Kesten's incipient infinite cluster and quasi-multiplicativity of crossing probabilities

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten's incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb Z^2\times\{0,\ldots,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption is fulfilled for $G=\mathbb Z^d$ if and only if $d<6$.


Introduction
Let G be an infinite connected bounded degrees graph with a vertex set V . Let ρ be the graph metric on V , and define for v ∈ V and positive integers m For x, y ∈ V and X, Y, Z ⊂ V , we write x ↔ y in Z if there is a nearest neighbor path of open edges such that all its vertices are in Z, X ↔ Y in Z if there exist x ∈ X and y ∈ Y such that x ↔ y in Z, and x ↔ Y in Z, if there exist y ∈ Y such that x ↔ y in Z. If Z = V , we omit "in Z" from the notation. We use instead of ↔ to denote complements of the respective events.
In this note we are interested in the existence and equality of the limits where E is a cylinder event. The question is highly non-trivial if P pc [|C(w)| = ∞] = 0. The seminal result of Kesten [16,Theorem (3)] states that if G is from a class of two dimensional graphs, such as Z 2 , then the above two limits exist and have the same value ν G,w (E). By Kolmogorov's extension theorem, ν G,w extends uniquely to a probability measure on configurations of edges, which is often called Kesten's incipient infinite cluster measure. It is immediate that ν G,w [|C(w)| = ∞] = 1. Kesten's argument is based on the existence of an infinite collection of open circuits around w in disjoint annuli and the properties that (a) each path from w to infinity intersects every such circuit and (b) by conditioning on the innermost open circuit in an annulus, the occupancy configuration in the region not surrounded by the circuit is still an independent Bernoulli percolation. These properties are no longer valid when one considers higher dimensional lattices. In fact, the existence of Kesten's IIC on Z d for d ≥ 3 is still an open problem. A partial progress has been recently made in sufficiently high dimensions by Heydenreich, van der Hofstad and Hulshof [13,Theorem 1.2], who showed using lace expansions the existence of the first limit in (1.1) under the assumption that n −2 P pc [0 ←→ S(0, n)] converges. Concerning low dimensional lattices, almost nothing is known there about critical and near critical percolation, and the existence of Kesten's IIC seems particularly hard to show. Several other constructions of incipient infinite clusters are obtained by Járai [15] for planar lattices and van der Hofstad and Járai [14] for high dimensional lattices.
The main result of this note is the existence and the equality of the two limits in (1.1) for graphs satisfying two assumptions: (A1) uniqueness of the infinite open cluster and (A2) quasimultiplicativity of crossing probabilities. While (A1) is satisfied by many amenable graphs, most notably Z d , (A2) can be expected only in low dimensional graphs. For instance, we argue below that (A2) holds for Z d if and only if d < 6. In our second result, we prove that (A2) is satisfied by slabs Z 2 × {0, . . . , k} d−2 (d ≥ 2, k ≥ 0), thus showing for these graphs the existence and equality of the limits in (1.1). We now state the assumptions and the main result, and then comment more on the assumptions. (A2) (Quasi-multiplicativity of crossing probabilities) Let v ∈ V and δ > 0. There exists c * > 0 such that for any p ∈ [p c , p c + δ], integer m > 0, a finite connected set Z ⊂ V such that Z ⊇ A(v, m, 4m), and sets X ⊂ Z ∩ B(v, m) and Y ⊂ Z \ B(v, 4m), (1.2) Theorem 1.1. Assume that the graph G satisfies the assumptions (A1) and (A2) for some choice of v ∈ V and δ > 0. Then, for any cylinder event E, the two limits in (1.1) exist and have the same value. If the assumptions (A1) and (A2) are satisfied at p = p c , then the first limit in (1.1) exists.
Before we discuss the strategy of the proof, let us comment on the assumptions.
Comments on (A1): The equivalence of the claims (1.3) and (1.4) follows from the inequalities where the second one is a consequence of the BK inequality.
It is elementary to see that (1.3) implies (A1). On the other hand, if (1.3) does not hold, then there exist v 0 ∈ V , ε 0 > 0 and m 0 ∈ N such that for all n > 4m 0 , and monotone decreasing in n. Thus, there exists p 0 ∈ [0, 1] such that P p 0 [E 2 (v 0 , m 0 , n)] ≥ ε 0 for all n > 4m 0 . By passing to the limit as n → ∞, we conclude that for p = p 0 , with positive probability there exist at least two infinite open clusters and (A1) does not hold.
Comments on (A2): 4. It follows from the Russo-Seymour-Welsh Theorem [19,21] that (A2) holds for two dimensional graphs, such as Z 2 , considered by Kesten in [16]. Russo-Seymour-Welsh ideas have been recently extended to slabs in [18,3], after the absence of percolation at criticality in slabs was proved by Duminil-Copin, Sidoravicius and Tassion [9]. In Lemma 3.2 of the present paper we prove that (A2) is fulfilled by slabs Z 2 × {0, . . . , k} d−2 (d ≥ 2, k ≥ 0), thus verifying the existence and equality of the limits (1.1) for slabs. 5. We believe that assumption (A2) holds for lattices Z d if d < 6, but does not hold if d > 6. Dimension d c = 6 is called the upper critical dimension above which the percolation phase transition should be described by mean-field theory, see, e.g., [7]. This was rigorously confirmed in sufficiently high dimensions by Hara and Slade [12,11].
It is easy to see that the mean-field behavior excludes (A2). Indeed, it is believed that above d c , the two point function decays as [11] proved it rigorously in sufficiently high dimensions. Given this asymptotics, Aizenman showed in [1,Theorem 4(2)] that for all m(n) ≤ n such that m(n) and Kozma and Nachmias [17] that P pc [0 ↔ S(0, n)] ≍ n −2 . Thus, the inequality cannot hold for large n.
The situation below d c is much more subtle. With the exception of d = 2, where planarity helps enormously, the (near-)critical behavior below d c is widely unknown. Let us nevertheless give a few words about why we think (A2) should hold below d c . It is believed that the number of clusters crossing any annulus A(0, m, 2m) is bounded uniformly in m if d < d c and grows at p = p c like m d−6 above d c , with log-correction for d = d c , and this dichotomy is intimately linked to the transition at d c from the hyperscaling to the mean-field; see [6,5]. Thus, it would be not unreasonable to expect that below d c , which is enough to establish (A2). We are not able to prove it yet or give a simpler sufficient condition for it. It would already be very nice if, for instance, (A2) was derived from the assumption that P p [∃! crossing cluster of A(0, m, 2m)] ≥ c or from the assumptions of [5].
We finish the introduction with a brief description of the proof of Theorem 1.1. Our proof follows the general scheme proposed by Kesten in [16] by attempting to decouple the configuration near w from infinity on multiple scales. The implementations are however rather different. Using (1.4) we identify a sufficiently fast growing sequence N i such that given w ↔ S(w, n), the probability that the annulus A(v, N i , N i+1 ) ⊂ B(w, n) contains a unique crossing cluster is asymptotically close to 1; see (2.2). Next, let an annulus A(v, N i , N i+1 ) contain a unique crossing cluster. We explore all the open clusters in this annulus that intersect the interior boundary S(v, N i ), call their union C i , and let D i be the subset of S(v, N i+1 + 1) of vertices connected by an open edge to C i ; see (2.3). Then, the configuration outside C i is distributed as the original independent percolation and every vertex from D i is connected by an edge to the same (crossing) cluster from C i . Thus, w ↔ S(w, n) if and only if (a) w is connected to D i (this event only depends on the edges intersecting S(v, N i ) ∪ C i ) and (b) D i is connected to S(w, n) outside C i (this only depends on the edges outside C i ). This allows to factorize P p [E, w ↔ S(w, n)]; see (2.4). The rest of the proof is essentially the same as that of Kesten [16]. We repeat the described factorization on several scales, obtaining in (2.6) an approximation of P p [E|w ↔ S(w, n)] in terms of products of positive matrices. Finally, we use (A2) to prove that the matrix operators are uniformly contracting, which is enough to conclude the proof; see (2.7) and the text below.
2 Proof of Theorem 1.1 We will prove the first claim of the theorem. The proof of the second one follows from the proof below by replacing everywhere p by p c . The general outline of the proof is the same as the original one of Kesten [16,Theorem (3)], but the choice of scales and the decoupling are done differently.
First of all, it suffices to prove that for any w ∈ V and a cylinder event E, Indeed, (2.1) implies the existence of the first limit in (1.1) and that ν p (E) is continuous. Since for any p > p c , , the existence of the second limit in (1.1) and its equality to the first one follows from the continuity of ν p (E). Actually, by the inclusion-exclusion formula, it suffices to prove (2.1) for all events E of the form {edges e 1 , . . . , e k are open}. Although our proof could be implemented for any cylinder event E, calculations are neater for increasing events.
Fix w ∈ V and an increasing event E. Also fix v ∈ V and δ > 0 for which the assumption (A2) is satisfied. Consider a sequence of scales N i such that N i+1 > 4N i for all i, B(v, N 0 ) contains w and the states of its edges determine E. We will write By (1.4), we can choose the scales N i so that ε i → 0 as i → ∞.
We first note that for n > N i+1 + N 0 , where c * is the constant in the assumption (A2). Indeed, by independence, where the last inequality follows from the assumption (A2).
We begin to describe the main decomposition step. Consider the random sets , and for all n > N i+1 + N 0 , Together with (2.2), this gives the inequality where the last step follows from the FKG inequality, since E is increasing. Define the constant C * = (c 2 * P pc [E]) −1 and for (U, R) ∈ Π i , let In this notation, (2.4) becomes and by replacing E above with the sure event, we also get Now we iterate. Let (U, R) ∈ Π i . We can apply a similar reasoning as in (2.2) and (2.4) to γ p (U, R, n) and obtain that for any j > i + 2 and n > N j+1 + N 0 ,

Then (2.5) becomes
Iterating further gives that for any ε > 0 and s ∈ N, there exist indices i 1 , . . . , i s such that i k+1 > i k + 2 and for all n > N is+1 + N 0 , where the two sums are over (U 1 , R 1 ) ∈ Π i 1 , . . . , (U s , R s ) ∈ Π is . We will prove that (A2) implies that there exists κ such that for all i, j > i + 2, all pairs (2.8) It follows from (2.8) and the fact that ξ ≤ 1 that for any m, n > N is+1 + N 0 and p ∈ [p c , p c + δ], which implies (2.1).
It remains to prove (2.7). Let j > i + 2. Consider the random sets : ∃ x ∈ X j , a neighbor of y, such that the edge x, y is open} .
Note that X j contains S j , the event {X j = X} depends only on the states of edges in A j−1 with at least one end-vertex in X, and either and let Γ j be the collection of all such pairs (X, By the assumption (A2), This easily implies (2.7) with κ = c −1 * . The proof of Theorem 1.1 is complete.
Remark 2.1. Instead of conditioning on the events {w ↔ S(w, n)}, one could condition on {w ↔ Y n in Z n }, where Z n ⊃ B(w, n) and Y n ⊆ Z n \ B(w, n), and obtain the same limits as in (1.1). This is immediate after observing that P p [E|w ↔ Y n in Z n ] satisfies inequalities (2.8) with the same ξ.

Quasi-multiplicativity for slabs
In this seciton we prove that the assumption (A2) is fulfilled by slabs Z 2 × {0, . . . , k} d−2 for any d ≥ 2 and k ≥ 0 and for any δ > 0 such that p c + δ < 1, thus proving , and An(m, n) = Q(n) \ Q(m − 1) the annulus of side lengths 2m and 2n. We will prove the following lemma.
Proof of Lemma 3.2. Instead of (3.1), it suffices to prove that there exists c > 0 such that for any m > 0, any finite connected Z ⊂ S such that Z ⊇ An(2m, 3m), and any X ⊂ Z ∩ Q(2m) and Y ⊂ Z \ Q(3m), Indeed, for Z as in the statement of the lemma, by (3.2), and P p [∂Q( 4 3 m) ↔ ∂Q(3m) in Z] ≥ P pc [∂Q( 4 3 m) ↔ ∂Q(3m)] ≥ c > 0, as proved in [3,18]. We proceed to prove (3.2). Let E be the event that there exists an open circuit (nearest neighbor path with the same start and end points) around Q(2m) contained in An(2m, 3m). It is shown in [18] that P p [E] ≥ P pc [E] > c > 0 for some c > 0 independent of m. Thus, by the FKG inequality, Consider an arbitrary deterministic ordering of all circuits in S, and for a configuration in E, let Γ be the minimal (with respect to this ordering) open circuit around Q(2m) contained in An(2m, 3m). For W ⊂ S, let Note that Thus, to prove (3.2), it suffices to show that for some C < ∞, This will be achieved using local modification arguments similar to those in [18]. In fact, for the above inequality to hold, it suffices to show that for some C < ∞, We write the event in the left hand side of (3.3) as the union of three subevents satisfying additionally It suffices to prove that the probability of each of the three subevents can be bounded from above by C · P p [X ↔ Y in Z]. The cases (b) and (c) can be handled similarly, thus we only consider (a) and (b).

Case (a):
We prove that for some C < ∞, Denote by E a the event on the left hand side. It suffices to construct a map f : E a → {X ↔ Y in Z} such that for some constant D < ∞, (1) for each ω ∈ E a , ω and f (ω) differ in at most D edges, (2) at most D ω's can be mapped to the same configuration, i.e., for each ω ∈ E a , |{ω ′ ∈ E a : f (ω ′ ) = f (ω)}| ≤ D. If so, the desired inequality is satisfied with C = D min(pc,1−pc−δ)) D . Take a configuration ω ∈ E a . Let U be the set of all points u ∈ Γ such that u is connected to X in Z by an open self-avoiding path that from the first step on does not visit {u}. For each u ∈ U , choose one such open self-avoiding path and denote it by π u . Similarly, let V be the set of all points v ∈ Γ such that v is connected to Y in Z by an open self-avoiding path that from the first step on does not visit {v}. For each v ∈ V , choose one such open self-avoiding path and denote it by π v .
Assume first that we can choose u ∈ U and v ∈ V such that {u} = {v}. For such ω's, the configuration f (ω) is defined as follows. We The constructed function f satisfies the requirement (1) with D = 4d (k + 1)k d−2 and the requirement (2) with D = 2 4d (k+1) d−2 . The proof of (3.4) is complete.
Case (b): We prove that for some C < ∞, Denote by E b the event on the left hand side. As in Case (a), (3.5) will follow if we construct a map f : , ω and f (ω) differ in at most D edges, (2) at most D ω's are mapped to the same configuration. Take a configuration ω ∈ E b . Let U be the set of all points u ∈ Γ such that u is connected to X in Z by an open self-avoiding path that from the first step on does not visit {u}. For each u ∈ U , choose one such open self-avoiding path and denote it by π u .
We first assume that there exists u ∈ U such that Y is connected to Γ in Z \ {u}. For such ω's, we define f (ω) as follows. We Assume next that for any u ∈ U , Y is not connected to Γ in Z \ {u}. Take u ∈ U . There exists v ∈ {u} such that v is connected to Y in Z by an open self-avoiding path that from the first step on does not visit {v}. Choose one such open self-avoiding path and denote it by π v . For such ω's, we define f (ω) exactly as in the first part of Case (a). We Notice that unlike in Case (a), it is allowed here that v ∈ Γ, but this makes no difference for the construction. Indeed, after closing edges as in (a), Y remains connected to Γ only if v ∈ Γ. Thus, after modifying ω according to The function f satisfies requirements (1) and (2), and the proof of (3.5) is complete.
Since the proof of Case (c) is essentially the same as the proof of Case (b), we omit it. Cases (a)-(c) imply (3.3). The proof of Lemma 3.2 is complete. Remark 3.3. (1) Theorem 3.1 and Remark 2.1 can be used to extend various results of Járai [15] to slabs. For instance, to prove that the local limit of the occupancy configurations around vertices in the bulk of a crossing cluster of large box are given by the IIC measures from Theorem 3.1. This will be detailed in [2].
(2) Using Lemma 3.2, one can show that the expected number of vertices of the IIC in Q(n) is comparable to n 2 P[0 ↔ ∂Q(n)].
(3) In [8], the so-called multiple-armed IIC measures were introduced for planar lattices, which are supported on configurations with several disjoint infinite open clusters meeting in a neighborhood of the origin. These measures describe the local occupancy configurations around outlets of the invasion percolation [8] and pivotals for open crossings of large boxes [2]. It would be interesting to construct multiple-armed IIC measures on slabs, but at the moment it seems quite difficult.