weaker bound: We divide the box Λ n in boxes of smaller length (denoted by 2t), and condition on the configuration outside certain open circuits in these smaller boxes. Conditioned on this information, the 'contributions' (to the sizes of certain open clusters) from the interiors of these circuits are independent random variables. This leads to a problem concerning the concentration function of a sum of independent random variables, to which a general ('classical') theorem is applied. The main difference with Járai's arguments is that we take t proportional to n, with a proportionality factor chosen as a suitable function of the 'parameters' k and δ in the theorem. This makes the arguments more powerful (and also somewhat more complicated). Moreover, the theorem on concentration functions we used (see Theorem 8 below) is somewhat stronger than the one used in Járai's arguments.
Furthermore, with essentially the same argument we can show that the probability that there exists a cluster with size in a given interval of length εs(n) goes to zero as ε → 0 uniformly in n: Theorem 2 For all x, δ > 0, there exists an ε > 0 such that, for all n ∈ N: P (∃u ∈ Λ n : xs(n) < |C n (u)| < (x + ε)s(n)) < δ. ( This last theorem is in some sense complementary to the result in an earlier paper [BC12], where we proved that, for any interval (a, b), the probability that |C (1) n |/s(n) ∈ (a, b) is bounded away from zero as n → ∞.

Preliminaries
First we need some more notation. For a cluster C n (u) we define its (left-right) diameter by diam(C n (u)) = max v,w∈Cn(u) For a box Λ n we define the spanning cluster by where L(Λ n ) = {−n} × [−n, n] ∩ Z 2 and R(Λ n ) = {n} × [−n, n] ∩ Z 2 . We use the notation A m,n for the annulus Λ n \Λ m and, for a vertex v ∈ Z 2 , the notation A m,n (v) for A m,n + v. In our proof of Theorem 1 and 2 we will use the following results from the literature, Theorems 3 -8 below. The first one is well known, see for example [BCKS01], [BK85]. [BK85]) There exist constants c 1 , c 2 , c 3 > 0, such that for all m ≤ n: As we already mentioned in the introduction, the largest clusters in Λ n are of order s(n). This is stated in the following result. and, In an earlier paper Borgs, Chayes, Kesten and Spencer showed exponential decay for the probability that there exists a cluster with large volume, but a small diameter: An easy consequence of Theorems 4 and 5 is the following.
Corollary 6 Let k ∈ N. For all δ > 0 there exist α > 0 and N ∈ N such that for all n ≥ N : In [J03] a version of Theorem 4 for the spanning cluster is given: In the proof of our main theorem we use the following inequality concerning the concentration function Q(X, λ) of a random variable X, which is defined by for λ > 0.
Let (X k ) k∈N be a sequence of independent random variables, and 0 <λ ≤ λ. Let a > 0 and let (b k ) k∈N be a sequence of real numbers such that, for all k ∈ N, There exists a universal constant where S m = X 1 + X 2 + · · · + X m .

Large clusters contain many good boxes
In the proof of our main theorem we need the following lemma, which is essentially already in [J03]. First some definitions. Recall the notation A m,n in the beginning of Section 2.. Let t ∈ 3N. (Later we will choose a suitable value for t). For any i, j ∈ Z we say that the box Λ t (2ti, 2tj) is 'good' if there is an open circuit in the annulus A 2 3 t,t (2ti, 2tj); in that case we denote the widest open circuit in that annulus by γ i,j . (Although γ i,j depends on t, we omit that parameter from the notation). For each vertex u we denote by G t (C n (u)) the set of good boxes in Λ n of which the corresponding γ i,j is contained in the open cluster of u. More precisely, Lemma 9 Let α > 0. For any δ, β > 0 there exist η > 0 and N ∈ N such that, for all n ≥ N and t ∈ (0, ηn) ∩ 3N Járai proved a somewhat stronger statement (see (3.15) in [J03]), but we only need this weaker statement and give a (short) proof.
Proof: The C i 's in this proof denote universal constants larger than 0. Their existence is important but their precise value does not matter for the proof. First note that we can cover the box Λ n by at most rectangles of width 1 4 αn and length 1 2 αn, such that every cluster with diameter at least αn crosses at least one of these rectangles in the easy direction. We consider one such rectangle, namely Q 0 := [0, 1 4 αn] × [0, 1 2 αn]. (The precise choice of Q 0 doesn't matter for our purpose). By RSW and the BK inequality we have that the probability that there are more than C 2 disjoint horizontal open crossings of Q 0 is less than Let R l denote the l-th lowest open crossing of Q 0 . We claim that there exist η ∈ (0, 1/2) and N ∈ N such that, for any n ≥ N , deterministic crossing r 0 of Q 0 , and t ∈ (0, ηn), where C n (r 0 ) denotes the open cluster which contains the crossing r 0 . From this claim we get (see (14) and (15)) that the l.h.s. of (13) is less than and the lemma follows. It remains to prove the claim concerning the inequality (16): The objects defined below involve a parameter i. We will always assume that i is such that the corresponding object is contained in [0, 1 4 αn] × [0, αn]. Consider all rectangles of the form A(i) := [2ti − t, 2ti + t] × [0, αn]. For every i we let j(i) be the smallest integer j for which the box Λ t (2ti, 2tj) is located above r 0 . Let E(i) be the event that Λ t (2ti, 2tj(i)) is good and γ i,j(i) is connected with r 0 inside A(i). The events E(i) are conditionally independent of each other (where we condition on the event R l = r 0 ), and, by RSW, each has probability larger than C 3 . Hence, when η is small enough (that is, n/t and thus the number of events E(i) is large enough), the probability that at most β of the E(i)'s occur is smaller than the r.h.s. of (16). This proves the claim and completes the proof of Lemma 9.
Remark: In one of the steps of Járai's proof (see the lines below our statement of Lemma 9), he shows that with large probability the l-th lowest crossing in Q 0 is contained in [0, 1 4 αn] × [0, 1 2 αn(1 − a)], for some constant a < 1. He used this to guarantee that the good boxes obtained are inside Q 0 . However, as the above arguments show, this (and hence the introduction of the extra constant a) is not needed in our argument.

Proof of Theorems 1 and 2 3.1 Gaps between sizes of clusters with large diameter
The following lemma will be used later to show that the conditions for Theorem 8 are satisfied in our situation. First we define, for each circuit γ, int(γ) as the interior of γ (that is, the bounded connected component of R 2 \ γ, where γ is seen as subset of the plane), and X γ := |{u ∈ int(γ) ∩ Z 2 : u ↔ γ}|.
Lemma 10 There exist universal constants χ, ξ > 0 and, for all t ∈ 3N and for any circuit γ in A 2 3 t,t , a value c(t, γ) ≥ 0 such that Proof: Fix some a ∈ (0, 1 2 ). Define the random variable Z = |{u ∈ A 1 3 t,t ∩ int(γ) : u ↔ γ}|. Let c(t, γ) be defined by By RSW, the probability that there is a closed dual circuit in A 1 3 t, 2 3 t is larger than some universal constant C 1 > 0. Moreover, if there is such a circuit, then X γ = Z. Hence, P(X γ ≤ c(t, γ)) is larger than or equal to the probability that there is such a circuit and that Z ≤ c(t, γ). By the above and FKG this is larger than C 1 a.
To prove (19) recall the notation (4) and define the random variable Y = |SC 1 3 t |. Theorem 7 implies that there exist constants C 2 , ξ > 0 such that, for all t, we have P(Y ≥ ξs(t)) > C 2 . Let E be the event that there is an open crossing in Λ 1 3 t from top to bottom and that this crossing is connected to γ. On E we have that X γ ≥ Z + Y , since the spanning cluster is connected to γ. By RSW, P(E) is larger than some universal constant C 3 . Hence P (X γ ≥ c(t, γ) + ξs(t)) ≥ P(E; Z ≥ c(t, γ); Y ≥ ξs(t)) ≥ C 3 (1 − a) C 2 , (20) where the last inequality uses FKG. This proves Lemma 10. Now we prove the following proposition, from which, as we show in the next subsection, Theorem 1 follows almost immediately. The set of clusters with diameter larger than αn is denoted by C α,n . More precisely, Proposition 11 For all α, δ > 0 there exist ε = ε(α, δ) > 0, N = N (α, δ) ∈ N such that, for all n ≥ N P(∃ distinct D 1 , D 2 ∈ C α,n : ||D 1 | − |D 2 || < εs(n)) < δ.
Proof: Let α, δ > 0 be given. By a standard RSW argument, the probability that |C α,n | ≥ 1 is smaller than some constant < 1 which depends only on α. Hence, by the BK inequality we can choose a κ = κ(α, δ) ∈ N such that, for all n: Let ξ and χ as in Lemma 10 and C as in Theorem 8. Take β so large that (For the time being, this property of β will play no role; it will become essential at (32) for a suitable choice of ε). Let η be as in Lemma 9 (but with δ/3 instead of δ in (13)). It is clear from that lemma that without loss of generality we may assume that η < α 2 .
For each n we take t = t(n) = 3⌊ 1 3 ηn⌋. Hence, by the above choice of η we have, for all sufficiently large n, Denote by W the event that there are at most κ clusters in Λ n with diameter at least αn and all these clusters have at least β good boxes. Note that the complement of W is the union of the event in the l.h.s. of (23) and the event in the l.h.s. of (26), and hence has probability smaller than 2δ/3. Therefore, to prove Proposition 11 it is sufficient to show that there exists ε > 0 such that for all sufficiently large n, We define (compare with (12)) G t,n = {(i, j) ∈ Z 2 : Λ t (2ti, 2tj) ⊂ Λ n is good }.
Recall that we denote the outermost open circuit in A 2 3 t,t (2ti, 2tj) (if it exists) by γ i,j . Denote the configuration on the edges in the set by ω H . To estimate the l.h.s. of (27) we condition first on the γ i,j 's and the configuration ω H . Therefore, letG be an arbitrary set of vertices (i, j) with Λ t (2ti, 2tj) ⊂ Λ n , and let, for each (i, j) ∈G,γ i,j be a (deterministic) circuit in A 2 3 t,t (2ti, 2tj). Let H be the analog of (28), with γ replaced byγ and letω be a configuration oñ H. We will consider the conditional distribution P(·|G t,n =G; γ i,j =γ i,j ∀(i, j) ∈ G t,n ; ω H =ω). Note that the information we condition on allows us to distinguish all the clusters in C α,n and their good boxes. (Here we used that (25) implies that no cluster of C α,n fits entirely in the interior of one of the above mentioned γ i,j 's). We may assume thatω is such that W holds. Let D 1 , D 2 be two open clusters in C α,n for the configurationω. Their sizes can be decomposed as follows: where a 1 = |D 1 ∩ H| and a 2 = |D 2 ∩ H|, and the X variables are as defined in (17). The terms a 1 and a 2 can be considered as 'fixed' (namely, determined byω), and the Xγ i,j 's as independent random variables. Therefore, and because there are at most κ 2 choices for D 1 and D 2 , to prove (27) it is enough to show that there exists ε > 0, which does not depend on a 1 , a 2 , G t (D 1 ), G t (D 2 ) and theγ i,j 's, such that On the event W we have that |G t (D 1 )| ≥ β. So we can mark β of the good boxes in G t (D 1 ), and condition (in addition to the earlier mentioned information) also on the values of X γ i,j for the remaining good boxes in G t (D 1 ) and all the good boxes in G t (D 2 ). Hence it is enough to show that there exists an ε > 0 such that uniformly in b ∈ N, γ 1 , · · · , γ β and G t (D 1 ), where the γ m 's are circuits in distinct annuli A 2 3 t,t (2ti, 2tj). We will do this by application of Theorem 8, where Lemma 10 (and our choice (24) for β) enables a suitable application of that theorem: From (24) it follows immediately that for all n there is an ε(n) such that By the lower bound in Theorem 3, the l.h.s. of (32) is bounded away from 0, uniformly in n. Hence, inf n ε(n) > 0. Take ε equal to this infimum. We get (with Q as in (11), where in the last inequality we used Lemma 10 and applied Theorem 8 (withλ = ξs(t), a = χ, m = β and λ = 2ε(n)s(n)). Note that the conditionλ ≤ λ in that theorem is satisfied because ξs(t) ≤ 2ε(n)s(n) by the first inequality in (32). Now, by the second inequality of (32) we have that the r.h.s. of (33) is at most δ . This shows (31) and completes the proof of Proposition 11.
Remark In the case of site percolation on the triangular lattice we can, in equation (32) and the line above it, skip the introduction of ε(n), and choose ε itself such that it is (for all sufficiently large n) between the l.h.s. and r.h.s. of (32). For that percolation model such ε exists because (see [GPS13], Proposition 4.9 and the last part of the proof of Theorem 5.1 in that paper) π(t)/π(n), and hence s(t)/s(n), has a limit as n → ∞ (with t/n fixed).
Further, by Proposition 11 there is an ε > 0 such that the probability that there are two clusters with diameter larger than αn of which the sizes differ less than εs(n) is smaller than δ/2. Hence the l.h.s. of (2) is less than δ/2 + δ/2.
This can be proved in practically the same way as Proposition 11. (And, in fact, a bit easier, because now we deal with single clusters instead of pairs of clusters. In particular the factor κ 2 is replaced by κ in the proof.)