Large deviation bounds for the volume of the largest cluster in 2D critical percolation

Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n)>x of the form exp(- C x^(2/alpha)) for x>1 and large n with alpha = 5/48 and C>0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d>2.


Introduction and statement of the main results
For a general introduction to the percolation model we refer to [Kes82], [Gri99], and [BR06]. Consider the critical bond percolation model on the lattice Z d for d ≥ 2. For n ∈ N let Λ n := {−n, −n + 1, . . . , n} d denote the hypercube (ball) centred at the origin with radius n. For v ∈ V (T) we write Λ n (v) := v + Λ n . Further let ∂A denote the (outer) boundary of A ⊆ Z d , that is ∂A := v ∈ Z d \ A : ∃u ∈ A such that u ∼ v .
We say that two sites v Herein the size of a cluster is measured by its number of vertices. Further, let C probability P pc (∂Λ m ↔ ∂Λ n ). We set π(n) := π(1, n). We will work under the following assumptions.
Assumption (I) (Quasi-multiplicativity). There exists a constant C 1 such that for all 0 ≤ k ≤ l ≤ m we have π(k, l)π(l, m) ≤ C 1 π(k, m). (1) Assumption (II). There exist constants C 2 > 0 and α < d such that for all n ≥ m ≥ 1 Assumption (I) and (II) hold for d = 2, as proved in [Gri99] and [Nol08]. Furthermore, Assumption (II) holds in high (d ≥ 19) dimensions, however, we do not expect Assumption (I) to hold in this case. See Remark vii) below for more details on this case. To our knowledge, it is an open question whether any of Assumption (I) or (II) is satisfied in dimensions 3 ≤ d ≤ 18.
In [BCKS99] the following bound was given: Theorem 1 (Proposition 6.3 of [BCKS99]). Suppose that Assumption (II) holds. Then there exist positive constants c 1 , c 2 such that for all x, n ≥ 0, We strengthen this result when both of Assumption (I) and (II) are satisfied: Theorem 2. Let d ≥ 2, and suppose that Assumptions (I) and (II) hold. There exist positive constants c 1 , c 2 depending only on d and the constants appearing in the assumptions, such that for all n, u > 1, Furthermore, for d = 2 there are constants c 3 , c 4 > 0 such that the lower bound The lower bound in Theorem 2 follows from standard RSW methods, nevertheless, for completeness we include its proof in Section 3.2. The upper bound above relies on Theorem 3 below, which is our main contribution. Let denote the set of vertices in Λ n which are connected to ∂Λ 2n .
Theorem 3. Let d ≥ 2, and suppose that Assumptions (I) and (II) hold. There is a constant c 1 such that for all n, u > 0 Consequently, for some positive constants c 2 , c 3 , we have The constants c 1 , c 2 , c 3 above only depend on d and the constants appearing in Assumptions (I) and (II).
A weaker version of Theorem 3 is proved in [BCKS99] as Lemma 6.1. Theorem 2 follows from Theorem 3 by arguments analogous to those in [BCKS99] which lead from [BCKS99, Lemma 6.1] to Theorem 1. Thus we only prove Theorem 3 and the lower bound in Theorem 2 here.
Remarks. i) Our motivation for studying the size of large critical clusters comes from the forest-fire processes described as follows. Let λ be some small positive number. At time 0 all the vertices of Z d are empty. As time goes on, empty vertices get occupied by a tree at rate 1, independently from each other. Vertices with trees get struck by lightning at rate λ independently from each other. When a tree gets struck by lightning, its forest (its connected component in Z d of vertices with trees) is ignited, that is, all of the trees are removed in this forest. Then trees occupy empty vertices with rate 1, and lightnings strike and so on. We are particularly interested in the case where λ > 0 is small. As we can see, a forest burns down at rate proportional to its size, thus a precise control of the size of critical clusters can be useful for the study of the processes above. ii) [BCKS99, Proposition 6.3] also treats the case where the percolation parameter p is different from p c . Our results extend to this case in an analogous way as in [BCKS99]. Furthermore, Assumptions (I) and (II), our results, as well as those in [BCKS99], in the case d = 2 hold for site/bond percolation on other lattices: As long as the lattice is invariant under a translation, a rotation around the origin with some angle and a reflection on one of the coordinate axes, the results above follow. Furthermore, these results remain valid for some inhomogeneous percolation models. See [Gri99] for more details. iii) The proof of Theorem 3 relies on the method presented in [KMS13]. However, the computation there only considers the case d = 2. As we will see below, the arguments in [KMS13] extend to the case d ≥ 3 in a straightforward way. Furthermore, by the results in [DCST13] the arguments in [KMS13] can also be adapted for the critical two dimensional FK percolation model with q ≥ 1. Hence statements analogous to those in Theorem 2 and 3 remain valid in such context. iv) Recall a ratio limit theorem, Proposition 4.9 of [GPS13] for the one arm events.
Combining it with Theorem 2 we get, for site percolation on the triangular lattice, with some universal constants c i for all x > 0 and n ≥ n 0 (x). v) The upper bound in Theorem 2 trivially extends to |C (l) n | the volume of the lth largest cluster. Furthermore, in dimension 2 the same lower bound with different constants also holds. Its derivation is analogous to that for the largest cluster, hence we omit it. vi) Theorem 3 gives upper bounds on the moments and the tail probability of V n /n 2 π(n), where, roughly speaking, V n counts the points in Λ n with one long open arm. Similar upper bounds can be achieved for the number of points with multiple disjoint arms. Let k ∈ N and σ ∈ {0, 1} k . Let π σ (m, n) denote the probability that ∂Λ m and ∂Λ n are connected by k disjoint arms, where in a counter-clockwise order of these arms the ith arm is open when σ i = 1 and dual closed otherwise. Suppose that Assumption (I) and (II) are satisfied when π is replaced by π σ with some constants C 1 , C 2 and for some α σ > 0 not necessarily smaller than d. We have two cases: when α σ < d, we get results analogous to Theorem 3. However, when α σ > d, by checking the computations in the proof of Theorem 3, one gets for some constant c 1 where V σ n denotes the multi-arm analogue of V n . A lower bound analogous to that in the second part of Theorem 2 hold in two dimensions when σ switches colours at most four times and α σ < 2. However, in this case the construction in the lower bound is more delicate, but we can apply the strong separation lemma [DS11, Lemma 6.2 and 6.3] to deduce the required lower bound. vii) Let us turn to the case d ≥ 19. Kozma and Nachmias [KN11,Theorem 1] proved that π(n) = O(n −2 ) building on the results in [Har08]. This combined with [Aiz97, Theorem 5] gives that |C (1) n | is of order n 4+o(1) . Hence the bounds in Theorem 1 and 2 are much weaker than those in [Aiz97, Theorem 5]. Nevertheless, we get some new conditional results which are interesting in dimensions below 19. viii) We note some results on the distribution of |C (l) n | for l ≥ 1. We already mentioned the results of [BCKS99] which are the most relevant for our purposes. The same authors in [BCKS01] describe the connection between the volume and the diameter of the largest critical and near-critical clusters. Járai [Jár03] showed, among other things, that the microscopic scale behaviour of the largest critical clusters can be described by that of the incipient infinite cluster. Finally, van den Berg and Conijn [vdBC12] proved that the probability of |C (1) n |/n 2 π(n) ∈ (a, b) is positive for all 0 < a < b for sufficiently large n. While in [vdBC13] they showed, roughly speaking, that the distribution of |C (1) n |/n 2 π(n) has no atoms for large n and that |C (l) n | − |C (l+1) n | = O(n 2 π(n)) for l ≥ 1.
Organization of the paper. In Section 2 we provide some more notation. We sketch the arguments of [KMS13] which are essential for the proofs of our results in Section 2.1. Building on these results, we prove Theorem 3 in Section 3.1. We conclude in Section 3.2 where we deduce the lower bound in Theorem 2.
Acknowledgements. The author thanks René Conijn and Rob van den Berg for fruitful discussions and for suggesting the problem. He is grateful to Markus Heydenreich for his advice on percolation in high dimensions.
2.1. The counting argument of [KMS13]. The proof of Theorem 3 is based on a counting argument found in [KMS13]. There the argument there is a strengthens the proof of [BCKS99, Lemma 6.1] and it is used to count certain passage points, which, roughly speaking, are the starting points six disjoint open and closed arms. Herein we give a sketch of the argument in the one arm case.
We give a bound on the probability of the event {V n ⊇ X}, but first some definitions. Let T 0 denote the empty graph on the vertex set X. Let us start blowing a ball at each point of X at unit speed. That is, at time t ≥ 0, we have the balls Λ t (x), x ∈ X.
For small values of t these balls are pairwise disjoint. As t increases, more and more of these balls intersect each other. Let r 1 , denote the smallest t when the first pair of balls touch. We pick one such pair balls in some deterministic way, with centres u 1 , v 1 ∈ X. We draw an edge e 1 between u 1 and v 1 and label it with l(e 1 ) := r 1 , and get the graph T 1 . Note that ||u 1 − v 1 || ∞ = 2r 1 . Then we continue with the growth process, and stop at time r 2 if we find a pair of vertices u 2 , v 2 ∈ X such that u 2 , v 2 are in different connected components of T 1 and Λ r 2 (u 2 ) and Λ r 2 (v 2 ) touch. Then we draw an edge e 2 between one such deterministically chosen pair with the label l(e 2 ) := r 2 and get T 2 . Note that it can happen that r 1 = r 2 . We continue with this procedure till we arrive to the tree T k−1 . Let R(X) denote the multiset containing r i for i = 1, 2, . . . , k − 1.
As we saw above, r 1 = 1 2 min u,v∈X,u =v ||u − v||. Furthermore, it is easy to see that for i = 1, 2, . . . , k −1 there are at least k +1−i vertices of X such that any pair of them is at least 2r i distance from other. This combined with the pigeon-hole principle provides the following observation: denote the boundary of the inner and outer faces of the sets G(B), respectively. Now we are ready to make a bound on the probability P(V n ⊇ X). Recall the definition of V n from (6). For all x ∈ V (B) we have The events {ib(B) ↔ ob(B)} are independent since they depend only on the state of the edges in G(B), which are pairwise disjoint subsets of Λ 2n . Hence Then, as in the proof of [KMS13,Proposition 14], an induction on the blobs leads to the following bound.
Proposition 5. Suppose that Assumption (I) and (II) holds. Then there is a constant C 3 = C 3 (c 1 , c 2 , α, d) such that for all X ⊆ Λ n Proposition 5 provides an upper bound on P pc (V n ⊇ X) as a function of R(X). To give a bound on E pc |Vn| k , we bound the number of sets X such that R(X) = R for fixed R. By arguments analogous to the proof of [KMS13, Proposition 15] we get the following.
Proposition 6. There is a universal constant C 4 such that for all multisets R with k − 1 elements we have where O(R) denotes the number of different ways the elements of R can be ordered.

Proof of Theorem 2 and 3
We start with the following consequence of Assumption (II).
Letr i denote the ith largest element ofR. Observation 4 provides an upper bound on E |Vn| k where in the sum in (12) we restrict to the terms such that r i ≤ n/2 l for all i with 2 dl ≤ i < 2 d(l+1) . We indicate this restriction by an additional tilde above the sum. Let j := log 2 d (k) and m = k − 1 − 2 dj . We arrive to the following bound: The multinomial term in (13) bounds the number of ways we can order k − 1 (not necessarily different) numbers when we do not distinguish between the largest 2 d − 1, the next (2 d − 1)2 d largest,..., and the next (2 d − 1)" d(j−1) largest of them. The product terms in (13) apply the above bounds on the range ofr i . Hence by Lemma 7, we have that We estimate the multinomial, and the two product terms separately. It is a simple computation to show that there is a constant C 7 = C 7 (d) such that and that for all k ≥ 1. We combine (14), (15), and (16) with the trivial bound π(n/ d √ k) k for the product of π's, and get with C 8 = C 5 C 6 C 2 7 . This finishes the proof of the first part of Theorem 3. Let us proceed to the proof of the second part. The statement is trivial for u > n, hence we assume u ∈ [1, n] in the following. For t ≥ 1 by (17) we get for some constant C 9 = C 9 (α, d). Note that the function x → (1 + x) 1/x is decreasing, and that u d Hence there is a constant C 10 such that for all K > 0 t Kn d π(n/u) = 1 + u d C 2 C 8 n d π(n/u) Then the Markov inequality, (18) and (19) with K = 2/(C 2 C 10 ) gives that  V(B(n, m)) that there is a vertical crossing of B(n, m). The following well-known statement fist appeared in [SW78], see also [Rus81].
Lemma 8 (RSW). There is a positive constant C 11 > 0 such that for all n ≥ 1 P pc (H(B(n, 2n))) ≥ e −C 11 . We say that an event A is increasing, if ω ∈ A then ω ∈ A for all ω ∈ Ω with ω ≥ ω, where ≥ is understood coordinate-wise. We recall the FKG -inequality [FGK71]: Lemma 9. (FKG) Let A, B be increasing events, then P pc (A ∩ B) ≥ P pc (A)P pc (B).
We start with the following lemma.
Proof of Lemma 10. Simple computation gives that E pc (|V n |) ≥ n 2 π(3n) ≥ C 2 3 −α n 2 π(n). This combined with Theorem 3 provides the desired constants C 12 and C 13 . Now we proceed to the proof of the lower bound in Theorem 2.
Proof of the lower bound in Theorem 2. For v ∈ Z 2 , we set B(v; n, m) := B(n, m)+ v, and Note that it is enough to prove (5) when u is an integer in [2, n]. We set n = n/u . Let D n (u) denote the event It is easy to check that on the event D n (u), all the vertices w ∈ Λ n−n with w ↔ ∂Λ 2n (w) belong to the same cluster. In particular, on D n (u) we have v∈Λ u−1 V n n v ≤ |C (1) n |.
Above we used Lemma 10 in (23) and in (24). Simple application of Assumption (II) finishes the proof of the lower bound of Theorem 2.