Sketch of proof for Propositions 3.15 and 3.16

The proofs of Propositions 3.15 and3.16 are very similar to those of Propositions 6.4 and 6.8 in [GM14], with only minor differences. Nevertheless, we sketch them below for com-pleteness. The estimates in the proofs are specific to the random-cluster model and will be important in Chapter4.

We keep the notationsGmixandGemixintroduced in the previous section.

Proof of Proposition3.15. We adapt the proof from Proposition 6.4 (more precisely, Lemma 6.7) of [GM14] to our case.

Recall the definition ofΣ^↓, the sequence of star-triangle transformations to consider here:

above the base level, we push down tracks ofG⁽²⁾below those ofG⁽¹⁾one by one, from the bottom-most to the top-most; below the base level, we proceed symmetrically. LetPbe a probability measure defined as follows. Pick a configurationωonGmix according toϕ_Gmix

; apply the sequence of star-triangle transformations Σ^↓ to it using the coupling described in Figure2.8, where the randomness potentially appearing in each transformation is inde-pendent ofωand of all other transformations. Thus, underPwe dispose of configurations on all intermediate graphs in the transformation fromGmix to Gemix. Moreover, in light of Proposition2.12,Σ^↓(ω)has lawϕ_Ge

mix

. We will prove the following statement

P

hΣ^↓(ω)∈ C(ρinn,(ρout+λ)n;λn)

ω∈ C(ρinn, ρoutn;n)i

>1−ρoute⁻ⁿ, (3.7) for any valuesρout> ρin>0,n>n₀,M >(ρout+λ)nandN >λn, whereλ, n₀>1will be chosen below. This readily implies Proposition3.15.

Fixρout, ρin, n, MandN as above. Chooseω₀∈ C(ρinn, ρoutn;n)and letγbe anω₀-open circuit as in the definition ofC(ρinn, ρoutn;n). As the transformations ofΣ^↓=σ_K◦ · · · ◦σ₁ are applied toω₀, the circuit γ is transformed along with ω₀. Thus, for each 06k 6K, (σ_k◦ · · · ◦σ₁)(γ)is an open path in(σ_k◦ · · · ◦σ₁)(ω₀)on the graph(σ_k◦ · · · ◦σ₁)(Gmix).

Since no star-triangle transformation ofΣ^↓affects the base,Σ^↓(γ)remains a circuit sur-rounding the segment of the base betweenx−ρinn,0andx_ρinn,0. Therefore, the only thing that is left to prove is that

P

hΣ^↓(γ)∈R

(ρout+λ)n;λn

ω=ω₀i

>1−ρoute⁻ⁿ. (3.8)

3.3. Proofs for16q64 Set

γ^(k)= (Σ^↓_N

1+k◦Σ^↓_−(N

1+k))◦ · · · ◦(Σ^↓_N

1+1◦Σ^↓_−(N

1+1)), whereΣ^↓_i =Σ_t0,ti◦ · · · ◦Σ_tN

1,t_i fori>0andΣ^↓_i =Σ_t−1,ti◦ · · · ◦Σ_t−N

1,t_ifori <0. The pathγ^(k) thus defined is the transformation ofγ after the firstktracks ofG⁽²⁾ above the base were sent down, and the symmetric procedure was applied below the base.

In [GM14], the vertices ofGmix visited byγ^(k) were shown to be contained in a region whose evolution withk= 0, . . . , N₂+ 1is explicit. This is done separately above and below the base level, and we focus next on the upper half-space.

LetH⁰={(i, j)∈Z×N:−(ρout+ 1)n6i−jandi+j6(ρout+ 1)nandj6n}. Then, H^k+1is defined fromH^kas follows. If(i, j)∈Z×Nis such that(i, j),(i−1, j)or(i+ 1, j) are inH^k, then(i, j)∈H^k+1. Otherwise, if(i, j−1)∈H^k, then(i, j)is included inH^kwith probabilityη∈(0,1), independently of all previous choices. We will see later how the value ofηis chosen using the bounded-angles property.

In consequence, the sets(H^k)₀6k6N are interpreted as a growing pile of sand, with a number of particles above everyi∈Z. At each stage of the evolution, the pile grows laterally by one unit in each direction; additionally, each column of the pile may increase vertically by one unit with probabilityη(see Figure3.11).

Figure 3.11 – One step of the evolution ofH: H^k is drawn in black, H^k+1contains the ad-ditional blue points (since they are to the left or right of vertices inH^k) and the red points (these are added due to the random increases in height).

Loosely speaking, [GM14, Lem. 6.6] shows that, ifηis chosen well, then all verticesx_i,j visited byγ^(k) have(i, j) ∈H^{k 2}. More precisely, the process(H^k)₀6k6N may be coupled with the evolution of(γ^(k))₀6k6N so that the above is true. This step is proved by induction onk, and relies solely on the independence of the star-triangle transformations and on the estimates of Figure3.5. Then, (3.8) is implied by the following bound on the maximal height ofH^N:

P

hmax{j: (i, j)∈H^λn}>λni

< ρoute⁻ⁿ (3.9) for someλ >0and allnlarge enough. The existence of such a (finite) constantλis guar-anteed by [GM13a, Lem. 3.11]. It depends onη, and precisely on the fact thatη <1[GM14, Lem. 6.7]. The choice ofη <1that allows the domination of(γ^(k))₀6k6N by(H^k)₀6k6N is done as follows.

We proceed in the same way as in the proofs of Lemmas 6.6 and 6.7 of [GM14]. We shall analyze the increase in height of portions ofγ^(k)as given by Figure3.5. Essentially, the only cases in whichγ^(k)increases significantly in height are depicted in the third and the last line of Figure3.5.

2This is not actually true, since there is a horizontal shift to be taken into account; let us ignore this technical detail here.

3. Universality of the random-cluster model

A t B

t^′

γ^(k)

Figure 3.12 – Star-triangle transformations between trackstandt⁰corresponding to the third line of Figure3.5. The trackstandt⁰have transverse anglesAandBrespectively. We assume that a portion of the pathγ^(k)reaches between the trackst andt⁰ as shown in the figure.

Moreover, if the dashed edge is open on the left, with probabilityη_A,B=y_π−Ay_π−(B−A)/q, the pathγ^(k)drifts upwards by 1 after the track exchange.

Let us examine the situation which appears in the third line of Figure 3.5and consider the notations as in Figure3.12. Using the notation of Figure3.12for the anglesAandB, the probability that the height of such aγ^(k)increases by 1 is given by

η_A,B=y_π−Ay_π−(B−A)

q = sin(rA) sin(r(B−A)) sin(r(π−A)) sin(r(π−(B−A)))

= cos(r(2A−B))−cos(rB) cos(r(2A−B))−cos(r(2π−B)), where we recall thatr= cos⁻¹(

√q 2 )6 ¹

3and that, due to theBAP(ε),A, B∈[ε, π−ε]. The same computation also applies to the last line of Figure3.5. Then,ηmay be chosen as

η:= sup

A,B∈[ε,π−ε]

η_A,B<1. (3.10)

The domination of the set of vertices of γ^(k) by H^k is therefore valid for this value of η, and (3.8) is proved for the resulting constantλ.

Remark 3.17. When we deal with the quantum model in Chapter4, it will be important to have a more precise estimate onη(ε). In particular, we will show that, in this special case, 1−η(ε)∼τ(q)εasε→0for some constantτ:=τ(q)depending only onq∈[1,4].

Proof of Proposition3.16. We adapt Proposition 6.8 of [GM14] to our case. FixnandN , M>

2n, and consider the graphGmix as described in the previous section. We recall the defini-tion ofΣ^↑, the sequence of star-triangle transformations we consider here: above the base level, we pull up tracks ofG⁽¹⁾ above those of G⁽²⁾ one by one, from the top-most to the bottom-most; below the base level, we proceed symmetrically.

As in the previous proof, write Pfor the measure taking into account the choice of a configurationω₀according to the random-cluster measureϕ_Gmixas well as the results of the star-triangle transformations inΣ^↑applied to the configurationω₀.

The events we are interested in only depend on the graph above the base level, hence we are not concerned with what happens below. For 06i 6N, recall from Section 3.2.1the

3.3. Proofs for16q64 notation

Σ^↑_i =Σ_ti,t2N+1◦ · · · ◦Σ_ti,tN+1,

for the sequence of star-triangle transformations moving the trackt_iofG⁽¹⁾aboveG⁽²⁾. Then Σ^↑=Σ^↑₀◦ · · · ◦Σ^↑_N.

First, note that ifω∈ C_v(n;n), we also haveΣ^↑_n+1◦ · · · ◦Σ^↑_N(ω)∈ C_v(n;n), since the two configurations are identical between the base andt_n. We will now write, for06k6n+ 1,

G^k=Σ^↑_n−k+1◦ · · · ◦Σ^↑_N(Gmix), ω^k=Σ^↑_n−k+1◦ · · · ◦Σ^↑_N(ω),

D^k={x_u,v∈G^k:|u|6n+ 2k+v,06v6N+n}, h^k= sup{h6N :∃u, v∈Zwithx_u,0 ^D

k,ω^k

←−−−→x_v,h}.

That is,h^kis the highest level that may be reached by anω^k-open path lying in the trape-zoidD^k. We note thatGⁿ⁺¹=Gemixandωⁿ⁺¹follows the law ofϕ_Ge

mix

.

With these notations, in order to prove Proposition3.16, it suffices to show the equivalent of [GM14, (6.23)], that is

P

hhⁿ⁺¹>δni

>c_nP

hh⁰>ni

, (3.11)

for someδ ∈(0,¹₂) to be specified below and explicit constantsc_n withc_n →1asn→ ∞.

Indeed,

P[h⁰>n]>P[ω⁰∈ C_v(n;n)] =ϕ_Gmix[C_v(n;n)].

Moreover, ifhⁿ⁺¹>δn, thenωⁿ⁺¹∈ C_v(4n;δn), and therefore we haveϕ_Ge

mix[C_v(4n;δn)]>

P(hⁿ⁺¹>δn).

We may now focus on proving (3.11). To do that, we adapt the corresponding step of [GM14] (namely Lemma 6.9). It shows that(h^k)₀6k6ncan be bounded stochastically from below by the Markov process(H^k)₀6k6n3

given by H^k=H⁰+

Xk

i=1

∆_i, (3.12)

whereH⁰=nand the∆_iare independent random variables with common distribution P(∆= 0) = 2δ, P(∆=−1) = 1−2δ, (3.13) for some parameterδto be specified later. Once the above domination is proved, the inequal-ity (3.11) follows by the law of large numbers.

The proof of (3.13) in [GM14] (see Equation (6.24) there) uses only the independence between different star-triangle transformations and the finite-energy property of the model.

Both are valid in our setting. We sketch this below.

Fix06k6n and let us analyse the(N−(n−k) + 1)^th step ofΣ^↑, that isΣ^↑_n−k. Write Ψ_j :=Σ_tn−k,tN+j◦· · ·◦Σ_tn−k,tN+1for06j6N. In other words,Ψ_jis the sequence of star-triangle transformations that applies toG^kand moves the trackt_n−kabovejtracks ofG⁽²⁾, namely t_N+1, . . . , t_N+j. Moreover,Ψ_N =Σ^↑_n−k; hence,Ψ_N(G^k) =G^k+1.

3To be precise, it is shown that for anyk, the law ofh_kdominates that ofH^k. It is not true that the law of the whole process(h^k)₀6k6ndominates that of(H^k)₀6k6n. This step uses [GM13a, Lem. 3.7].

3. Universality of the random-cluster model

N+n

0

n−k+j=: ˜j

2(n+ 2k) 2(2(n+k) +N)

D^k_j

Dk+1

Dk

Figure 3.13 – The evolution ofD^k(red) toD^k+1(blue) via intermediate stepsD_j^k (black).

LetD_j^kbe the subgraph ofΨ_j(G^k)induced by verticesx_u,vwith06v6N+nand

|u|6











n+ 2k+v+ 2 ifv6ej, n+ 2k+v+ 1 ifv=ej+ 1, n+ 2k+v ifv>ej+ 2.

,

where we letej =n−k+j. Note thatD^k ⊆D₀^k ⊆ · · · ⊆D_N^k ⊆D^k+1, see Figure3.13 for an illustration. Letω^k_j =Ψ_j(ω^k). Ifγ is aω^k_j-open path living inD_j^k, thenΣ_t

n−k,t_N+j+1(γ)is a ω_j+1^k -open path living inD_j+1^k . This is a consequence after a careful inspection of Figure3.5, where blue points indicate possible horizontal drifts. Define also

h^k_j = sup{h6N :∃u, v∈Zwithx_u,0 ^D

k j,ω^k_j

←−−−→x_v,h}.

Then,h^k6h^k₀andh^k_n6h^k+1. As in [GM14], we need to prove that for06j6N−1,

h^k_j+1=h^k_j ifh^k_j ,ej,ej+ 1, (3.14)

h^k_j+1−h^k_j = 0or1 ifh^k_j =ej, (3.15)

h^k_j+1−h^k_j =−1or0 ifh^k_j =ej+ 1, (3.16) P(h^k_j+1>h|h^k_j =h)>2δ ifh=ej+ 1. (3.17) The four equations above imply the existence of a processH^k as in (3.13).

As explained in [GM14], (3.14), (3.15) and (3.16) are clear because the upper endpoint of a path is affected byΣ:=Σ_{tn−k,tN+j+1} only if it is at heightejorej+ 1. And the behavior of the upper endpoint can be analyzed using Figure3.6. More precisely,

• when it is at heightej+ 1, the upper endpoint either stays at the same level or drifts downwards by 1;

• when it is at heightej, it either stays at the same level or drifts upwards by 1.

Hence, the rest of the proof is dedicated to showing (3.17).

We start with a preliminary computation. Fixjand letP_j be the set of pathsγofΨ_j(G^k), contained inD_j^k, with one endpoint at height0, the other at heighth(γ), and all other vertices with heights between1andh(γ)−1.

Assume that inΣ, the additional rhombus is slid from left to right and defineΓ to be the left-most path ofP_j reaching heighth^k_j ⁴. (Such a path exists due to the definition ofh^k_j.) This

4OtherwiseΓ should be taken right-most.

3.3. Proofs for16q64 choice is relevant since later on, we will need to use negative information in the region on the left of the pathγ. Moreover, forγ, γ⁰ ∈ P_j, we writeγ⁰ < γifγ⁰ ,γ,h(γ⁰) =h(γ)and γ⁰does not contain any edge strictly to the right ofγ.

Denote byΓ =Γ(ω^k_j)the ω^k_j-open path of P_j that is the minimal element of {γ ∈ P_j : h(γ) =h^k_j, γ isω^k_j-open}. Given a pathγ∈ P_j, we can write{Γ =γ}={γisω^k_j-open} ∩N_γ whereN_γ is the decreasing event that

(a) there is no γ⁰ ∈ P_j withh(γ⁰) > h(γ), all of whose edges not belonging to γ are ω^k_j-open;

(b) there is no γ⁰ < γ with h(γ⁰) = h(γ), all of whose edges not belonging to γ are ω^k_j-open.

LetF be a set of edges disjoint fromγ, writeC_Ffor the event that all the edges inFare closed. We find,

P[C_F|Γ =γ] =

P[N_γ∩C_F|γis open] P[N_γ|γis open]

>P[C_F|γis open]

>ϕ_K¹[C_F], (3.18)

where the second line is given by the FKG inequality due to the fact thatP[· |γis open]is still a random-cluster measure and bothN_γ andC_F are decreasing events. And in the last line, we compare the boundary conditions, whereK is the subgraph consisting of rhombi containing the edges ofF.

z z z

e1

e2

e3 e4 z⁰ z⁰ z⁰ z⁰

e5

1 1

ye1ye4

q

Figure 3.14 – Three star-triangle transformations contributing toΣslid the gray rhombus from left to right. The dashed edges are closed, the bold edges are open and the state of dotted edgee₂does not really matter. The first and last passages occur with probability1, and the second with probabilityy_e1y_e4/q.

Now, we are ready to show (3.17). Letγ∈ P_j withh(γ) =ej+ 1and assumeΓ(ω_j^k) =γ. Now, it is enough to show that

P

hh(Σ(γ))>ej+ 1 Γ =γi

>2δ. (3.19)

Letz=x_u,ej+1denote the upper endpoint ofγand letz⁰denote the other endpoint of the unique edge ofγleading toz. Eitherz⁰=x_u+1,ej orz⁰=x_u−1,ej. In the second case, it is always the case thath(Σ(γ))>ej+ 1.

Assume thatz⁰=x_u+1,ej as in Figure3.14and consider edgese_ifori= 1, . . . ,4as follow, e₁=hx_u,ej+1, x_u−1,ej+2i, e₂=hx_u−1,ej+2, x_u−2,ej+1i,

e₃=hx_u−2,ej+1, x_u−1,eji, e₄=hx_u−1,ej, x_u,ej+1i.

Let us now analyse the star-triangle transformations that affecte₁, . . . , e₄; these are depicted in Figure3.14. We note that conditioning on the eventC_F∩ {Γ =γ}, whereF={e₃, e₄}, we have:

3. Universality of the random-cluster model

(a) The edgee₁must be closed due to the conditioning{Γ =γ}.

(b) Whichever the state ofe₂is, the edgee₅is always closed.

(c) The second passage occurs with probabilityy_e1y_e4/q. (d) The third passage is deterministic.

Thus,

P

hh(Σ)>ej+ 1 Γ =γi

>y_e1y_e4

q ·P[C_F|Γ =γ].

Moreover, the preliminary computation (3.18) gives that

P[C_F|Γ =γ]>ϕ_K¹[C_F] = (1−p_e3)(1−p_e4),

whereKconsists only of the two rhombi containinge₃ande₄and we use the fact that in the random-cluster measureϕ¹_K, these edges are independent (the number of clusters is always equal to 1). Finally,

P

hh(Σ)>ej+ 1 Γ =γi

>y_e1p_e4(1−p_e3)

q >y_π−εp_π−ε(1−p_ε)

q >2δ, (3.20)

where

δ=1 2min

(y_π−εp_π−ε(1−p_ε)

q ,1

)

>0. (3.21)

To conclude, we have P[hⁿ>δn]

P[h⁰>n] >P[Hⁿ>δn]

P[H⁰>n] >P[Hⁿ>δn|H⁰>n] =:c_n(δ),

and sinceHⁿ/n→2δ asn→ ∞due to the law of large numbers, we know thatc_n→1as n→ ∞.

3.3.3 Doubly-periodic isoradial graphs

Now that the RSW property is proved for isoradial square lattices, we transfer it to arbitrary doubly-periodic isoradial graphsG. We do this by transforming a finite part ofG(as large as we want) into a local isoradial square lattice using star-triangle transformations. The approach is based on the combinatorial result Proposition3.9.

Proposition 3.18. Any doubly-periodic isoradial graphGsatisfies the RSW property.

Proof. LetGbe a doubly-periodic isoradial graph with grid(s_n)_n∈Z and(t_n)_n∈Z. Fix a con-stantd >1as given by Proposition3.9applied toG. In the below formula,C^hp

h (n;n)denotes the horizontal crossing event in the half-plane rectangular domainR^hp(n;n) :=R(−n, n; 0, n). We will show that

ϕ⁰_Λ(6dn)h C^hp

h (n;n)i

=ϕ⁰_Λ(6dn)h

C_h(−n, n; 0, n)i

>δ, (3.22)

for some constantδ >0which does not depend onn. Moreover, a careful inspection of the forthcoming proof shows thatδ only depends the bounded angles parameterε >0and on

3.3. Proofs for16q64 the size of the fundamental domain ofG. The same estimate is valid for the dual model, since it is also a random-cluster model on an isoradial graph withβ= 1.

The two families of tracks(s_n)_n∈Zand(t_n)_n∈Zplay symmetric roles, therefore (3.22) may also be written

ϕ⁰_Λ(6dn)h

C_v(0, n;−n, n)i

>δ. (3.23)

The two inequalities (3.22) and (3.23) together with their dual counterparts imply the RSW property by Lemma3.10⁵.

The rest of the proof is dedicated to (3.22). In proving (3.22), we will assumento be larger than some threshold depending onGonly; this is not a restrictive hypothesis.

Let(σ_k)₁6k6K be a sequence of star-triangle transformations as in Proposition3.9such that ineG:= (σ_K◦ · · · ◦σ₁)(G), the region enclosed bys−4n,s_4n,t−2n andt_2n has a square lattice structure. Recall that all the transformationsσ_k act horizontally betweens−6dn and s_6dnand vertically betweent_nandt−n.

Consider the following events foreG. LetCebe the event that there exists an open circuit contained in the region betweens−2n ands_2n and betweent−n/2 andt_n/2 surrounding the segment of the base betweens−nands_n. LetCe^∗be the event that there exists an open circuit contained in the region betweens−3nands_3nand betweent−3n/2andt_3n/2surrounding the segment of the base betweens−2nands_2n.

LetGebe the subgraph ofeGcontained betweens−4n,s_4n,t−2nandt_2n. ThenGeis a finite section of a square lattice with4n+ 1horizontal tracks, but potentially more than 8n+ 1 vertical ones. Indeed, any track ofGthat intersects the base betweens−4n,s_4nis transformed into a vertical track ofGe.

Write(es_n)_n∈Zfor the vertical tracks ofGe, with

es₀coinciding withs₀(this is coherent with the notation in the proof of Proposition3.9). Then, the tracks(s_n)_n∈Zare a periodic subset of(es_n)_n∈Z, with period bounded by the number of tracks intersecting a fundamental domain ofG. It follows that there exist constantsa, bdepending only onG, not onn, such that the number of tracks(es_n)_n∈Zbetween any two trackss_iands_j(withi6j) is between(j−i)a−b and(j−i)a+b.

By the above discussion, for some constantc >1andnlarge enough (larger than some n₀depending only onaandb, therefore only on the size of the fundamental domain ofG), the eventsCeandCe^∗may be created using crossing events as follow:

He₁∩He₂∩Ve₁∩Ve₂⊆eC and He^∗ events are shown in Figure3.15.

5The conditions of Lemma3.10differ slightly from (3.22) and (3.23) in the position of the rectangle and the domain where the measure is defined. Getting from one to the other is a standard application of the comparison between boundary conditions.

3. Universality of the random-cluster model

Notice that all events above depend only on the configuration inGe. LetϕG_edenote some infinite-volume measure onG.e By the RSW property for the square latticeGe(that is, by Corollary3.12), the comparison between boundary conditions and the FKG inequality,

ϕeG for someδ₁>0independent ofn. Moreover, by the same reasoning,

ϕ_eG

LetPbe the probability that consists of choosing a configuration

ωeoneGaccording toϕ

eG, then applying the inverse transformationsσ_K⁻¹, . . . , σ₁⁻¹to it. Thus,ω:= (σ₁⁻¹◦ · · · ◦σ_K⁻¹)(eω) is a configuration onGchosen according to some infinite-volume measureϕG.

Let ωe ∈ eC ∩Ce^∗, and write since the transformations do not affect the base,γ^∗ surrounds the segment of the base be-tweens−2nands_2nwhileγsurrounds the segment of the base betweens−nands_nbut only traverses the base betweens−2nands_2n.

WriteCfor the event that a configuration onGhas an open circuit contained inR(6dn; 2n), surrounding the segment of the base betweens−nands_nand traversing the base only between s−2nands_2n. Also, setC^∗to be the event that a configuration onGhas a dually-open circuit contained inR(6dn; 2n), surrounding the segment of the base betweens−2nands_2n.

3.3. Proofs for16q64 BothCandC^∗are reminiscent of the eventseCandCe^∗, in spite of small differences. Indeed, the discussion above shows that if

ωe∈eC ∩Ce^∗, thenω∈ C ∩ C^∗. Thus,

For a configurationωonG, writeΓ^∗(ω)for the exterior-most dually-open circuit as in the definition ofC^∗(that is contained inR(6dn; 2n)and surrounding the segment of the base betweens−2nands_2n), if such a circuit exists. LetInt(Γ^∗)be the region surrounded byΓ^∗, seen as a subgraph ofG.

It is standard thatΓ^∗may be explored from the outside and therefore that, conditionally onΓ^∗, the random-cluster measure inInt(Γ^∗)isϕ_Int(Γ^{0 ∗}₎.

Observe that forω∈ C ∩ C^∗, due to the restrictions over the intersections with the base, any circuit in the definition ofCis surrounded by any in the definition ofC^∗. Thus, if for ω∈ C^∗, the occurence ofConly depends on the configuration insideInt(Γ^∗). Therefore,

ϕG the before last line, we used the fact thatInt(γ^∗)⊆R(6dn; 2n), whereR(6dn; 2n)is defined using tracks inG, and the comparison between boundary conditions to say that the free boundary conditions on∂Int(γ^∗)are less favorable to the increasing eventCthan those on the more distant boundary∂R(6dn; 2n).

Due to the previous bound onϕG

hC ∩ C^∗i

, we deduce that ϕ⁰_R(6dn;2n)h

Ci

>δ₁δ₂.

Finally, notice that any circuit as in the definition of C contains a horizontal crossing of R^hp(n;n). We conclude from the above that

ϕ_R(6dn;2n)⁰ h

R^hp(n;n)i

>δ₁δ₂.

This implies (3.22) by further pushing away the unfavorable boundary conditions.

Dans le document Universality of the random-cluster model and applications to quantum systems (Page 91-100)