Slab percolation for the Ising model revisited

In this note, we give a new and short proof for a theorem of Bodineau stating that the slab percolation threshold $\hat{p}_c$ for the FK-Ising model coincides with the standard percolation critical point $p_c$ in all dimensions $d\geq3$. Both proofs rely on the positivity of the surface tension for $p>p_c$ proved by Lebowitz&Pfister. The key difference is that while Bodineau's proof is based on a delicate dynamic renormalization inspired by the work of Barsky, Grimmett&Newman, our proof utilizes a technique of Benjamini&Tassion to prove the uniqueness of macroscopic clusters via sprinkling, which then implies percolation on slabs through a rather straightforward static renormalization.


Introduction
We study the supercritical phase of the FK (also known as random-cluster) model with cluster weight q ≥ 1 on Z d , d ≥ 3.This class of percolation models was introduced by Fortuin & Kasteleyn [FK72] and is intimately linked to the q-state Potts model for integers q ≥ 2 -see [Gri06,Dum17] for an account on these models and their connections.For q = 2 this model is sometimes called the FK-Ising model.
We define the slab percolation threshold pc = pc (q, d) := inf p ∈ [0, 1] : ∃L ≥ 0 such that inf It is clear that pc ≥ p c .The slab percolation threshold was introduced in the seminal work of Pisztora [Pis96], where a powerful coarse graining technique was developed to describe the behavior of the model assumption that p > pc .This technique has then found multiple applications in the study of fundamental features of supercritical percolation, such as the Wulff crystal construction [Bod99, CP00, CP01], the structure of translation invariant Gibbs measures for the Ising model [Bod06], the exponential decay of truncated Ising correlations [DGR20] and the existence of long range order for the random the field Ising model [DLX22], to cite just a few.It is conjectured that pc (q, d) = p c (q, d) for all q ≥ 1 and d ≥ 3, which implies that Pisztora's coarse graining and its consequences are valid up to the critical point.This has only been proved for q = 1 (corresponding to Bernoulli percolation) by Grimmett & Marstrand [GM90] and for q = 2 (corresponding to the Ising model) by Bodineau [Bod05] -see however [DT19] for a weaker result for integer q.In this note, we give a new proof for the Ising case.
where Λ N := {−N, . . ., N } d .We can then consider the associated critical point It is easy to see that pc ≥ p c .Lebowitz & Pfister [LP81] proved that in the Ising case q = 2 one has τ p > 0 for all p > p c , where τ p is defined similarly but with wired boundary conditions on R(L, δL) instead -see Appendix A. Using a weak mixing property, one can compare wired and free boundary conditions, thus leading to the following.
For the sake of completeness, we include in Appendix A the proof of positivity of the (wired) surface tension from [LP81] along with the comparison between boundary conditions leading to Theorem 1.2.The latter follows the same lines as [Bod05, Theorem 3.1], with a slight simplification due to the fact that in our definition of pc the (free) boundary conditions are allowed to be at a macroscopic distance from the support of the relevant (dis)connection event, which is not the case in [Bod05].
Theorem 1.1 then follows readily from Theorem 1.2 and the following result, which concerns all FK models with q ≥ 1.
Theorem 1.3.One has pc (q, d) = pc (q, d) for all q ≥ 1 and d ≥ 3. Theorem 1.3 is similar to [Bod05, Theorem 2.2], but our proof is completely different.We also stress that, due to the aforementioned difference between our definition of pc and that of [Bod05], our result is slightly stronger.Indeed, the proof of [Bod05, Thorem 2.2] relies on a delicate dynamic renormalization scheme inspired by the work of Barsky, Grimmett & Newman [BGN91] on Bernoulli percolation in the half space, which requires connections to go up to the boundary of the domain (with free boundary conditions).Our approach on the other hand is based on static renormalization, for which a macroscopic distance from the boundary is typically harmless.
Our proof of Theorem 1.3 goes as follows.First, we observe that the surface order exponential cost for disconnection given by the condition p > pc implies that the clusters in Λ L touching ∂Λ L are ℓ-dense in Λ L with high probability for ℓ = C(log L) Then we adapt a technique of Benjamini & Tassion [BT17] to prove that all the clusters crossing an annulus get connected to each other with high probability after an ε-Bernoulli sprinkling -see also [DS23] for a very similar use of this technique for Voronoi percolation and [DGRS23, DGRST23] for more sophisticated arguments in the context of Gaussian Free Field level sets and random interlacements.Finally, we use this local uniqueness property to perform a standard static renormalization argument implying percolation in the slab.
Remark 1.4.For simplicity, we chose to focus on nearest-neighbor interactions, but it is straightforward to adapt all the proofs to finite range interactions as well.
Acknowledgements.I would like to thank Hugo Duminil-Copin, Ulrik Thinggaard Hansen and the anonymous referees for their helpful comments on earlier drafts of this paper.This research was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 851565).

From disconnection to slab percolation
In this section we prove Theorem 1.3.We fix q ≥ 1 and d ≥ 3 and henceforth omit them from the notation.As explained in Section 1, the proof is split into two parts, which are done in separate subsections.

Uniqueness with sprinkling
We start with the following lemma, which is an easy consequence of the FKG inequality.
Lemma 2.1.If τp > 0, then there exists δ > 0 and such that for infinitely many L one has Proof.Since τp > 0, there exist C ′ ≥ 1 and δ ′ > 0 such that for infinitely many L one has Therefore, by the FKG inequality together with (2.2), we can find an i 0 ∈ {1, . . ., m} such that By comparison between boundary conditions we obtain which does not depend on x i0 , so the result follows with Ber(ε) be independent random variables, and denote its joint distribution on Ω 2 Λ by ψ ξ Λ,p,ε .The following proposition is the heart of our proof.It is inspired by the work of Benjamini & Tassion [BT17], where it was proved that any "everywhere percolating" subgraph of Z d becomes connected after an ε-Bernoulli sprinkling.We adapt their proof to the case of a very dense subgraph.
Proposition 2.2.Let Unique(L) be the event that there is a cluster in ω ∩ Λ L crossing Λ L \ Λ L/8 and that every cluster of ω ∩ Λ L crossing Λ L/2 \ Λ L/4 are connected to each other in (ω ∪ γ) ∩ Λ L/2 .For p > pc there exists δ > 0 such that for all ε > 0 one has Proof.We follow closely the proof and notation of [DS23, Proposition 4.1], where a similar result is proved for Voronoi percolation.Fix p > pc and ε > 0. Let δ ′ > 0 be given by Lemma 2.1 and L ′ such that (2.1) holds.Set By monotonicity on boundary conditions and the inequality (2.1), we obtain for all x ∈ Λ δL and all ξ.By union bound, one concludes that the event It remains to prove that Unique(δL) happens with high probability under ψ ξ (uniformly in ξ) conditionally on A L .First, notice that the existence of a cluster of ω ∩ Λ δL crossing Λ δL \ Λ δL/8 is automatically implied by A L , so we only need to focus on the uniqueness part.Consider the set of boundary clusters It is enough to prove that, with high probability conditionally on A L , all the clusters in . We will add γ to ω progressively on each annulus For every 0 where Finally, we set Notice that it is enough to prove that U ⌊δ √ L/4⌋ = 1 with high probability conditionally on A L , which follows from the following lemma.
Lemma 2.3.There exists c > 0 such that for every ξ and every 0 ≤ i ≤ ⌊δ √ L/4⌋ − 8, one has By Lemma 2.3 together with a union bound, the following event occurs with high probability On this event, U ⌊δ √ L/4⌋ > 1 would imply U 0 ≥ 2 ⌊δ √ L/32⌋−1 , which contradicts the fact that U 0 ≤ CL d−1 .This yields that U ⌊δ √ L/4⌋ = 1 on this event, thus concluding the proof.It remains to give the following proof.
Proof of Lemma 2.3.We fix ξ and write ψ instead of ψ ξ Λ L ,p,ε for simplicity; all estimates will be uniform on ξ.Fix 0 ≤ i ≤ ⌊δ √ L/4⌋ − 8.We start by finding a sub-annulus of V i \ V i+8 such that most clusters in it are crossing.For every η ⊃ ω and j ∈ {0, 1, 2, 3}, let In the definition above, we abuse the notation by identifying the equivalence class of cluster C ∈ U i (η) with its associated η-cluster.Since (U j i (η)) j are disjoint subsets of U i (η), we can find j ∈ {0, 1, 2, 3} such that . We fix such a j for the rest of the proof and focus on the annulus We now further restrict to one of the two sub-annuli V i+2j \ V i+2j+1 and V i+2j+1 \ V i+2j+2 as follows.If at least one of the clusters in U j i (η i ) touches V i+2j+1 , we "wire" all clusters in U j i (η i ) (i.e.we treat their union as a single element) and focus on the annulus V i+2j \ V i+2j+1 .Otherwise, we forget about all the clusters in U j i (η i ) and focus on the annulus V i+2j+1 \ V i+2j+2 .More precisely, consider the family where C := ∪ C∈U j i (ηi) C if U j i (η i ) ̸ = ∅ and C = ∅ otherwise, and the annulus Notice that in any case, every element of Ũ contains a crossing of the annulus A, and that every box Λ ℓ (x) ⊂ A, x ∈ ℓZ d , intersects at least one element of Ũ. Define the following partition of Ũ.Let U 0 be This cannot occur on the event , where here we abuse the notation again by identifying Ũ1 and Ũ2 with the union of the associated η i -clusters.As a conclusion, we have where the sum in Ũ runs over all the (finitely many) possibilities for Ũ which are compatible with A L .Fix a set Ũ and a partition Ũ = Ũ1 ⊔ Ũ2 as in (2.4).Recall that by construction both Ũ1 and Ũ2 contain a crossing of the annulus A and their union is ℓ-dense in A. Therefore, by splitting A into annuli of thickness 10ℓ, one can find vertices x 1 , . . ., x n ∈ ℓZ d , n ≥ c d √ L/ℓ, such that the boxes D k := Λ 2ℓ (x k ) ⊂ A, j ∈ {1, . . ., n}, are disjoint and each one intersects both Ũ1 and Ũ2 .In particular, on each of these boxes, there exists a path of length at most 4dℓ between Ũ1 and Ũ2 which is fully open in γ with probability at least ε 4dℓ .Notice that the event { Ũ = Ũ } ∩ A L is (ω, η i )-measurable and therefore independent of γ ∩ (V i \ V i+8 ).By independence, we conclude that for some constant c > 0, where we have used that nε 4dℓ ≫ L 1/4 .Combining (2.4), (2.5) and the fact that , for some c ′ > 0 and L large enough, thus concluding the proof.

A Positivity of the surface tension
In this Appendix, we prove Theorem 1.2.The proof relies on specific properties of the Ising model, namely the Ginibre inequality, which is used in the proof of Theorem A.1 below, and the uniqueness of infinite volume measure for its FK representation on the half space with positive boundary wiring, which is used in the proof of Proposition A.2 below (see Lemma A.3). Since we will only work with the FK-Ising model (q = 2), we henceforth omit q from the notation.We start by introducing the Ising model.Given a finite Λ ⊂ Z d , and inverse temperature β ≥ 0 and a boundary field η ∈ R ∂Λ , we define the Hamiltonian on Σ Λ := {−1, 1} Λ given by where for y ∈ ∂Λ we write σ y = η y .Given an inverse temperature β ≥ 0, consider probability measure where Z η Λ,β = σ∈ΣΛ e −βH η Λ (σ) is the normalizing partition function -see e.g.[FV17] for an introduction to the Ising model.We denote by + the boundary field η ≡ 1 and by ± the boundary field given by η β denote the infinite volume measure on Σ = {−1, 1} Z d obtained as the weak limit of ⟨•⟩ + Λ,β as Λ ↑ Z d .The critical inverse temperature is given by β c := inf{β ≥ 0 : ⟨σ 0 ⟩ + β > 0}.Finally, the surface tension at β is defined as It is classical that the limit in (A.1) exists and that it is the same if taken jointly in L and M for any ]).One has τ β > 0 for every β > β c .
The Ising model at inverse temperature β and the FK-Ising with p = 1 − e −2β are coupled together through the so-called Edwards-Sokal coupling -see [Gri06,Dum17]  Therefore, in order to prove Theorem 1.2, it remains to compare wired and free boundary conditions, which is the subject of the following proposition.

Figure 1 :
Figure 1: The gluing procedure of Lemma 2.3.The blue and red clusters represent Ũ1 and Ũ2 , respectively.The dark red clusters represent the set C: we are therefore in the case where C ∩ V i+2j+1 ̸ = ∅, and A is the grey annulus.The black squares represent the boxes (D k ) n k=1 , of radius 2ℓ = o(log L), where Ũ1 and Ũ2 meet.