Random-cluster model on isoradial graphs

• Ifsintersects with some (all) tracks inTeand some others (all) inTe⁰, keepsapart.

Finally, let

T = [

a,b∈Z

τ_a,bTe and T⁰= [

a,b∈Z

τ_a,bTe⁰. We claim that the families of tracksT andT⁰form a grid ofG.

First, note that in the construction, we can replace “some” by “all” due to Corollary2.3.

Then, by definition, it is easy to see that each translationK_a,bofKintersects at least one track ofT andT⁰(it intersectsτ_a,bet₀andτ_a,bes₀). Let us check that the tracks inT do not intersect each other. If there aret₁, t₂∈ T intersecting each other, then by Corollary2.4, we can find two tracks

et₁and

et₂ both inTe intersecting each other by translation. This contradicts the construction ofTe. The same property holds for the tracks inT⁰.

Let us check that all the tracks ofGnot inT intersect all those ofT. Taket₁ a track not inT andt₂∈ T. Consideret₁andet₂two translations oft₁andt₂respectively such that both go throughK. Hence, by the construction, they should intersect each other, because otherwise they would have been put both intoT. Sot₁andt₂intersect each other as well by Corollary2.4. To show that all tracks ofGnot inT⁰ intersect all those ofT⁰, the proof is the same.

Finally, it is always possible to order tracks inT and those inT⁰according to intersections with some referential trackss₀∈ T⁰ andt₀∈ T. The periodicity of the tracks follows from the periodicity of the graphG.

For the last part of the lemma, we note that in a square lattice, any three different (hor-izontal or vertical) tracks have either0(3 of the same type) or2(2 of one type and 1 of the other type) intersections between them. If there is a grid(s_n)_n∈Zand(t_n)_n∈Znot containing all the tracks ofG. Takeswhich is not one of thes_nort_n. Then, betweens₀,t₀andsthere are three intersections, which is not possible.

To conclude this part about the construction of a grid in a doubly-periodic isoradial graph (Proposition2.5), we note that Lemma2.2provides us with an alternative. Write(ρ_i)₁6i6`the set of (distinct) asymptotic directions seen as unit complex numbers. There is only a finite number of them due to the double periodicity ofG. Denote by T^(ρi)(G)the set of tracks whose asymptotic direction isρ_i. Then it is easy to see that

T(G) = [`

i=1

T^(ρi)(G).

Pick up two distinctρ_iandρ_j, then the familiesT^(ρi)andT^(ρj)form a grid ofG.

2.4 Random-cluster model on isoradial graphs

2.4.1 Definition

Previously, we defined the random-cluster model on any finite graph and extended the def-inition to infinite graphs with specific boundary conditions. Here we define the model on (infinite) isoradial graphs with respect to what we call theisoradial parameters.

Fix the cluster weightq>1and an additional parameterβ >0. We define the family of edge parametersp(β) = (p_e(β))_e∈Evia the relation

p_e(β) = y_e(β) 1 +y_e(β),

2. Isoradial graphs and star-triangle transformation

wherey_e(β)is defined according to the value ofqandθ_e, the subtended angle of the edgee as shown earlier in Figure2.4:

if16q <4, y_e(β) =β√

qsin(r(π−θ_e))

sin(rθ_e) , wherer= 1 πcos⁻¹

√q 2

; ifq= 4, y_e(β) =β2(π−θ_e)

θ_e ; ifq >4, y_e(β) =β√

qsinh(r(π−θ_e))

sinh(rθ_e) , wherer= 1

πcosh⁻¹

√q 2

. (2.2)

The parameterspdefined by these formulae are calledisoradial probability parameters. The associated infinite-volume random-cluster measure will be denoted by ϕ^ξ_G,β,q, where the boundary conditionsξcan be either0(free) or1(wired), as given in the previous sections.

Proposition 2.6. We have the following duality relations for the random-cluster model:

ϕ_G^ξ_,β,q^(d)=ϕ¹_G^−∗_,1/β,q^ξ , where ξ= 0,1.

In particular, atβ= 1, the random-cluster model is self-dual up to the boundary conditions, that isϕ_G⁰_,1,q^(d)= ϕ¹_G^∗_,1,qandϕ_G¹_,1,q^(d)= ϕ_G^0∗_,1,q.

Proof. According to Proposition1.1, it is enough to check thaty_θ(β)y_π−θ(¹_β) =q. This is immediate by (2.2).

2.4.2 Loop representation

The loop representation was already mentioned in Section1.1.5for any finite graph. Here, we give a more precise description for isoradial graphs.

Given a finite graphG = (V , E) and a random-cluster configurationω ∈ {0,1}^E on it, we connect midpoints of edges inEby avoiding intersection with the edges given byω. In other words, we connect midpoints to separate vertices inV andV^∗and thus create interface between different connected components. Figure2.6illustrates how this is done locally for each rhombus. In consequence, we obtain theloop representationof the model.

θ^∗

θ

sin(rθ) sin(rθ^∗)

Figure 2.6 – The local lines connecting midpoints of edges ofEare drawn in red. On the left, the primal edge is open and on the right, the dual one is open.

2.4. Random-cluster model on isoradial graphs

We continue the calculation from (1.5). We have ϕ^ξ_G,1,q[ω]∝Y

e∈E

y_e y_e^∗

!ω(e)/2

·√ q^lξ(ω)

∝Y

e∈E

sin(rθ^∗) sin(rθ)

ω(e)

·√ q^lξ(ω)

∝Y

e∈E

sin(rθ^∗)^ω(e)sin(rθ)^ω∗(e∗)·√ q^lξ(ω),

whereθ^∗=π−θ, as indicated in Figure2.6. This is theloop representationof the random--cluster model on the isoradial graphG.

We note that when we are on the quare lattice, we have θ = θ∗, which means that ϕG^ξ,1,q[ω]is proportional to

√q^lξ(ω).

2.4.3 Uniqueness of the measure for doubly-periodic isoradial graphs Here, we will determine some conditions under which we can deduce the uniqueness of the infinite-volume random-cluster measure. From what precedes, without any further infor-mation, we could have more than one infinite-volume random-cluster measures. However, using the same techniques as for the square lattice [Gri06], we can show that this measure is actually unique, except for a set of parameters which is at most countable, for any dou-bly-periodic isoradial graph.

Proposition 2.7. There exists a subsetD_q of(0,+∞)at most countable such that ifβ <D_q, thenϕ⁰_G,β,q=ϕ_G¹_,β,q. Thus, the infinite-volume random-cluster measure is unique for values of βoutside of a countable set.

Before showing the proposition, we need to introduce the notion of thefree energy. Proposition2.5provides us with two families of train tracks(s_n)_n∈Zand(t_n)_n∈Z which form a grid. Due to the periodicity of G, there existM, N ∈Nsuch that the regionK = [0, M)×[0, N)is a fundamental domain ofG. Moreover, the number of edges on its boundary can be bounded|∂_EK|6c(M+N)for somec >0.

Fork∈N, letG_k= [−2^kM,2^kM)×[−2^kN ,2^kN)and writeV_k andE_kits vertex set and edge set. Note that|∂G_k| 62^k+1c(M +N)and thatG_k+1contains four copies of G_k with edges connecting them, thus|E_k+1|= 4|E_k|.

Fork∈N,ξ= 0,1andβ >0, define the following quantities Z_k^ξ(β) :=Ze_G^ξ

k,β,q= X

ω∈{0,1}Ek

Y

e∈E_k

y_e(β)^ω(e)q^kξ(ω), (2.3) f_k^ξ(β) = 1

|E_k|lnZ_k^ξ(β). (2.4)

Lemma 2.8. For allβ >0, the limitsf_k⁰(β)andf_k¹(β)exist and coincide whenk→ ∞. Write f(β) = limf_k⁰(β) = limf_k¹(β). Moreoever, the functionβ7→f(β)is convex.

Proof. Cut G_k+1 into four disjoint copies ofG_k, denoted by G¹_k, G²_k,G³_k andG⁴_k and let F be the set of edges connecting boundaries of differentGⁱ_k. We can boundZ_k+1⁰ from below by taking into account only the configurations with all the edges inF closed. Given such

2. Isoradial graphs and star-triangle transformation

Then, the statement of the lemma follows because the pointwise limit of convex functions is still convex. Due to the continuity of the functionsβ 7→f_k^ξ(β) (polynomial inβ), it is sufficient to show that their derivatives are increasing inβ.

We start by writingZ_k^ξ(β)differently, Z_k^ξ(β) = X

where we use the fact that

dπ_e(β)

dβ = ¹_β in the second line. Moreover, since β 7→ p_e(β) is increasing for alle∈E_k, from Proposition1.7, the last term is also increasing inβ. The proof is complete.

Note that here we use the infinite-volume measure ϕ^ξ_β to define h^ξ(β). By translational invariance of the measure by periods ofG, we also have, for allk∈N,

h^ξ(β) = 1

|E_k| X

e∈E_k

ϕ_β^ξ[eis open].

The following proposition gives a criterion in terms ofh^ξ for the uniqueness of the in-finite-volume measure.

Lemma 2.9. Ifh¹(β) =h⁰(β), thenϕ¹_β=ϕ_β⁰and the infinite-volume random-cluster measure is unique.

2.5. Known results Proof. Knowing that for allk >0, the measure ϕ_k,β¹ dominates ϕ_k,β⁰ stochastically. There exists a couplingP_k with marginals ω₀ ∼ ϕ_k,β⁰ , ω₁ ∼ ϕ¹_k,β and P_k(ω₀ 6 ω₁) = 1. Take the weak limits and we have a couplingPwith marginalsω₀∼ϕ⁰_βandω₁∼ϕ_β¹such that P(ω₀6ω₁) = 1.

Then, for an increasing eventAdepending on a finite set of edges, sayF, and forklarge enough such thatF⊂E_k, we have

Thus, we obtain the equalityϕ¹_β(A) =ϕ⁰_β(A)for all increasing eventsAdepending on finitely many edges. From the construction of these two infinite-volume measures as mentioned in Section1.1.8, we conclude thatϕ_β¹andϕ_β⁰are equal.

Now we are ready to prove Proposition2.7.

Proof of Proposition2.7. A standard argument says that a convex function is differentiable at all but countably many points. Moreover, from the limit given by Lemma2.8, atβ >0where f is differentiable, its derivative is given by the limit of(f_k^ξ)⁰(β),ξ= 0or1. From (2.5), also haveh¹(β) =h⁰(β), which implies uniqueness of the infinite-volume measure atβ ac-cording to Lemma2.9. Moreover, the convexity off implies that it is not differentiable at most on a countable set.

2.5 Known results

It is shown in [GM14] that the percolation on isoradial graphs is critical for parameters de-fined in (2.2) atβ= 1. The main idea is based on the star-triangle transformations which are going to be introduced in the next section. In Section3, we generalize this method to the random-cluster model with cluster-weightq>1to show the criticality is also atβ= 1.

2. Isoradial graphs and star-triangle transformation

When it comes to critical behaviors, the interface of the Ising model on isoradial Do-brushin domains converges to a conformally invariant limit as mentioned before for the case of Z². This result is given in [CS12,CDCH⁺14] and the authors use the discrete complex analysis developped in [Ken02,CS11].