E l e c t ro n ic J o f P r o b a bi l i t y
Electron. J. Probab. 23 (2018), no. 63, 1–25. ISSN: 1083-6489 https://doi.org/10.1214/17-EJP86
The phase transitions of the random-cluster and Potts
models on slabs with
q ≥ 1
are sharp
Ioan Manolescu
*Aran Raoufiï
†Abstract
We prove sharpness of the phase transition for the random-cluster model withq ≥ 1 on graphs of the formS := G × S, whereG is a planar lattice with mild symmetry assumptions, andSa finite graph. That is, for any such graph and anyq ≥ 1, there exists some parameterpc = pc(S , q), below which the model exhibits exponential
decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.
Keywords: random-Cluster model; Potts model; sharp phase transition; percolation models. AMS MSC 2010: 60K35.
Submitted to EJP on April 7, 2016, final version accepted on July 25, 2017.
1
Introduction
In the last few years, a variety of results concerning the phase transition of the random-cluster model (or FK-percolation) on planar lattices have emerged; see [3, 8, 5, 6]. The first are specific to the self-dual setting of the square lattice, while the third, still in preparation, extends the results of the first two to isoradial graphs. Finally, the fourth paper – a companion to the present paper – proves the sharpness of the phase transition of random-cluster models on generic planar graphs with sufficient symmetry.
These recent advances offer an understanding of planar random-cluster models that approaches that of Bernoulli percolation. However, contrary to the case of Bernoulli percolation, for which exponential decay in the subcritical phase was proved for lattices of any dimension (see [1, 15] and [10] for a recent short proof), the phase transition of the random-cluster model in dimensions above two is still not known to be sharp. We take a first step in this direction by proving the result for slabs, that is finite planar “slices” of a d-dimensional lattice.
Percolation on slabs has already been considered in the literature, most notably in the paper [13], where it was shown that the critical point of percolation onZ2× {0, 1, 2, . . . , N }(d−2) tends decreasingly to that of
Zd as N → ∞. There is work in
progress on the same type of result for the random-cluster model with integerq[9]. Let
*Université de Fribourg, Switzerland. E-mail: [email protected] †Institut des Hautes Études Scientifiques, France. E-mail: [email protected]
us also mention that Bernoulli percolation on slabs has recently been shown to exhibit a continuous phase transition [7]; a result which is also long sought for lattices of general dimension. Arguments similar to those of [9] also appear in [16] and [2].
The present paper blends the method of [6] with the techniques of [7]. It is intended as a complement to [6], focusing essentially on the new elements needed to treat the case of slabs.
Next we briefly introduce the model. For more details on the random-cluster model, we refer the reader to the monograph [12].
Consider a finite graphG = (VG, EG). The random-cluster measure with edge-weight
p ∈ [0, 1]and cluster-weightq > 0onGis a measureφp,q,Gon configurationsω ∈ {0, 1}EG.
For such a configurationω, an edgeeis said to beopen (inω) ifω(e) = 1, otherwise it isclosed. The configurationωcan be seen as a subgraph ofGwith vertex setVGand
edge-set{e ∈ EG: ω(e) = 1}. Acluster is a connected component of the subgraphω. Let
o(ω),c(ω)andk(ω)denote the number of open edges, closed edges and clusters inω, respectively. The probability of a configuration is then equal to
φp,q,G(ω) =
po(ω)(1 − p)c(ω)qk(ω)
Z(p, q, G) ,
whereZ(p, q, G)is a normalising constant called the partition function.
Fix for the rest of the paper a connected planar locally-finite graphG = (VG, EG), which is invariant under the action of some lattice Λ ' Z ⊕ Z, under reflection with respect to the line{(0, y), y ∈ R} and rotation by some angleθ ∈ (0, π)around 0. For simplicity we will assume in the present paper thatθ = π/2and thatG is invariant under translations by the vectors(1, 0)and(0, 1).
In addition letS = (VS, ES)be a finite graph and define the “slab”S = G × S. That
isS is the graph with verticesVS = VG × VS and edgesES connecting two vertices
(u, v) ∈ VS and(u0, v0) ∈ VS if eitheru = u0and(v, v0) ∈ ES or(u, u0) ∈ EG andv = v0.
Maybe the most common such example is forG = Z2andS = {1, . . . , n}, in which case
S is a slice of thicknessnof the three dimensional latticeZ3.
Forp ∈ [0, 1]andq ≥ 1, random-cluster measures with parametersp, qmay be defined on the infinite graphS by taking weak limits of measures on sequences of nested finite graphsGn tending toS (see [12, Ch. 4] or [4, Sec 4.5] for a detailed account). We call
such limitsinfinite-volume measures. For a pair of parametersp, q, more than one such infinite-volume measure may exist; the two most notable infinite-volume measures are the free and wired ones, denoted byφ0
p,q,S andφ 1
p,q,S, respectively. These are ordered
in that, forp < p0andq ≥ 1,
φ0p,q,S ≤stφ1p,q,S ≤stφ0p0,q,S,
where≤stdenotes stochastic domination. Moreover,φ0p,q,S andφ1p,q,S are the extremal
measures with parameters pandq, in the sense that, ifφp,q,S is an infinite volume
measure with these parameters, then
φ0p,q,S ≤stφp,q,S ≤stφ1p,q,S.
While it is possible to have values of pfor which the infinite volume measure is not unique, i.e. for whichφ0
p,q,S 6= φ 1
p,q,S, only at most countably many such values of p
exist for any fixedq ≥ 1. Forp, qsuch thatφ0
p,q,S = φ 1
p,q,S, we will denote the unique
infinite-volume measure byφp,q,S.
Theorem 1.1.Fixq ≥ 1. There existspc = pc(S ) ∈ [0, 1]such that
• forp < pc, there existsc = c(p,S ) > 0such that for anyx, y ∈S,
• forp > pc, there exists a.s. an infinite open cluster underφ0p,q,S.
The equivalent of Theorem 1.1 is also valid for planar random-cluster models with finite range interactions; we define these next. LetJ : VG× VG → [0, +∞)be a function with the property that there exists a constant M ≥ 1such that, if dG(x, y) > M, then J (x, y) = 0(wheredG is graph distance onG). Moreover, suppose thatJ has the same symmetries asG. Infinite-volume random-cluster measuresφβ,q,G ,J with parameters
β > 0 and q ≥ 1 may be defined as before as weak limits of measures φβ,q,Gn,J on
sequences of finite subgraphsGntending towardG, whereφβ,q,Gn,J is defined as
φβ,q,Gn,J(ω) = Q x,y∈VGn(e βJ (x,y)− 1)ω(e)qk(ω) Z(β, q, Gn, J ) ,
Z(β, q, Gn, J ) being a normalising constant. The same remarks about the different
infinite-volume measures as in the case of slabs apply here.
Theorem 1.2.Fixq ≥ 1. There existsβc = βc(G , J) ∈ [0, 1]such that
• forβ < βc, there existsc = c(p,G , J) > 0such that for anyx, y ∈G,
φ1p,q,G ,J[xandyare connected by a path of open edges] ≤ exp(−c|x − y|), • forβ > βc, there exists a.s. an infinite open cluster underφ0p,q,G ,J.
The proof of Theorem 1.2 is a direct adaptation of that of Theorem 1.1. In what follows we will only prove Theorem 1.1; we leave the details of the adaptation of the proof to the second theorem to the interested reader.
Results for the Potts model. The above results have direct consequences for Potts model. Consider an integer q ≥ 2 and introduce the polyhedron Ωq ⊂ Rq−1 with q
elements defined by the property that for anya, b ∈ Ωq
a · b = (
1 ifa = b, − 1
q−1 otherwise,
where·denotes the scalar product onRq−1.
Let G = (VG, EG) be a finite graph and β > 0. The q-state Potts model on Gat
inverse-temperatureβ > 0 with free boundary conditions is defined as follows. The energy of a configurationσ = (σx: x ∈ VG) ∈ ΩVqGis given by the Hamiltonian
HG(σ) := −
X
{x,y}∈EG
σx· σy. (1.2)
The probabilityµβ,q,Gof a configurationσis defined by
µβ,q,G(σ) :=
exp[−βHG(σ)]
Z(G, β, q) , (1.3)
whereZ(G, β, q)is defined in such a way that the sum of the weights over all possible configurations equals 1.
As for the random-cluster model, the q-state Potts measure with free boundary conditionsµβ,q,S on the infinite graphS may be defined by taking the weak limit of
measuresµβ,q,Gn on sequences of nested finite graphsGnconverging toS.
The Edward-Sokal coupling between the measures φ0p,q,S and µβ,q,S where p =
1 − exp(−q−1q β)yields the following relation for any two verticesx, y ∈S φ0p,q,S[xandyare connected by a path of open edges] = µβ,q,S(σx· σy).
Corollary 1.3.Fixq ≥ 2. There existsβc= βc(S ) ∈ [0, ∞)such that
• forβ < βc, there existsc = c(β,S ) > 0such that for anyx, y ∈S,
µβ,q,S(σx· σy) ≤ exp(−c|x − y|),
• forβ > βc, there existsc0= c0(β,S ) > 0such that for anyx, y ∈S,
µβ,q,S(σx· σy) ≥ c0.
Likewise, Theorem 1.2 may be translated for the Potts model. IfJ is a function as before, define the Hamiltonian of the weighted Potts model on a finite sub-graphGofG by
HG,J(σ) := −
X
x,y∈VG
J (x, y) σx· σy,
and the associated measureµβ,q,G,J by (1.3). Infinite volume measuresµβ,q,G ,Jmay also
be defined as above.
Corollary 1.4.Fixq ≥ 2. There existsβc= βc(G , J) ∈ [0, ∞)such that
• forβ < βc, there existsc = c(β,G , J) > 0such that for anyx, y ∈G,
µβ,q,G ,J(σx· σy) ≤ exp(−c|x − y|),
• forβ > βc, there existsc0= c0(β,G , J) > 0such that for anyx, y ∈G,
µβ,q,G ,J(σx· σy) ≥ c0.
We will not discuss further this adaptation to the Potts model. For background on the Potts model and its coupling to the random-cluster model we direct the reader to [12]. Deriving the two corollaries from Theorems 1.1 and 1.2 through the Edward-Sokal coupling is straightforward.
2
Notation and preparatory remarks
Notation. In the rest of the paperq ≥ 1will be fixed and we drop it from the notation. We will only work with infinite volume measures onS, hence we will equally dropS from the notation forφ.
Thusφp will denote any infinite volume measure onS with
edge-weightpand cluster-weightq.
It will be apparent in the proofs that we always allow ourselves to alterpin a small open interval. We can therefore assume that all the values ofpmentioned hereafter are such thatφp is theunique infinite-volume measure.
If Ais a subgraph ofG, then we defineA = A × S and regard this as a subgraph ofS.
Letu, v ∈S be two vertices,D ⊂S be a subgraph andω ∈ {0, 1}ES be a
configura-tion. We writeu←−→ vω,D for the event that there exists anω-open path, i.e. a self-avoiding chain of adjacentω-open edges, linkinguandv and contained inD. For setsA, B of vertices ofS, writeA←−→ Bω,D if there existsu ∈ Aandv ∈ Bsuch thatu←−→ vω,D holds. When no confusion is possible, the configurationωwill be omitted from the notation. If Dis omitted, it is assumed equal toS.
Fora < bandc < d, we identify[a, b] × [c, d]with the subgraph ofG induced by the vertices contained in [a, b] × [c, d]. We call a rectangle, a subgraph ofS of the form R = [a, b] × [c, d](note that a rectangle is not planar, it has “thickness”S).
For a rectangle R = [a, b] × [c, d], if we set A = {a} × [c, d] and B = {b} × [c, d] (respectivelyA = [a, b] × {c} andB = [a, b] × {d}), the eventA ←−→ Bω,R is denoted by Ch([a, b] × [c, d])(respectivelyCv([a, b] × [c, d])) and if it occurs we say thatRis crossed
horizontally (respectively vertically). An open path fromAtoBis called a horizontal crossing (respectively vertical crossing). Whena = 0andc = 0, we simply writeCh(b, d)
andCv(b, d) for the events above. Whenb − a > d − c, horizontal crossings are called
crossings in the hard direction, while vertical ones are crossings in the easy direction. The terms are exchanged whenb − a < d − c.
Forγ = (γ1, . . . , γm)andχ = (χ1, . . . , χm)two paths ofS, we say thatγandχoverlap
at some pointg ∈G if there existiandj such thatγi, χj ∈ {g}.
Forg ∈G, letBR(g)(and∂BR(g)) be the set of vertices at distance less than or equal
toR(equal toR, respectively) fromz. For a pointz = (g, n) ∈S defineΛR(z) = BR(g)
and∂ΛR(z) = ∂BR(g). We callΛR(z)the box of sizeRaroundz.
Strategy of the proof. Define
pc:= infp ∈ (0, 1) : φp(xis in an infinite cluster) > 0
˜ pc:= supp ∈ (0, 1) : lim n→∞− 1 nlogφp(0 ↔ ∂Λn) > 0 .
Forp < ˜pcwe say thatφpexhibits exponential decay since connection probabilities decay
exponentially with the distance; forp > pc,φpis supercritical, in that it contains a.s. an
infinite cluster. It is immediate thatp˜c ≤ pc. We wish to prove thatpc= ˜pc(this is simply
another way of stating the main result), and we therefore focus on the inequalityp˜c ≥ pc.
As mentioned before, we adapt the argument of [6], which consists of three steps: • First it is proved that, forp > ˜pc, the crossing probabilities underφp of 2n × n
rectangles in the easy direction are bounded away from0uniformly inn.
• Building on this, in the second step, it is showed that theφp-crossing probabilities
of2n × nrectangles in the hard direction are also bounded away from0uniformly inn.
• Finally, in the third step, assuming thatp˜c < pc, it is showed that forp ∈ (˜pc, pc),
φp(Ch(2n, n)) → 1, as n → ∞. The first step then implies that the dual of φp0
exhibits exponential decay for anyp0 ∈ (p, pc), and this contradicts the fact that
p < pc.
While the first step is not specific to planar lattices, the next two steps make use of planarity, namely by “gluing” crossings and invoking duality. In Sections 4 and 5 of the present paper, we adapt the arguments used in the last two steps to the setting of slabs. An essential element is the “gluing” lemma discussed in Section 3. Loosely speaking, the gluing lemma says that if open paths come close to each other, then they connect with positive probability. Such a statement already appeared in [7] in the context of percolation on slabs.
Adapting the final step requires particular attention, since there is no duality available in the model. To overcome this difficulty, we deduce bounds on the speed of convergence of φp(Ch(2n, n)) to 1 for p ∈ (˜pc, pc). These bounds are obtained by and fed into a
quantitative version of the gluing lemma. Roughly stated, this version guarantees that, whenever overlapping paths occur with high probability, they also connect with high probability.
The quantitative version of the gluing lemma is paramount for our approach: in order to obtain a contradiction in the third step, we show that for anyp > ˜pc an infinite cluster
exists. We construct such a cluster by gluing together an infinite number of crossings of rectangles, with increasingly high probability.
Differential inequalities. For an event A and a configuration ω let HA(ω) be the
Hamming distance betweenωandA, that is the minimal number of edges whose state needs to be altered to obtain from ω a configurationω0 ∈ A. ThusHA is a random
variable taking non-negative integer values. Moreover, IfAis an increasing event, then HAis a decreasing random variable.
The following lemma is the integrated form of the differential inequality of [14], as written in [6, Rem. 2.4]. This is the cornerstone of our approach.
Lemma 2.1.LetAbe an increasing event depending only on the state of finitely many edges. Then, for0 < p < p0 < 1,
φp0(A) ≥ φp(A) exp4(p0− p)φp0(HA), (2.1)
whereφp0(HA)is the expectation ofHAunderφp0. Similarly, ifAis decreasing,
φp0(A) ≤ φp(A) exp − 4(p0− p)φp(HA). (2.2)
The following lemma taken from [12, Thm. 3.45] will also be useful.
Lemma 2.2.Let 0 < p < p0 < 1. For any non-empty increasing event A, and any non-negative integerk, φp(HA≤ k) ≤ Ckφp0(A), (2.3) where C = q 2(1 − p) (p0− p)[p0+ q(1 − p)].
In both statements above,φpandφp0 denote any infinite volume measure onS with
edge-weightspandp0, respectively, and cluster-weightq. The first lemma is obtained by integrating betweenpandp0 the differential inequality of [14] on finite subgraphs ofS, then taking the limit as the subgraphs tend toS. We may perform this procedure for values ofpandp0 for which the infinite-volume measure is unique. Since these values are dense in[0, 1]and by the monotonicity ofφp, we obtain (2.1) and (2.2) for all values
0 < p < p0 < 1and any infinite-volume measures. The same argument may be applied to deduce the second lemma, for which the finite-volume version is ensured by [12, Thm. 3.45].
3
Gluing Lemma
One of the main challenges in percolation in dimensions higher than 2 is that Jordan’s theorem does not apply. As a consequence, it is difficult to connect open paths together. Indeed, contrary to planar graphs, on non-planar graphs such as slabs, paths may overlap without intersecting. The gluing lemma is a tool to overcome this obstacle for slabs or for models with finite range interactions. Here we will only present it in the context of slabs.
Lemma 3.1 (Gluing Lemma).LetDbe a subset ofG andA1, A2, B1, B2⊂ D. Suppose
that the following deterministic topological condition is satisfied:
Any two pathsχ, γ ⊂ DconnectingA1toA2andB1toB2, respectively, intersect.
(3.1) In addition letD0be a subset ofG containingDandA
0be a subset ofD0. DefineA
as the event that there exists an open clusterC ⊂ D0 intersectingA
0and that contains a
pathχ ⊂ DconnectingA1andA2. LetBbe the event thatB1is connected toB2by an
open path contained inD. Finally letX be the event that there exists an open cluster C0⊂ D0 that intersectsA
0and contains a pathγ ⊂ DconnectingB1toB2. Then the two
A0 A1 A 2 B1 γ A0 A1 A 2 B2 B1 γ B2
Figure 1: The typical use of Lemma 3.1 seen from “above”. The grey area isD which formsD0with the additional white rectangle. The blue path ensures the occurrence ofB; the red cluster is the one in the definition ofA. The two overlap but do not necessarily connect. (In the figure any path fromA0toA2insideD
0
intersectsA1; this is simply an
artefact of the figure, not an assumption of the gluing lemma.)
Left: a configuration inY(1) but not inY(2). Right: a configuration in bothY(1) and
Y(2). The overlap points are marked.
(i) There exists a constantc > 0, only depending onp,G andS, such that
φp(X ) ≥ c φp(A )φp(B). (3.2)
(ii) There exists a constantβ > 0, only depending onp,G andS, such that φp(X ) ≥ φp(A )φp(B) − (1 − φp(A ))
β
. (3.3)
The first statement may be understood as follows. If two open paths necessarily overlap, then they have a positive probability of being connected to each other. The second statement is a quantitative version of the first, useful in Section 5. It essentially states that ifA occurs with high probability, then the overlapping paths connect with high probability. We believe that the second statement and the ideas of its proof are interesting in their own right and may be useful in future works regarding statistical mechanics on non-planar graphs.
The asymmetry between the events A and Bis purely technical. Indeed, in our applications of the gluing lemma,A0will act as an “anchor” and the lemma will allow us
to extend the cluster ofA0.
Initially a version of this lemma appeared in [7] in the context of Bernoulli percolation. Its proof does not essentially use independence; it relies on the finite-energy property, a property shared by the random-cluster model. The property states that for any configurationω0and any edgee
p
p + q(1 − p) ≤ φpω(e) = 1
ω(f ) = ω0(f ), ∀f 6= e ≤ p. (3.4)
The second part of the lemma, although similar in spirit, requires additional work. We give a full proof of the two parts below. To help legibility, we start with the simpler statement (i), we then discuss the additional elements needed to obtain (ii).
Lemma 3.1(i). SetY = (A ∩ B) \ X. In addition, fori = 1, 2, letY(i)⊂Y be the event
that there exists an open cluster inDthat contains a crossing fromA1toA2, does not
Y(2)are not necessarily disjoint, butY(1)∪Y(2)=Y. The rest of the proof is dedicated
to bounding the probability ofY(1).
Let be an ordering of the oriented edges of D. This induces a lexicographical ordering of the paths contained inD, which we will denotelex.
Forω ∈ B, letγ = γ(ω) be the minimal open path (forlex) contained in D, from
B1 toB2. We call a pointz ∈ γ(ω)anoverlap point if there exists a cluster as in the
definition ofY(1) that intersects{z}.
We now define a map Ψ : Y(1) → X as follows. For ω ∈ Y(1), because of the
topological condition (3.1), there exists at least one overlap pointz ∈ D. We choose arbitrarily one such overlap pointz = z(ω).
We defineΨ(ω)by modifying the configurationωinside the regionΛ2(z)as follows.
Letγi andγj be the first and last points, respectively, ofΛ1(z)visited byγ. Leta0be a
point of∂Λ1(z), connected toA0by an open path(a0, . . . , am), witha1, . . . , am∈ D0\Λ1(z).
The existence of such a point is guaranteed by the fact thatz is an overlap point. In Ψ(ω), edges with no endpoint inΛ1(z)have the same state as inω. All edges with exactly
one endpoint inΛ1(z) are declared closed, with the exception of (γi−1, γi), (γj, γj+1)
and(a0, a1), which are open (note that since ω /∈ X, these three edges are distinct).
The edges with both endpoints in Λ1(z) are closed, with the exception of two open
edge-disjoint pathsg = (g0, . . . , gk) ⊂ Λ1(z)andh = (h0, . . . h`) ⊂ Λ1(z)such that
• g0= γi,gk = γj
• h0= gtfor some1 ≤ t ≤ k − 1andh`= a0
• (gt, gt+1) (gt, h1)andgt∈ {z}(wheretis such thatgt= h0).
The existence of such a modification may easily be checked and we do not give additional details here. See Figure 2 for an illustration. It is immediate thatΨ(ω)is indeed inX.
In order to compareφp(X )toφp(A ∩ B), we will use the following simple relation
φp Y(1) = X ω∈Y(1) φp(Ψ(ω)) φp(ω) φp(Ψ(ω)) ≤ sup ω∈Y(1) φp(ω) φp(Ψ(ω)) · sup σ∈Im(Ψ) |Ψ−1(σ)| · X σ∈Im(Ψ) φp(σ). (3.5)
SinceωandΨ(ω)only differ inΛ2(z), the finite energy property (3.4) implies
sup ω∈Y(1) φp(ω) φp(Ψ(ω)) ≤ q min{p, 1 − p} |Λ2| ,
where|Λ2|denotes the number of edges ofΛ2. Moreover, sinceΨtakes values inX, we
haveP
σ∈Im(Ψ)φp(σ) ≤ φp(X ).
Let us now boundsupσ∈Im(Ψ)|Ψ−1(σ)|. Fixσ ∈ Im(Ψ)andω ∈ Ψ−1(σ). Recall that γ(σ)is the minimalσ-open path contained inD, fromB1toB2. (Such a path necessarily
exists sinceσ ∈X.) We claim that, due to the nature of the modification applied toω in order to obtainΨ(ω) = σ and to the fact thatlexis lexicographical,γ(σ)coincides
withγ(ω)up to the first time it entersΛ1(z)and after the last time it exitsΛ1(z). More
precisely, we claim thatγ(σ)is the concatenation ofγ[0,i](ω),(g0, . . . , gk)andγ[j,n](ω),
(wherenis the length ofγ(ω)andi, j and(g0, . . . , gk)are defined above). This fact is
essential, and we give a detailed explanation below.
Let χ be the concatenation of γ[0,i](ω), (g0, . . . , gk) and γ[j,n](ω) and suppose that
γ(σ) 6= χ. Sinceχis open inσ, it must be thatγ(σ) ≺lexχ. Letτ = inf{i ≥ 0 : χi6= γi(σ)}.
There are three possible situations; we analyse them separately and show that each leads to a contradiction.
γi(ω) γj(ω) = gk hℓ h1 gt gt+1 B1 B2 A0
Figure 2: The local modification performed onω in and aroundΛ1(z) to obtainΨ(ω).
The blue path isgand the red ish. Note that(gt, gt+1) (h0, h1). The central axis in the
image isz.
Supposeτ > i + k, i.e.γ(σ)andχdiffer after exitingΛ1(z). Thenγ(σ)can not visit
Λ1(z)again, since the boundary of Λ1(z) has only threeσ-open incoming edges, and
two of them have already been visited byγ(σ). Hence the latter part of γ(σ), namely γ[τ,|γ(σ)|](σ), is open inω as well as inσ. Moreoverγ[τ,|γ(σ)|](σ) ≺lexχ[τ,|χ|]= γ[j,|γ(ω)|](ω),
which contradicts the minimality ofγ(ω).
Suppose i < τ ≤ i + k, i.e. thatγ(σ) and χ differ when in Λ1(z). Then the only
possibility is thatγτ(σ) = h1whileχτ= gt+1. Since(gt, gt+1) (h0, h1), this contradicts
the minimality ofγ(σ).
Finally supposeτ < i. In particular
γ[0,τ +1](σ) ≺lexχ[0,τ +1]= γ[0,τ +1](ω).
The minimality ofγ(ω)then implies thatγ(σ)is notω-open, hence it uses an edge with at least one endpoint inΛ1(z). This occurs after timeτ, sinceγ[0,τ ](σ) = γ[0,τ ](ω)does
not intersectΛ1(z).
Letτ0 > τ be the first timeγ(σ)visits the boxΛ1(z). Thenγτ0(σ) ∈ {γi(ω), γj(ω), a0}
(these are the only points of∂Λ1(z)accessible byσ-open edges from the outside). It is not
possible thatγτ0(σ) = a0, sincea0is not connected toB1inD \ Λ1(z)in the configuration
σ(we use the fact that σ = ω outsideΛ1(z) and that ω ∈ Y(1)). Thusγτ0(σ) ∈ γ(ω).
In other words,γ(σ)separates fromγ(ω)at timeτ, then later joinsγ(ω)again before visitingΛ1(z). Let us show that this is impossible.
Letτ00 be the first time afterτ whenγτ00(σ) ∈ γ(ω). The above discussion implies
thatτ < τ00 ≤ τ0 andγ
[0,τ00](σ) ≺lexγ(ω). This contradicts the minimality ofγ(ω), since
γ[0,τ00](σ)is open inωand represents a more optimal first section for a connection from
B1toB2inD.
This concludes the proof ofγ(σ) = χ. Let us return to the analysis ofΨ−1(σ). Note thatgtis the unique pointx ∈ γ(σ)that is connected by aσ-open path toA0in
D0\ γ(σ). Thusgtis determined byσ, and so isz, the first coordinate ofgt. Sinceωand
σdiffer only insideΛ2(z), we obtain the bound
|Ψ−1(σ)| ≤ 2|Λ2(z)| for allσ ∈Im(Ψ).
It follows from (3.5) and the above bounds that φp(Y(1)) ≤
2q
min{p, 1 − p} |Λ2|
The same bound applies toφp(Y(2)), and combining the two yields φp(A )φp(B) − φp(X ) ≤ φp(Y ) ≤ φp(Y(1)) + φp(Y(2)) ≤ 2 2q min{p, 1 − p} |Λ2| φp(X ), which leads to (3.2).
The idea for the proof of the second statement is that, if A has high probability, then typically there must be a large number of overlap points, otherwise the connection betweenA0, A1 andA2 could easily be broken (this is proved in Lemma 3.3). Using
this fact, we may associate to a configuration ω ∈ (A ∩ B) \ X not one, but many configurationsσ ∈X. This in turn implies, using Lemma 3.2 below, thatX has much higher probability than(A ∩ B) \ X.
Several technical difficulties occur in this argument, and the proof requires some new ingredients. In particular, the ordering of the edges used for defining the minimal pathγ(ω)needs to be random.
LetOdenote the set of total orderings of oriented edges ofD0andµbe the uniform
measure on O. Setν = φp⊗ µto be the measure on{0, 1}E(S )×O obtained as the
product ofφpandµ.
Lemma 3.2.Suppose we have two eventE , F ⊆ {0, 1}E(S )×O, and a mapΨfromE to
2F. Suppose that the following statements are true: 1. If(ω, ) ∈E and(ω0, 0) ∈ Ψ(ω, ), then=0.
2. There existst > 0such that for each(ω, ) ∈E, we have|Ψ(ω, )| ≥ t.
3. There existsssuch that for each(ω0, ) ∈F there exists a finite setS(ω0, )of edges with|S(ω0, )| ≤ s, such that the configurations inΨ−1(ω0, ) := {ω : (ω0,
) ∈ Ψ(ω, )}differ fromω0 only insideS(ω0, ). Then the following statement is true:
ν(E ) ≤ 1 t 2q min{p, 1 − p} s ν(F ). (3.6)
Proof. The lemma is a generalization of [7, Lem. 7], and the proof is similar. Due to the finite energy property (3.4) and to the third condition, for all (ω, ) ∈ E and (ω0, ) ∈ Ψ(ω, ), we have ν(ω, ) = µ()φp(ω) ≤ q min{p, 1 − p} s · µ()φp(ω0) = q min{p, 1 − p} s · ν(ω0, ). By summing over(ω, ) ∈E and(ω0, ) ∈ Ψ(ω, ), we obtain:
ν(E ) ≤1 t q min{p, 1 − p} s X (ω,)∈E X (ω0,)∈Ψ(ω,) ν(ω0, ) =1 t q min{p, 1 − p} s X (ω0,)∈F |Ψ−1(ω0, )| · ν(ω0, ) ≤ 1 t q min{p, 1 − p} s X (ω0,)∈F 2sν(ω0, ) = 1 t 2q min{p, 1 − p} s ν(F ).
Lemma 3.1(ii). Letω ∈Band ∈O. As before, letlex be the lexicographical order
A
0A
1A
2B
2B
1γ
(2)γ
(1)Figure 3: The red cluster ensures the occurrence of the eventA. The blue paths areγ(1)
andγ(2). Not all intersections between the blue and red paths are overlap points, only the marked ones are. Out of these, only the doubly marked point is inW. The encircled region contains a(2)-almost-overlap point.
lex-minimal open path ofDfromB1toB2, andγ(2)(ω, )thelex-minimal open path of
DfromB2toB1.
As in the previous proof, setY = (A ∩ B) \ X and considerω ∈Y. In the previous proof, we have defined overlap points. Since in the present proof we will need to work withγ(1)andγ(2)simultaneously, we will define(1)-overlap points and(2)-overlap points.
Fori = 1, 2, letW(i)= W(i)(ω, )be the set of pointsz ∈G, such that{z}intersectsγ(i)
and also intersects an open clusterCofDwith the following properties: • Ccontains a path fromA1toA2,
• Cdoes not intersectBi,
• Cis connected toA0inD0.
Call the points ofW(i)(i)-overlap points. Obviously a point can be simultaneously both a
(1)and(2)-overlap point. See Figure 3 for an illustration.
Since ω ∈ Y, any crossing in D between A1 and A2 as in the definition of A
necessarily contains at least one overlap point of each type.
We also introduce the following related notion. Fori = 1, 2, we say a pointz ∈ Dis an(i)-almost-overlap point if there existsz0∈ Λ1(z)ands, s0∈ S such that
• (z, s) ∈ γ(i),
• (z0, s0)is connected toA0inD0\ {z},
• (z0, s0)is not connected toBiinD,
• (z, s0) /∈ γ(i).
LetU(i)(ω, )denote the set of(i)-almost-overlap points. It will be useful to note that for
i = 1, 2,W(i)(ω, ) ⊂ U(i)(ω, ). To be precise, an(i)-almost-overlap point is a(i)-overlap
point if in addition to the conditions above,z = z0 and(z, s0)is connected to bothA1and
A2inD.
Our aim is to boundφp(Y ) = ν(Y × O). To do this we will splitY in three events.
Since these will depend on the (random) order, we will henceforth work with couples (ω, ).
Lemma 3.3), and defineα = −c log(φp(Ac)). Define the following events: Y≤α= n (ω, ) : ω ∈Y , |U(1)(ω, )| ≤ αand|U(2)(ω, )| ≤ αo, Y(1) >α =(ω, ) : ω ∈Y , |U (1)(ω, )| > α , Y(2) >α =(ω, ) : ω ∈Y , |U (2)(ω, )| > α .
Note that indeedY × O = Y≤α∪Y (1) >α ∪Y
(2)
>α, but that the two latter events are not
necessarily disjoint. We start by bounding the probability of the first event.
Lemma 3.3.Provided that the constantc > 0in the definition ofαis small enough, we have
ν Y≤α ≤
q
φp(Ac).
The idea behind this lemma is that, for(ω, ) ∈Y≤α, the connection betweenA1and
A2inDis fragile, since it only has few overlap points withγ(1) andγ(2). Thus, it is easy
to break this connection, and this leads to an upper bound onν(Y≤α)in terms ofφp(Ac).
Proof. Define a mapΨ :Y≤α→Ac×O as follows. Take(ω, ) ∈Y≤α. LetW (ω, )be
the set of pointsz ∈ W(1)∪ W(2)such that{z}is connected toA
0without using other
points ofW(1)∪ W(2). It is essential to remark that, sinceω ∈Y,W (ω, ) 6= ∅.
LetΨ(ω, ) = (σ, )withσequal toωfor all edges with no end-point inW (ω, ). The edges with at least one end-point inW (ω, )are declared closed inσ, unless they are part ofγ(1) orγ(2), in which case they remain open.
Let us show that Ψ(ω, ) ∈ Ac×O, that is σ /∈ A. Suppose that this is not the
case, and thatσ ∈A. We know thatσ ∈B(B1 andB2being united byγ(1)andγ(2));
moreoverσ ≤ ωsoσ /∈X. We therefore conclude thatσ ∈Y. Then, by the topological condition (3.1), inσthere exists at least one overlap pointz0, which is connected toA0
inD0. Letχbe a σ-open path inD0 fromz
0 toA0. Denote(z1, s1)the last point onχ
such thatz1is an overlap point. Thenz1∈W (ω, )and hence the edges emanating from
(z1, s1)should be closed inσ. This contradicts the fact that(z1, s1)is connected toA0in
σ. We have therefore shown thatσ /∈A.
We now use Lemma 3.2 to bound the probability of the event under study. Condition 1 is satisfied by definition; condition 2 is satisfied with t = 1. We focus on the third condition.
Let(σ, ) ∈Im(Ψ)andω ∈ Ψ−1(σ, ). Since any open edge ofσis also open inωand γ(1)(ω)andγ(2)(ω)are both open inσ, we haveγ(i)(σ) = γ(i)(ω)fori = 1, 2. Moreover, in
going fromωtoσ, we do not create new almost-overlap points, i.e.U(i)(σ, ) ⊂ U(i)(ω, ).
Finally we observe that, by definition ofΨ, all points where modifications were made when going fromωtoσ, are almost-overlap points ofσ. In conclusion,ωandσonly differ in the vicinity of points inU(1)(σ, ) ∪ U(2)(σ, ), and there are at most2αof these. It
follows that the third condition of Lemma 3.2 is satisfied withs = 2αK, whereKis the number of edges ofΛ1.
Using the definition ofαand Lemma 3.2, we obtain ν(Y≤α) ≤ 2q min{p, 1 − p} 2αK ν(Ac×O) = φp(Ac) 1−2cK logmin{p,1−p}2q ,
which implies the lemma provided thatc ≤h4K logmin{p,1−p}2q i
−1
.
Remark 3.4. Given a configurationσin the image ofΨ, the almost-overlap points ofσ, rather than simply the overlap points, are the places where modifications may have been performed when constructingσfrom one of its pre-images. This explains the necessity of introducing the additional notion of almost-overlap point.
We will now focus on bounding the probabilities ofY>α(i)fori = 1, 2. More specifically
we will prove the following.
Lemma 3.5.There exists a constantβ > 0depending onp,G andS only, such that, for φp(Ac)small enough
ν Y≥α(i) ≤ φp(Ac)β, (3.7)
fori = 1, 2.
By symmetry we can concentrate on boundingφp(Y (1)
>α). To simplify notation, we will
henceforth omit the index(1).
The idea behind this lemma is that, for(ω, ) ∈Y>α, the multitude of almost-overlap
points gives many opportunities forγ to connect toA0. Thus, the probability of Y>α
should be much smaller than that ofX, and this will ultimately yield the bound (3.7). To make this heuristic rigorous, we will define a multi-valued mapΨ :Y>α→ 2X ×O
and apply Lemma 3.2. As suggested above, the functionΨwill consist in connectingA0
toγby modifying the configuration locally around certain almost-overlap points; we say we will perform a connecting surgery at these points. Not all almost-overlap points are suited to perform the connecting surgery, and we start by identifying those who are.
Fix inS an arbitrary system of geodesics uniting any pair of pointss, s0∈ S; such a system always exists sinceS is connected. We may then talk of the segment between sands0, which we denote by[s, s0]. As mentioned in the introduction, one may think of S = {0, . . . , k}, in which case the segment between sands0 > sis simply[s, s0] =
{s, s + 1, . . . , s0}. For(ω, ) ∈Y × O, we call a pointz ∈ Dagood almost-overlap point,
if it is an almost-overlap point (withz0, sands0 as in the definition of(1)-almost-overlap points) and in addition
• there is notstrictly betweensands0 such that(z, t) ∈ γand
• ifγj= (z, s)and ift ∈ Sis the first point afterswhen going fromstos0along[s, s0],
then(γj, γj+1) ((z, s), (z, t)).
LetV (ω, )be the set of good almost-overlap points. The following lemma states that generally a positive proportion of almost-overlap points are good.
Lemma 3.6.For any configurationω0∈Y and pathγ0,
νh|V (ω0, )| ≥ 14|U (ω0, )| ω = ω0; γ(ω0, ) = γ0 i ≥1 4, (3.8) whenever the conditioning is not void.
Remark 3.7. It is for the above lemma alone that the random ordering is necessary. Indeed, for a fixed ordering, there is no guarantee that enough good almost-overlap points exists.
Proof. Before we start the proof, let us mention that the set of almost-overlap points U (ω, )only depends on ω andγ(ω, ), not otherwise on. The set of good almost-overlap points does however depend further on.
Fixω0 ∈ Y and a pathγ0. LetU0 be the setU (ω0, ) for an orderingsuch that
γ(ω0, ) = γ0. (Such an ordering exists if the conditioning in (3.8) is not degenerate.)
We will prove that, for eachz ∈ U0,
νz ∈ V (ω, ) ω = ω0, γ(ω, ) = γ0 = µz ∈ V (ω0, ) γ(ω0, ) = γ0 ≥ 1 2. (3.9) In other words, when averaging over the choice of the order, any almost-overlap points is good with probability at least1/2. This implies (3.8) through a direct application of Markov’s inequality.
Fixz ∈ U0 as above. Let z0 ∈ Λ1(z) ands, s0 ∈ S closest to each other, as in the
definition of almost-overlap point, i.e. with • (z, s) ∈ γi,
• (z, s0) /∈ γ,
• (z0, s0)is connected toA
0inD0\ {z},
• (z0, s0)is not connected toB1inD.
Lett ∈ S be the first point after s when going from s to s0 along [s, s0] and let f = ((z, s), (z, t)). Lete = (γi, γi+1)ande1, . . . , ek be the oriented edges emanating fromγi,
other thane, such that there exists anω-open path inD \ γ[0,i] from(z, t)toB2starting
withei.
Underµ(. | γ(ω0, ) = γ0)the ordering of the oriented edges emanating fromγi is
uniform among orderings such thate eifori = 1, . . . , k. With this in mind, we notice
that:
• iff ∈ {e1, . . . , ek}, thenµ(e f | γ(ω0, ) = γ0) = 1.
• iff /∈ {e1, . . . , ek}, thenµ(e f | γ(ω0, ) = γ0) = k+1k+2.
Equation (3.9) follows from the above.
Lemma 3.5. LetY>α0 =Y>α∩|V (ω, )| > α/4 . By Lemma 3.6ν(Y>α0 ) ≥ 1
2ν(Y>α), and
we will focus on boundingν(Y>α0 ). In order to do this we will define a mapΨ :Y>α0 → 2X ×O and apply Lemma 3.2 to it.
Fix a real numberc0∈ (0, 1/2)which we will identify later and letj = dc0αe. Consider
a pair(ω, ) ∈Y>α0 . Forz1, . . . , zj∈ V (ω, ), we defineωz1,...,zj as follows.
By definition ofV, for eachzk there exists a pair of distinct pointssk, s0k ∈ Sand a
pointzk0 such that (a) (zk, sk) ∈ γ,
(b) (zk0, s0k)is connected toA0inD0\ {zk},
(c) γdoes not intersect{zk} × (sk, s0k],
(d) Λ1(zk) × (sk, s0k)is not connected toA0inD0\ {zk},
(e) ifγi= (zk, sk)andtk is the first point ofSwhen going fromsk tos0kalong[sk, s0k],
then(γi, γi+1) ((zk, s), (zk, t)).
Note that conditions (a),(b),(c) and (e) are exactly those of the definition ofV. Condi-tion (d) may be assumed by takings0k as close tosk as possible.
We choose pointssk, s0kandtkas above, following some deterministic ordering when
several choices are possible. Thenωz1,...,zj is identical to ω except for the following
edges, for eachk:
• all edges of{zk} × [sk, s0k]are open,
• ifzk6= zk0, then((zk, s0k), (zk0, s0k))is open,
• all edges of the form((zk, t), (z0, t))witht ∈ (sk, s0k)are closed.
We say we obtain ωz1,...,zj fromω by performing a connecting surgery at each point
zk. Observe that, for any choice of z1, . . . , zj ∈ V (ω, ), ωz1,...,zj ∈ X. Indeed the
connecting surgery does not close the pathγ(ω, ), soωz1,...,zj ∈B, and in addition any
one connecting surgery ensures thatγ(ω, ) D
0
←→ A0. We setΨ(ω, ) =(ωz1,...,zj, ) :
z1, . . . , zj∈ V (ω, ≺)
.
Let us now verify the conditions of Lemma 3.2. The first condition of the lemma is satisfied by definition. Since|V (ω, )| > α/4 for all(ω, ≺) ∈Y0
>α,|Ψ(ω, ≺)| > α/4
j
. Thus the second condition is satisfied witht = α/4j
B1 B2 A0 B1 B2 A0 B1 B2 A0
Figure 4: Left diagram: the marked vertical line corresponds to an almost-overlap point, but not an overlap point. The points(z, s)and(z0, s0)are marked in red and the point (z, s0)is marked by a circle. Middle diagram: the red vertical line corresponds to an overlap point. Right diagram: the red line does not correspond to an almost-overlap point, since the two paths come close to each other at the same horizontal level.
Let us now study the third condition. The first thing to notice is that, for(ω, )as above andz1, . . . , zj ∈ V (ω, ), we haveγ(ω, ) = γ(ωz1,...,zj, ). The proof of this fact
follows the same line as the corresponding step in the proof of Lemma 3.1(i). Let us only mention that the connection surgery used here is such thatγ(ω, )is open inωz1,...,zj.
Moreover,z1, . . . , zj were chosen asgood almost-overlap points, so that the pathγ(ω, )
is the minimal continuation of a crossing fromB1 toB2 at every pointzk. Indeed, if
γ(ω, )j = (zk, sk), then there are only threeωz1,...,zj-open edges emanating from(zk, sk)
and(γ(ω, )j, γ(ω, )j+1)is preferable to the first edge in the link between(zk, sk)and
(z0k, s0k).
Fix now(σ, ) = (ωz1,...,zj, ) ∈ Ψ(ω)for some(ω, ) ∈Y>αandz1, . . . , zj ∈ V (ω, ).
Then((z1, s1), . . . , (zj, sj))are the only points(z, s)onγ(σ, )that are connected toA0
by aσ-open path only intersectingγ(σ, )at(z, s). Thus,ωandσonly differ on the set S(σ) = ∪jk=1Λ1(zk). We insist that, since(z1, s1), . . . , (zj, sj)are determined byσ, so is
the setS(σ). Thus the third condition of Lemma 3.2 is satisfied, with s = j|Λ1|. The
lemma then implies
ν (Y>α0 ) ≤ α/41 j 2q min{p, 1 − p} j|Λ1| · ν (X × O) .
LetQ = (min{p,1−p}2q )|Λ1|. Note thatQis a constant depending onp,G andS only and
recall thatj was chosen asj = dc0αe. Sinceν(X × O) ≤ 1, using Stirling’s formula we obtain ν (Y>α0 ) ≤ Q j α/4 dc0αe ≤Q · (2c 0)c0c0α · (1 − 2c0)(12−c 0)α , (3.10)
for α large enough. By choosing c0 ∈ (0, 1/2) such that (2c0)c0 ≤ Q−1 and setting
c1= (1 − 2c0)( 1 2−c 0) ∈ (0, 1), we deduce ν (Y>α0 ) ≤ cα1,
forαlarge enough. In order to obtain the conclusion of Lemma 3.5, recall thatα = −c log(φp(Ac))for some constantc depending onp,G andS only, and thatαmay be
considered large since we restrict ourselves to small values ofφp(Ac).
Let us now conclude the proof of Proposition 3.1(ii). Note that the sought bound is only relevant whenφp(Ac)is small. We will therefore prove the bound assumingφp(Ac)
is small enough for Lemma 3.5 to hold. The result may be extended to any value of φp(Ac), with a possibly altered constantβ.
Recall thatY × O = Y≤α∪Y>α(1)∪Y (2)
>α. Lemmas 3.3 and 3.5 bound theν-probability
of the three events on the right hand side; we can combine them to obtain: φp(Y ) = ν(Y ) ≤ ν(Y≤α) + ν(Y
(1) >α) + ν(Y
(2)
>α) ≤ 3φp(Ac)max{β,1/2}.
The above yields (3.3) through basic algebra.
4
Bounds for crossing probabilities
As mentioned in the introduction, the first step in the argument of [6] applies to non-planar graphs. We state it here without proof:
Proposition 4.1 ([6] Prop 3.1).Forp > ˜pc,
lim inf
n→∞ φp(Cv(2n, n)) > 0.
The object of this section is the following result, which corresponds to the second step in [6]. It may be understood as a Russo-Seymour-Welsh type result, with the remark that it requires increasing the value of the edge-weight.
Proposition 4.2.Ifp ∈ (0, 1)is such that lim inf
n→∞ φp(Cv(2n, n)) > 0, (4.1)
then for anyp0> p,
lim inf
n→∞ φp
0(Ch(2n, n)) > 0. (4.2)
An immediate consequence of the two above statements is the main result of this section:
Corollary 4.3.Forp > ˜pc,
lim inf
n→∞ φp(Ch(2n, n)) > 0.
Let us now focus on the proof of Proposition 4.2, the core of which lies in the following lemma.
Lemma 4.4.Let0 < p1< p2< p3< 1, and suppose that
infφp1(Cv(2n, n)) : n ∈ N > δ > 0.
There exist constantsc0, c1> 0, depending only onp1, p2andp3, such that ifn, I ∈ Nare
such that1 ≤ I ≤ n/400and
I2φp3(Ch(2n, n)) c1/I ≤ c0δ, then φp2(HCh(2n,n/2)) ≥ 2I− 1 2 δ. (4.3)
In [6] it was shown that a similar statement implies Proposition 4.2 (with slightly different formulations). This step adapts readily to the present context, and we do not give more details here; the interested reader is referred to [6, Proof of Prop. 4.1]. The rest of the section is dedicated to proving Lemma 4.4. We start with some notation.
LetA1, . . . , AK be subsets of vertices of some rectangleRofS. For a configuration
ω ∈ {0, 1}S, we say that the subsets are separated in R if Ai ω,R
←−→ Aj fails for all
1 ≤ i < j ≤ K. That is, if they are contained in distinct clusters of the configuration ω restricted to R. We say that the subsetsA1, . . . , AK arestrongly separated inR if
A1, . . . , AK are separated inR. Here we have abusively used the notationAifor the set
Ai= {(u, v) ∈ VS : ∃v0 ∈ VSsuch that(u, v0) ∈ Ai}.
In other words, the sets are strongly separated if there is no open path in the rectangle whose projection onG crosses the projection of two distinct sets.
It is easy to check that, ifωis a configuration containingKstrongly separated vertical crossings of some rectangleR, then
HCh(R)(ω) ≥ K − 1.
Indeed, letA1, . . . , AK be vertical crossings ofR, strongly separated inR in the
con-figurationω. In particularA1, . . . , AK are disjoint connected sets crossingRvertically.
Hence we may order them from left to right – we will assume this is already the case. Fix a self-avoiding pathγcontained inRand crossing it horizontally; orient it from left to right. It intersects each setAi at least once. But for eachi, since Ai andAi+1 are
strongly separated inR,γmust contain at least oneω-closed edge between any point of intersection withAiand the first following intersection withAi+1. This implies thatγ
contains at least one closed edge in the region betweenAi andAi+1for everyi, hence
at leastK − 1closed edges overall. This implies the desired bound.
Lemma 4.4. FixnandIsatisfying the assumptions of the lemma and setv = 100I1 (we will specify the values of the constantsc0andc1later in the proof; it will be apparent
that they do not depend onnandI).
In light of the above observation, to prove Lemma 4.4 we aim to show the existence of2I strongly separated vertical crossings of[0, 2n] × [0, n/2]. The proof follows the lines
of [6, Lemma 4.3] with the essential difference that the crossings need to be strongly separated rather than simply separated.
We start of with series of claims forφp3, similar to those in the proof of [6, Lemma
4.3]. In the present context, the proof of these claims will require the gluing lemma 3.1(i). Once the claims established, we use them to show that, with positiveφp3-probability,
there exist2I separated crossings of[0, 2n] × [0, n/2]. Finally, we deduce that there exist
2I strongly separated crossing with positiveφ
p2-probability, using Lemma 2.2.
In what follows, the constantc > 0is that of (3.2); it only depends onp3andS. Define
α = supnφp3 Ch(d(2 + v)ke, 2k) : k ∈ [ n 8, n 2] o . (4.4)
Claim 4.5.Forαdefined as above, we have α ≤ 1 c h φp3(Ch(2n, n)) iv/28 ≤1 c h φp3(Ch(2n, n)) i2c1/I , (4.5)
where andc1= 1/5600(this is constantc1that appears in Lemma 4.4).
Proof of Claim 4.5. Choosek ∈ [n8,n2]achieving the maximum in (4.4). We will show by induction onj ≥ 1that
φp3
Ch((2 + jv)k, 2k) ≥ (cα)2j.
Applying this toj = 14/v, we obtain
φp3 Ch(2n, n) ≥ φp3 Ch(16k, 2k) ≥ (cα)
(2 + v)k (2 + v)k (2 + (j + 1)v)k 2k (2 + v)k D D0 A0 A1 A2 B1 B2 D D0 A0 B2 B1 A2 A1
Figure 5: Left: The first application of the gluing lemma allows us to obtain the eventH. Middle: The eventH; the vertical crossing is contained in the square to the left.Right: The second application of the gluing lemma allows us to combineH with a horizontal crossing of[0, (2 + jv)k] × [0, 2k]to generate a horizontal crossing of the longer rectangle [0, (2 + (j + 1)v)k] × [0, 2k].
which implies (4.5) readily.
Forj = 1the statement is a direct consequence of the definition ofα. Suppose the statement holds for somej ≥ 1. LetH be the event that there exists an open cluster in[jvk, (2 + (j + 1)v)k] ×[0, 2k]which intersects{(2 + (j + 1)v)k}×[0, 2k]and contains a vertical crossing of the rectangle[jvk, (2 + jv)k] × [0, 2k].
Apply the gluing lemma 3.1(i) with domainsD0 = [jvk, (2 + (j + 1)v)k] × [0, 2k]and D = [jvk, (2 +jv)k] × [0, 2k] for the two events Ch([jvk, (2 + (j + 1)v)k] × [0, 2k]) and
Cv([jvk, (2+jv)k]×[0, 2k])(i.e. forA0= {(2+(j +1)v)k}×[0, 2k],A1= {(2+jv)k}×[0, 2k], A2 = {jvk} × [0, 2k] andB1 = [jvk, (2 + jv)k] × {0}, B2 = [jvk, (2 + jv)k] × {2k}). We obtain that φp3(H) ≥ cφp3 C h([jvk, (2 + (j + 1)v)k] × [0, 2k])φp3 C v([jvk, (2 + jv)k] × [0, 2k]) ≥ cα2.
where the first inequality is the conclusion of the gluing lemma and the second is due to the invariance under translation and rotation.
Apply now the gluing lemma with the domainsD0= [0, (2 + (j + 1)v)k] × [0, 2k]and D = [jvk, (2+ jv)k] × [0, 2k]for the eventsH andCh (2 + (j + 1)v)k, 2k
(i.e. forA0=
{(2 + (j + 1)v)k} × [0, 2k], A1 = [jvk, (2 + jv)k] × {0} A2 = [jvk, (2 + jv)k] × {2k}and
B1 = {0} × [0, 2k], B2 = {(2 + jv)k} × [0, 2k]). The conclusion of the gluing lemma,
together with the bound onφp3(H)and the induction hypothesis, yield
φp3
C
h((2 + (j + 1)v)k, 2k) ≥ cφp3[H]φp3
C
h([0, (2 + jv)k] × [0, 2k]) ≥ (cα)2(j+1).
which is the desired conclusion.
Fix an integer k ∈ [n4,n2] andu ∈ [v, 1/12]such that ku ∈ Z. The following three claims are concerned with crossings of the rectangleR(k) = [−(1 + u)k, (1 + u)k] × [0, 2k]. Claims 4.6-4.9 are equivalent to those used in [6, Proof of lemma 4.3]; Claim 4.10 however is specific to the case of slabs and requires special attention. We give the proof of all claims for completeness.
Claim 4.6.Let E (k) be the event that there exists a vertical open crossing ofR(k), with the lower endpoint not contained in[−3uk, 3uk] × {0}, or the higher endpoint not contained in[−3uk, 3uk] × {2k}. Then
Proof of Claim 4.6. Let β be the φp3-probability that there exists a vertical open
crossing of R(k), with the lower endpoint in [−(1 + u)k, −3uk] × {0}. By the defini-tion ofα, the probability of crossing[−(1 + u)k, (1 − 2u)k] × [0, 2k]vertically is at most α. Thus, with probability β − α, there exists a vertical crossing ofR(k)with an end-point in [−(1 + u)k, −3uk] × {0}which intersects the vertical line {(1 − 2u)k} × [0, 2k]. By reflection with respect to {−3uk} × [0, 2k], with probability β − α, there exists an open path in [−(1 + 4u)k, (1 − 5u)k] × [0, 2k], between [−3uk, (1 − 5u)k] × {0} and {−(1 + 4u)k} × [0, 2k].
When combining the two events above using the first part of the gluing lemma, we obtain
φp3 Ch[−(1 + 4u)k, (1 − 2u)k] × [0, 2k] ≥ c(β − α)
2.
The above event has probability less thanα(by definition ofα), henceβ ≤ (α +pα/c). By considering the other possibilities for the lower and higher endpoints, the claim
follows.
Claim 4.7.LetF (k)be the event that there exists a vertical open crossing ofR(k)that does not intersect the vertical line{(1 − 2u)k} × [0, 2k]. Then
φp3(F (k)) ≤ 2α.
Proof of Claim 4.7. Any vertical crossing ofR(k)not intersecting{(1 − 2u)k} × [0, 2k] is either contained in [−(1 + u)k, (1 − 2u)k] × [0, 2k] or in [(1 − 2u)k, (1 + u)k] × [0, 2k]. Both these rectangles are crossed vertically with probability less thanα, and the claim
follows.
Claim 4.8.LetG (k)be the event that there exists an open path inR × [0, (2 − 11u)k] between[−3uk, 3uk] × {0}and the vertical segment{(1 − 2u)k} × [0, (2 − 11u)k]. Then
φp3(G (k)) ≤ α + pα/c.
Proof of Claim 4.8. Letβ = φp3(G (k)). SupposeG (k)occurs and letγbe an open path
inR × [0, (2 − 11u)k]between[−3uk, 3uk] × {0}and{(1 − 2u)k} × [0, (2 − 11u)k]. There are two possibilities forγ. Eitherγcrosses the line{−(1 − 8u)k} × [0, (2 − 11u)k], or it does not.
The first situation arises with probability at most αsince it induces a horizontal crossing of the rectangle[−(1 − 8u)k, (1 − 2u)k] × [0, (2 − 11u)k].
Thus the second situation arises with probability at leastβ − α. Then, by symmetry with respect to {3uk} × R, with probability at least β − αthere exists an open path connecting[3uk, 9uk] × {0}to{−(1 − 8u)k} × [0, (2 − 11u)k]. Hence, by the first part of the gluing lemma,[−(1 − 8u)k, (1 − 2u)k] × [0 , (2 − 11u)k]is crossed horizontally with probability no less thanc (β − α)2. This is less than or equal toαby its definition, and
the claim follows.
In the claims above we have introduced the eventsE (k), F (k)andG (k). In addi-tion, defineG (k)e as the symmetric ofG (k)with respect to the lineR × {k}, i.e. the event that there exists an open path inR × [11uk, 2k]between[−3uk, 3uk] × {2k}and {(1 − 2u)k} × [11uk, 2k]. The bound of Claim 4.8 applies toG (k)e as well.
All four events revolve around the rectangle R(k). In the following, we will use translates of these events (byz ∈G), and we will say for instance thatE (k)occurs in some rectangleR(k) + zifE (k)occurs for the translate of the configuration by−z. Claim 4.9.Except on an eventH (k), with φp3(H (k)) ≤
96
upα/c, any open vertical
crossing ofS(k) = [0, 2n] × [−k, k], contains two separated vertical crossings ofS((1 − 11u)k) = [0, 2n] × [−(1 − 11u)k, (1 −11u)k].
Proof of Claim 4.9. The rectangle[0, 2n] × [−k, k]is the union ofRj= [juk, (2 + (j + 2)u)k] ×[−k, k], for0 ≤ j ≤ J, where J := 1 u( n k − 2) − 2 ≤ 6/u.
LetH (k)be the union of the following events for0 ≤ j ≤ J:
• the rectangle[juk, (2 + (j + 1)u)k] × [−k, k]contains a horizontal open crossing, • E (k)occurs in the rectangleRj,
• F (k)occurs in the rectangleRj,
• at least one ofG (k)andG (k)˜ occurs in the rectangleRj.
Using a simple union bound and the estimates of Claims 4.6-4.8, we obtain
φp3(H (k)) ≤
96pα/c
u . (4.6)
Consider a configuration not in H (k)containing a vertical open crossing γ of S(k). We are now going to explain why such a crossing necessarily contains two separated crossings ofS((1 − 11u)k).
Since none of the rectangles[juk, (2 + (j + 1)u)k] × [−k, k]is crossed horizontally,γ is contained in one of the rectanglesRj. Fix the corresponding indexj. Parametrizeγby
[0, 1], withγ0being the lower endpoint.
Since E (k) does not occur inRj, γ0 and γ1, are contained in [(1 + (j − 2)u)k, (1+
(j + 4)u)k] × {−k} and [(1 + (j − 2)u)k, (1 + (j + 4)u)k] × {k}, respectively. Moreover, sinceF (k)does not occur inRj,γ crosses the vertical line{(2 + (j − 1)u)k} × [−k, k].
Lettandsbe the first and last times thatγintersects this vertical line.
Since G (k)does not occur inRj, γ intersects the line[0, 2n] × {(1 − 11u)k}before
timet. Likewise, sinceG (k)˜ does not occur,γintersects the line[0, 2n] × {−(1 − 11u)k} after times. This implies thatγcontains at least two disjoint crossings ofS((1 − 11u)k). Callγ1the first one (in the order given byγ) andγ2the last one.
The above holds for any vertical crossing γ of S(k), hence the crossings γ1 and γ2 are necessarily separated inS((1 − 11u)k). Indeed, if they were connected inside
S((1 − 11u)k), thenF (k)would occur.
Claim 4.10.LetI (k)be the event that there exists an open vertical crossing ofS(k), which does not contain two strongly separated vertical crossings ofS((1 − 11u)k). Then
φp2(I (k)) ≤
C0 u
√ α, whereC0 > 0is a constant depending only onp2andp3.
Proof of Claim 4.10. Let ω ∈ I (k) \ H (k) and γ be an ω-open vertical crossing of S(k)which does not contain two strongly separated vertical crossings ofS((1 − 11u)k). Letγ1be the first subpath ofγcrossingS((1 − 11u)k)vertically, and letγ2be the last
(whenγis oriented from bottom to top). By choice ofω inI (k) \ H (k),γ1andγ2are
separated inS((1 − 11u)k), but not strongly separated. Hence there exists a third open pathχinS((1 − 11u)k)that overlaps with bothγ1andγ2. Fix such a pathχand overlap
pointsu, v ∈G betweenχandγ1andγ2, respectively. Then, ifω0 is the configuration obtained formωby opening all the edges in{u}and{v}, we haveω0 ∈H (k). Indeed,
theω0-open path obtained by followingγ1up tou, thenχtovand finallyγ2fromvto the
top ofS((1 − 11u)k)crossesS((1 − 11u)k)vertically, but does not contain two sub-paths separated inS((1 − 11u)k). Thus
and consequentlyI (k) ⊂ {HH (k)≤ 2|ES|}. Lemma 2.2 implies that
φp2 I (k) ≤ φp2 HH (k)≤ 2|ES| ≤ C
2|ES|φ
p3 H (k),
whereC = q2(1−p2)
(p3−p2)[p2+q(1−p2)].By inserting the bound (4.6) onφp3(H (k))into the above,
we obtain the desired result withC0=96C√2|ES |
c .
Getting back to the proof of the lemma. Letki = b(1 − 22vi)n/2cfor0 ≤ i ≤ I. We
will investigate vertical crossings of the nested stripsS(ki) = [0, 2n] × [−ki, ki]. Note that
S(k0)is contained in a translation of the rectangle[0, 2n] × [0, n], and thatS(kI)contains
a translation of the rectangle[0, 2n] × [0, n/2].
Fix a sequence(ui)i, withui∈ [v, 2v]andkiui∈ Zfor0 ≤ i < I. The existence ofui
is due to the fact thatv ≥ n4 (sinceI ≤ n/400). Define the eventsI (ki)of Claim 4.10
for these values ofui. Except on the eventS I−1
i=0I (ki), any configuration with a vertical
crossing ofS(k0)has2I strongly separated vertical open crossings ofS(kI).
By the union bound, claims 4.5 and 4.10 and the definitions ofuandv, we obtain
φp2 I−1 [ i=0 I (ki) ! ≤C 0√α u I ≤ 100C0I2 √ c φp3(Ch(2n, n)) c1/I ≤100C 0c 0 √ c δ, where the last inequality is due to the choice ofI. We may choosec0=
√
c/(200C0) > 0, so that the right-hand side is smaller thanδ/2. But S(k0) is crossed vertically with
φp2-probability at least δ, hence, with φp2-probability at least δ/2, S(kI) contains 2
I
strongly separated vertical crossings. By the observations made before the proof, we have φp2HCh(2n,n/2)≥ 2 I − 1 ≥ φ p2HCh(S(kI)) ≥ 2 I− 1 ≥ δ 2, which directly implies the desired result.
5
Proof of Theorem 1.1
The previous section showed that for p > ˜pc, crossing probabilities in the hard
direction for2n × nrectangles are bounded away from0, uniformly inn. The following two results show us that these probabilities actually tend rapidly to1asn → ∞, for any p > ˜pc.
We start with a lemma taken from [6, Cor. 5.2] and which is valid in all dimensions. It is an integrated form of the result of [11]. We do not give the proof here, as it is identical to the one in [6].
Lemma 5.1.For any0 < p < p0< 1, there existsc = c(p) > 0such that, forn ≥ 1, φp(Ch(2n, n)) 1 − φp0(Ch(2n, n)) ≤ φp0(0 ↔ ∂Λn)
c(p0−p)
. (5.1) The above lemma, along with Proposition 4.2, imply that, forp ∈ (˜pc, pc)(if such ap
exists),limnφp(Ch(2n, n)) = 1. The following proposition tells us that, for such a value of
p,φp(Ch(2n, n))actually converges to1faster than any polynomial.
Proposition 5.2.Fixp < p0and∆ > 0. Suppose thatlimn→∞φp(Ch(2n, n)) = 1. Then,
fornsufficiently large,
φp0(Ch(2n, n)) ≥ 1 − n−∆.
Lemma 5.3.There existsβ > 0such that, for anyp < p0 andN > n, 1 − φp0(Ch(2N, N )) 1 − φp(Ch(2N, N )) ≤ exp −2(p0− p)N n φp(Ch(2n, n))2N/n− 2 N n 1 − φp(Ch(2n, n)) β . Lemma 5.3. We prove this lemma by bounding the expected (underφp) Hamming
dis-tance to the decreasing eventCh(2N, N )cand applying (2.2).
The Hamming distance to Ch(2N, N )c is clearly larger or equal to the number of
edge-disjoint horizontal crossings of[0, 2N ] × [0, N ]. Thus
φp(HCh(2N,N )c) = φp number of disjoint horizontal crossing of[0, 2N ] × [0, N ]
≥ N n
φp(Ch(2N, n)).
The second inequality is due to the fact that the horizontal crossings of rectangles[0, 2N ] ×[in, (i + 1) n]for0 ≤ i < bN/ncare disjoint and to the invariance of the measure under translation.
Let us now boundφp(Ch(2N, n))from below. By the same induction as in Claim 4.5,
using the quantitative gluing lemma 3.1(ii)2dN
netimes, we obtain φp Ch(2N, n) ≥ φp Ch(2n, n) 2N/n − 2 N n 1 − φp(Ch(2n, n)) β , (5.2)
Whereβ > 0is given by Lemma 3.1(ii). Using (2.2) and the fact that dN ne ≤
N 2n, the
lemma follows.
Proposition 5.2. Fixp < p0and∆ > 0as in the proposition. Fixε > 0such thatp + < p0. We first introduce two increasing sequences(nk)k≥k0 ∈ Nand(pk)k≥k0∈ [p, p
0]such that
φpk(Ch(2nk, nk)) > 1 − e
−2k
,
(The indices start fromk0only for a mater of a more clear notation.)
Fork ≥ 1, setv(k) = (1 − e−2k)2·4k− 2 · 4ke−β2k
. The sequencev(k)tends to1ask tends to infinity, so we may fix an indexk0such thatv(k) > 1/2for allk ≥ k0. Setpk0 = p
and choosenk0 ∈ Nsuch thatφp(Ch(2n, n)) > 1 − e
−2k0
for alln ≥ nk0(the choice ofnk0
is possible by hypothesis). Now define, fork ≥ k0,
nk+1= nk4k,
pk+1= pk+
2k−1.
We will now prove by induction thatφpk(Ch(2nk, nk)) ≥ 1 − e
−2k
for all k ≥ k0. The
statement is true fork0by choice ofn0. Suppose it is true for somek ≥ k0. Then, based
on the Lemma 5.3, 1 − φpk+1(Ch(2nk+1, nk+1)) ≤ exp −2(pk+1− pk) nk+1 nk φpk(Ch(2nk, nk)) 2nk+1 nk − 2 nk+1 nk 1 − φpk(Ch(2nk, nk)) β ≤ exph− 2 2k−14 kv(k)i≤ e−2k+1 , and the induction is complete.
By monotonicity ofp 7→ φp, we deduce that1 − φp0(Ch(2nk, nk)) ≤ e−2 k for allk ≥ k0. Since nk = nk04 k0+···+(k−1) ≤ n k04
k2, it follows that for all k ≥ k
0 sufficiently large
n∆ k ≤ e
2k
, hence1 − φp0(Ch(2nk, nk)) ≤ n−∆
k , which is the desired statement forn = nk.
It remains to prove the statement for values ofnin between the scales(nk)k≥k0. Fix
nsuch thatnk< n < nk+1for somek ≥ k0. Based on (5.2), we have
φp0(Ch(2n, n)) ≥ φp0(Ch(2nk+1, nk)) ≥ v(k).
In order to obtain the desired result, it suffices to show thatv(k) > 1 − n−∆k+1> 1 − n−∆for
ksufficiently large. Recall thatnk ≤ nk04
k2. As a consequence2 · 4ke−β2k
≤ 1 2(nk04
k2)−∆
for sufficiently largek. Moreover, we have
1 − e−2k2·4 k ≥ exp−2 · 4 k e2k ≥ exp−1 2 nk04 k2−∆ ≥ 1 −1 2 nk04 k2−∆
for sufficiently largek. The first and last inequality are due to the fact that, for sufficiently smallx, we havee−x≥ 1 − x ≥ e−2x; the second inequality comes from direct asymptotic
estimates. Hencev(k) > 1 − (nk04
k2)−∆> 1 − n−∆
k+1> 1 − n
−∆forklarge enough.
We are finally ready to put the different elements together to prove our main result. Theorem 1.1 . Recall the definitions ofpcandp˜cand that we are aiming to provepc ≤ ˜pc.
We proceed by contradiction and assumepc> ˜pc. Then there exist parametersp˜c< p0<
p1< p2< pc. Corollary 4.3 implies that{φp0(Ch(2n, n)) : n ≥ 1}is bounded away from0.
Sincep1< pc,φp1(0 ↔ ∂Λn) → 0asn → ∞, and Lemma 5.1 yields
φp1 Ch(2n, n) −−−−→n→∞ 1.
Proposition 5.2 with∆ = 1implies that there existsn0such that for alln > n0we have
φp2 Ch(2n, n) ≥ 1 −
1
n. (5.3)
Recall the exponentβ appearing in the second part of the gluing lemma (βmay be taken small with no loss of generality; will assumeβ < 1for computational purposes). Now choosen1> n0such that
2−n1+ 4 X
k≥n1
2−β2k < 1.
Forn ≥ n1, letHnbe the event that[0, 2n1] × {0}is connected to[0, 2n] × {2n}inside
the domain[0, 2n]2. That is
Hn = [0, 2n1] × {0} [0,2n]2
←−−−→ [0, 2n] × {2n}.
Let us estimate the difference between φp2(Hn+1) and φp2(Hn)for some n ≥ n1.
Fix such a value n and let A be the event that there exists an open crossing in [0, 2n+1] × [0, 2n]between{0} × [0, 2n]and{2n+1} × [0, 2n]that is connected in[0, 2n+1]2
to[0, 2n+1] × {2n+1}. (See the left of Figure 6 for an illustration.) The second part of the
gluing lemma and the estimate (5.3) imply that φp2(A ) ≥ (1 − 2
−(n+1))2− 2−β(n+1)≥ 1 − 2−n− 2−β(n+1)≥ 1 − 2 · 2−βn.
We may now apply the gluing lemma using the events A and Hn. That is, apply it
with D = [0, 2n+1] × [0, 2n], D0 = [0, 2n+1]2, A
2n+1
Figure 6:Left: the eventA. Right: the gluing ofA etHnto obtainHn+1.
A2= {2n+1} × [0, 2n],B1= [0, 2n1] × {0}andB2= [0, 2n] × {2n}. Then we obtain
φp2 Hn+1 = φp2 B1 D0 ←→ A0 ≥ φp2(Hn)φp2(A ) − 1 − φp2(A ) β ≥ φp2(Hn) − 2 · 1 − φp2(A ) β ≥ φp2(Hn) − 4 · 2 −β2n . Finally, as a consequence of (5.3),φp2(Hn1) > 1 − 2
−n1. We may therefore deduce that,
for alln ≥ n1, φp2(Hn) ≥ φp2(Hn1) − 4 X n1≤k<n 2−β2k ≥ 1 − 2−n1− 4 X k≥n1 2−β2k> 0,
due to our choice ofn1. Observe now that this implies
φp2 [0, 2
n1] × {0} ↔ ∞ = lim
n→∞φp2 [0, 2
n1] × {0} ↔ ∂Λ2n ≥ lim
n→∞φp2(Hn) > 0. (5.4)
By finite energy property and the FKG inequality, we deduce that the origin belongs to an infinite open cluster with positiveφp2-probability. This contradicts the choice ofp2,
and the theorem is proved.
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