Behavior under star-triangle transformations in the Ising model

is only modified locally since the random-cluster measure is preserved (Proposition2.13). In particular, whenq= 2, we are in the case of the FK-Ising model and the parafermionic ob-servable given in (2.11) can be rewritten as

F(e) :=F_Ω,β,q(e) =ϕ_Ω,β,q

exp

−i

2W(a, e) 1e∈γ

, (2.12)

which is calledfermionic observable becauseσ evaluates to ¹

2. The fermionic observable is s-holomorphic [CS12] and actually, this can be translated into a local linear relation between

2. Isoradial graphs and star-triangle transformation

a^⋄_b a^⋄ a^⋄_w

b^⋄_w b^⋄_b b^⋄

Figure 2.12 – A configuration on a Dobrushin domain. Solid lines represent primal open edges and dashed lines represent dual open edges. Loops separating different clusters are drawn in red and the interface separating the primal cluster connected to∂ and the dual cluster connected to∂^∗is drawn in orange.

the observable before and after a local star-triangle transformation, independently of the boundary condition.

To be more precise, let us consider the notations as in Figure2.13. On the left, we have the star graph?and on the right, the triangle graph4. Since the star-triangle transformation preserves the random-cluster measure outside of this hexagon, the observable takes the same values on the boundary of both?and4.

b1

Figure 2.13 – A local star-triangle transformation with notations that we use in Proposi-tions2.14.

Proposition 2.14. WriteG = (g₁, g₂, g₃)^T and H = (h₁, h₂, h₃)^T two column matrices con-sisting of the values of the observable at the triangle mid-edges and the star mid-edges (cf. Fig-ure2.13). There exist two3×3matricesP andQsuch that

H=P G and G=QH.

Moreover, we haveP =Qwhich are given by the following formula P =Q= 1

2.7. Observable on isoradial graphs wherec_]= cos(₂^])ands_]= cos(₂^])for]∈ {a, b, c}.

Proof. We apply Lemma2.15. WriteF = (f₁, f₂, f₃, f₄, f₅, f₆)^T, then in the star graph?, we can write the vectorF in terms ofG; in the triangle graph4, we can writeF in terms ofH. These relations are given by the explicit relationF=DMG=D⁰M⁰H, where

D= i·Diag(s⁻_a¹, s⁻_a¹, s⁻_b¹, s⁻_b¹, s⁻_c¹, s⁻_c¹), D⁰= i·Diag(s⁻_b¹, s⁻_c¹, s⁻_c¹, s⁻_a¹, s⁻_a¹, s⁻_b¹), computa-tion shows thatQis of the form (2.13). To conclude, we use the fact thatQis invertible and is an involution.

Lemma 2.15. Letf be a s-holomorphic function defined on mid-edges. Given a rhombus whose vertices are denoted byv₁,v₂,v₃andv₄(two are primal and two others are dual) in the coun-terclockwise order with angleαbetween −−−−→

v₁v₂ and −−−−→

v₁v₄. Letf_i=f([v_i−1v_i])fori= 1,2,3,4 withv₀ =v₄. Then, for the both possible ways of choosing primal and dual vertices, we can writef₃andf₄in terms off₁andf₂,

Proof. We just need to show one of the possible way of choosing primal and dual vertices. Say, v₁andv₃are primal andv₂andv₄are dual as shown in Figure2.14. Letf =f₁+f₃=f₂+f₄ which is the corresponding value of the functionf at the center of the rhombus. Write θ= arg(i(v₂−v₁)). Then, we have

2. Isoradial graphs and star-triangle transformation This gives a formula off in terms off₁andf₂,

f = i

s_α(ν_αf₂−ν_αf₁).

Then, we use the fact thatf₃=f −f₁andf₄=f −f₂to conclude.

Chapter 3

Universality of the random-cluster model

3.1 Results

The square lattice embedded so that each face is a square of side-length

√

2is an isoradial graph. We will denote it abusively byZ²and call it theregular square lattice. The edge-weight associated to each edge ofZ² by (2.2) is

√q 1+√

q. This was shown in [BD12] to be the critical parameter for the random-cluster model on the square lattice. Moreover, the phase transition of the model was shown to be continuous whenq∈[1,4][DCST17] and discontinuous when q >4[DGH⁺16]. The following two theorems generalise these results to periodic isoradial graphs.

Theorem 3.1. Fix a doubly-periodic isoradial graphGand16q64. Then,

• ϕ_G¹_,1,q[0↔ ∞] = 0andϕ⁰_G,1,q=ϕ¹_G,1,q;

• there exista, b >0such that for alln>1, n^−a6ϕG⁰,1,q

h0↔∂B_ni 6n^−b;

• for anyρ >0, there existsc=c(ρ)>0such that for alln>1, ϕ_R,1,q⁰ h

C_h(ρn, n)i

>c,

whereR= [−(ρ+ 1)n,(ρ+ 1)n]×[−2n,2n]andC_h(ρn, n)is the event that there exists a path inω∩[−ρn, ρn]×[−n, n]from{−ρn} ×[−n, n]to{ρn} ×[−n, n].

The last property is called thestrong RSW property(or simply RSW property) and may be extended as follows: for any boundary conditionsξ,

c6ϕ_R,1,q^ξ h

C_h(ρn, n)i

61−c, (3.1)

for anyn>1and some constantc >0depending only onρ. In words, crossing probabilities remain bounded away from 0 and 1 uniformly in boundary conditions and in the size of the box (provided the aspect ratio is kept constant). For this reason, in some works (e.g. [GM14]) the denominationbox crossing propertyis used.

The strong RSW property was known for Bernoulli percolation on the regular square lattice from the works of Russo and Seymour and Welsh [Rus78,SW78], hence the name. The

3. Universality of the random-cluster model

term strong refers to the uniformity in boundary conditions; weaker versions were developed in [BD12] for the square lattice. Hereafter, we say the model has the strong RSW property if (3.1) is satisfied.

The strong RSW property is indicative of a continuous phase transition and has numerous applications in describing the critical phase. In particular, it implies the first two points of Theorem3.1. It is also instrumental in the proofs of mixing properties and the existence of certain critical exponents and subsequential scaling limits of interfaces. We refer to [DCST17]

for details.

Theorem 3.2. Fix a doubly-periodic isoradial graphGandq >4. Then,

• ϕ_G¹_,1,q[0↔ ∞]>0;

• there existsc >0such that for alln>1,ϕ⁰_G,1,q[0↔∂B_n]6exp(−cn).

Note that the above result is also of interest for regular graphs such as the triangular and hexagonal lattices. Indeed, the transfer matrix techniques developed in [DGH⁺16] are specific to the square lattice and do not easily extend to the triangular and hexagonal lattices.

The strategy of the proof for Theorems 3.1and 3.2is the same as in [GM14]. There, Theorem 3.1was proved forq = 1(Bernoulli percolation). The authors explained how to transfer the RSW property from the regular square lattice model to more general isoradial graphs by modifying the lattice step by step. The main tool used for the transfer is the star-triangle transformation.

In this article, we will follow the same strategy, with two additional difficulties:

• The model has long-range dependencies, and one must proceed with care when han-dling boundary conditions.

• Forq64, the RSW property is indeed satisfied for the regular square lattice (this is the result of [DCST17]), and may be transferred to other isoradial graphs. This is not the case forq >4, where a different property needs to be transported, and some tedious new difficulties arise.

The results above may be extended to isoradial graphs which are not periodic but satisfy the so-called bounded angles property and an additional technical assumption termed the square-grid propertyin [GM14]. We will not discuss this generalisation here and simply stick to the case of doubly-periodic graphs. Interested readers may consult [GM14] for the exact conditions required forG; the proofs below adapt readily.

A direct corollary of the previous two theorems is that isoradial random-cluster models are critical forβ= 1. This was already proved forq >4in [BDCS15] using different tools.

Corollary 3.3. FixGa doubly-periodic isoradial graph andq>1. Then, for anyβ,1, one hasϕ¹_G,β,q=ϕ_G⁰_,β,qand

• whenβ <1, there existsc_β>0such that for anyx, y∈V, ϕ¹G,β,q[x↔y]6exp(−c_βkx−yk);

• whenβ >1,ϕG⁰,β,q[x↔ ∞]>0for anyx∈V.

For 1 6 q 6 4, arm exponents at the critical point β = 1 are believed to exist and to be universal (that is they depend on q and the dimension, but not on the structure of the underlying graph). Below we define the arm events and effectively state the universality of the exponents, but do not claim their existence.

3.2. Switching between isoradial graphs