Scaling limit of isoradial dimer models and the case of triangular quadri-tilings

(1)

Scaling limit of isoradial dimer models and the case of triangular quadri-tilings

Béatrice de Tilière

¹

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland Received 13 March 2006; accepted 19 October 2006

Available online 29 January 2007

Abstract

We consider dimer models on planar graphs which are bipartite, periodic and satisfy a geometric condition calledisoradiality, defined in [R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2) (2002) 409–439]. We show that the scaling limit of the height function of any such dimer model is a Gaussian free field.Triangular quadri-tilingswere introduced in [B. de Tilière, Quadri-tilings of the plane, math.PR/0403324, Probab. Theory Related Fields, in press]; they are dimer models on a family of isoradial graphs arising from rhombus tilings. By means of two height functions, they can be interpreted as random interfaces in dimension 2+2. We show that the scaling limit of each of the two height functions is a Gaussian free field, and that the two Gaussian free fields are independent.

Résumé

On considère le modèle de dimères sur des graphes planaires, bipartis, périodiques et satisfaisant une condition géométrique appeléeisoradialité, définie dans [R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2) (2002) 409–439]. Nous montrons que la limite d’échelle de la fonction de hauteur d’un tel modèle est décrite par un champ libre gaussien. Lesquadri-pavages triangulairesont été introduits dans [B. de Tilière, Quadri-tilings of the plane, math.PR/0403324, Probab. Theory Related Fields, in press] ; ils constituent une famille de modèles de dimères sur des graphes engendrés par des pavages par losanges. Grâce à deux fonctions de hauteur, ils sont interprétés comme des interfaces discrètes en dimension 2+2.

Nous montrons que la limite d’échelle de chacune des fonctions de hauteur est décrite par un champ libre gaussien, et que ces deux champs libres sont indépendants.

Keywords:Dimer model; Gausian free field; Scaling limit; Height function; Quadri-tilings

E-mail address:beatrice.detiliere@math.unizh.ch (B. de Tilière).

1 Supported by Swiss National Fund grant 47102009.

doi:10.1016/j.anihpb.2006.10.002

(2)

1. Introduction

1.1. Height fluctuations for isoradial dimer models 1.1.1. Dimer models

The setting for this paper is thedimer model. It is a statistical mechanics model representing diatomic molecules adsorbed on the surface of a crystal. An interesting feature of the dimer model is that it is one of the very few statistical mechanics models where exact and explicit results can be obtained, see [9,10] for an overview. Another very interesting aspect is the alleged conformal invariance of its scaling limit, which is already proved in the domino and 60^◦-rhombus cases [11,12,14]. Theorem 1 of this paper shows this property for a wide class of dimer models containing the above two cases.

In order to give some insight, let us precisely define the setting. Thedimer modelis in bijection with a mathe- matical model called the 2-tiling modelrepresenting discrete random interfaces. Thesystemconsidered for a 2-tiling model is a planar graphG.Configurationsof the system, or 2-tilings, are coverings ofGwith polygons consisting of pairs of edge-adjacent faces ofG, also called 2-tiles, which leave no hole and don’t overlap. Thesystemof the corresponding dimer model is the dual graphG^∗ ofG.Configurationsof the dimer model areperfect matchings ofG^∗, that is set of edges covering every vertex exactly once. Perfect matchings ofG^∗determine 2-tilings ofGas explained by the following correspondence. Denote byf^∗ the dual vertex of a face f of G, and consider an edge f^∗g^∗ofG^∗. We say that the 2-tile ofGmade of the adjacent facesf andgis the 2-tilecorrespondingto the edge f^∗g^∗. Then 2-tiles corresponding to edges of a dimer configuration form a 2-tiling ofG. Let us denote byM(G^∗) the set of dimer configurations ofG^∗.

As for all statistical mechanics models, dimer configurations are chosen with respect to theBoltzmann measure defined as follows. Suppose that the graphG^∗ is finite, and that a positive weight functionν is assigned to edges ofG^∗, then each dimer configurationMhas anenergy,E(M)= −

e∈Mlogν(e). The probability of occurrence of the dimer configurationMchosen with respect to the Boltzmann measureμ¹is:

μ¹(M)= e^−E^(M) Z(G^∗, ν)=

e∈Mν(e) Z(G^∗, ν) ,

whereZ(G^∗, ν)is the normalizing constant called thepartition function. Using the bijection between dimer configurations and 2-tilings,νcan be seen as weighting 2-tiles, andμ¹as a measure on 2-tilings ofG. When the graph G^∗is infinite, aGibbs measureis defined to be a probability measure onM(G^∗)with the following property: if the matching in an annular region is fixed, then matchings inside and outside of the annulus are independent, moreover the probability of any interior matchingMis proportional to

e∈Mν(e). From now on, let us assume that the graph Gsatisfies condition(∗)below:

(∗)

The graphGis infinite, planar, and simple (Ghas no loops and no multiple edges); its vertices are of degree3.Gissimply connected, i.e. it is the one-skeleton of a simply connected union of faces; and it is made of finitely many different faces, up to isometry.

1.1.2. Isoradial dimer models

This paper actually proves conformal invariance of the scaling limit for a sub-family of all dimer models called isoradial dimer models, introduced by Kenyon in [13]. Much attention has lately been given to isoradial dimer models because of a surprising feature: many statistical mechanics quantities can be computed in terms of thelocalgeometry of the graph. This fact was conjectured in [13], and proved in [4]. The motivation for their study is further enhanced by the fact that the yet classical domino and 60^◦-rhombus tiling models are examples of isoradial dimer models. Last but not least their understanding allows us to apprehend a random interface model in dimension 2+2 called thetriangular quadri-tiling modelintroduced in [3], see Section 1.2.1.

Let us now define isoradial dimer models. Speaking in the terminology of 2-tilings,isoradial2-tiling modelsare defined on graphsGsatisfying a geometric condition calledisoradiality: all faces of an isoradial graph are inscribable in a circle, and all circumcircles have the same radius, moreover all circumcenters of the faces are contained in the closure of the faces. The energy of configurations is determined by a specific weight function called the critical weight function, see Section 2.1 for definition. Note that ifGis an isoradial graph, an isoradial embedding of the

(3)

dual graphG^∗is obtained by sending dual vertices to the center of the corresponding faces. Hence, the corresponding dimer model is called anisoradial dimer model.

1.1.3. Height functions

LetGbe an isoradial graph, whose dual graphG^∗is bipartite. Then, by means of theheight function, 2-tilings ofG can be interpreted as random discrete 2-dimensional surfaces in a 3-dimensional space that are projected orthogonally to the plane. In physics terminology, one speaks of random interfaces in dimension 2+1. The height function, denoted byh, is anR-valued function on the vertices of every 2-tiling ofG, and is defined in Section 3.

1.1.4. Gaussian free field in the plane

The Gaussian free field in the plane is defined in Section 4. It is a random distribution which assigns to functions ϕ1, . . . , ϕk∈C_c,0^∞(R²)(the set of compactly supported smooth functions ofR², which have mean 0), a real Gaussian random vector(F ϕ₁, . . . , F ϕ_k)whose covariance function is given by

E(F ϕ_iF ϕ_j)=

R²

g(x, y)ϕ_i(x)ϕ_j(y)dxdy,

whereg(x, y)= −_2π¹ log|x−y|is the Green function of the plane (defined up to an additive constant).

1.1.5. Statement of result

Let Gbe an isoradial graph, whose dual graph G^∗ is bipartite. Suppose moreover that G^∗ is doubly periodic, i.e. that the graph G^∗ and its vertex-coloring are periodic. Then by Sheffield’s theorem [17], there exists a two- parameter family of translation invariant, ergodic Gibbs measures. Note that the two parameters of the Gibbs measures correspond to the normalized height change (horizontal and vertical) of dimer configurations on a family of toroidal graphs, which form a natural exhaustion ofG^∗. Let us denote byμthe unique measure of this two parameter family which has minimal free energy per fundamental domain. From now on, we assume that dimer configurations ofG^∗ are chosen with respect to the measureμ.

Let us multiply the edge-lengths of the graphGbyε >0, this yields a new graphG^ε. Leth^ε be the unnormalized height function on 2-tilings ofG^ε. An important issue in the study of the dimer model is the understanding of the fluctuations ofh^ε, as the meshεtends to 0. This question is answered by Theorem 1 below. Define:

H^ε:C^∞_c,0 R²

−→R,

ϕ −→H^εϕ=ε²

v∈V (G^ε)

a(v^∗)ϕ(v)h^ε(v),

whereV (G^ε)denotes the set of vertices of the graphG^ε, anda(v^∗)is the area inG^∗of the dual facev^∗of a vertexv.

Theorem 1.Consider a graphGsatisfying the above assumptions, thenH^εconverges weakly in distribution to √¹ π

times a Gaussian free field, that is for everyϕ₁, . . . , ϕ_k∈C_c,0^∞(R²),(H^εϕ₁, . . . , H^εϕ_k)converges in law(asε→0) to √¹

π(F ϕ₁, . . . , F ϕ_k), whereF is a Gaussian free field.

• As a direct consequence of Theorem 1, we obtain convergence of the height function of domino and 60^◦-rhombus tilings chosen with respect to the uniform measure to a Gaussian free field. Note that this result is slightly different than those of [11,12,14] since we work on the whole plane, and not on simply connected regions.

• The method for proving Theorem 1 is essentially that of [11], except Lemma 19 which is new. Nevertheless, since we work with a general isoradial graph (and not the square lattice), each step is adapted in a non-trivial way.

1.2. The case of triangular quadri-tilings 1.2.1. Triangular quadri-tiling model

An exciting consequence of Theorem 1 is that it allows us to understand height fluctuations in the case of a random interface model in dimension 2+2, called thetriangular quadri-tiling model. It is the first time this type of result can be obtained on such a model.

(4)

Fig. 1. Four type of quadri-tiles (left). Triangular quadri-tilings correspond to two superimposed dimer models (right).

Let us start by defining triangular quadri-tilings. Consider the set of right triangles whose hypotenuses have length 1, and whose interior angle isπ/3. Color the vertex at the right angle black, and the other two vertices white.

Aquadri-tileis a quadrilateral obtained from two such triangles in two different ways: either glue them along a leg of the same length matching the black (white) vertex to the black (white) one, or glue them along the hypotenuse.

There are four types of quadri-tiles classified as I, II, III, IV, each of which has four vertices, see Fig. 1 (left). Atri- angular quadri-tilingof the plane is an edge-to-edge tiling of the plane by quadri-tiles that respects the coloring of the vertices, seeT of Fig. 1 for an example. LetQdenote the set of all triangular quadri-tilings of the plane, up to isometry.

In [3], triangular quadri-tilings ofQare shown to correspond to two superimposed dimer models in the following way, see also Fig. 1. Define a lozenge to be a 60^◦-rhombus. Then triangular quadri-tilings are 2-tilings of a family of graphsLwhich are lozenge-with-diagonals tilings of the plane, up to isometry. Indeed letT ∈Qbe a triangular quadri-tiling, then on every quadri-tile ofT draw the edge separating the two right triangles, this yields a lozenge- with-diagonals tilingL(T )called theunderlying tiling. Moreover, the lozenge tilingL(T )obtained fromL(T )by removing the diagonals, is a 2-tiling of the equilateral triangular latticeT.

Note that lozenge-with-diagonals tilings and the equilateral triangular lattice are isoradial graphs. Assigning the critical weight function to edges of their dual graphs, we deduce that triangular quadri-tilings ofQcorrespond to two superimposedisoradialdimer models.

1.2.2. Height functions for triangular quadri-tilings

Triangular quadri-tilings are characterized bytwoheight functions in the following way. LetT ∈Qbe a triangular quadri-tiling, then the first height function, denoted byh₁, assigns to vertices ofT the “height” ofT interpreted as a 2-tiling of its underlying lozenge-with-diagonals tilingL(T ). The second height function, denoted byh2, assigns to vertices ofT the height ofL(T )interpreted as a 2-tiling ofT. An example of computation is given in Section 3.3.

By means ofh₁andh₂, triangular quadri-tilings are interpreted in [3] as discrete random 2-dimensional surfaces in a 4-dimensional space that are projected orthogonally to the plane, i.e. in physics terminology, as random interfaces in dimension 2+2.

(5)

1.2.3. Statement of result

The notion of Gibbs measure can be extended naturally to the setQof all triangular quadri-tilings, see Section 2.4.

In [4], we give an explicit expression for such a Gibbs measureP, and conjecture it to be of minimal free energy per fundamental domain among afour-parameter family of translation invariant, ergodic Gibbs measures. Let us assume that triangular quadri-tilings ofQare chosen with respect to the measureP.

Corollary 2 below describes the fluctuations of the unnormalized height functions h^ε₁ andh^ε₂. Suppose that the equilateral triangular lattice has edge-lengths 1, and letT^ε be the latticeTwhose edge-lengths have been multiplied byε. Observe that vertices ofTare vertices ofL, for every lozenge-with-diagonals tilingL∈L. Fori=1,2, and for ϕ∈C_c,0^∞(R²)define:

H_i^εϕ=ε²

v∈V (T^ε)

√3

2 ϕ(v)h^ε_i(v).

Corollary 2.Under the joint law, the two height functionsH₁^ε, H₂^εconverge weakly in distribution to two independent Gaussian free fieldsF₁andF₂.

1.3. Outline of the paper

• Section 2: statement of the explicit expressions of [3] for the Gibbs measuresμandP, that are used in the proof of Theorem 1 and Corollary 2.

• Section 3: definition of the height function on vertices of 2-tilings of isoradial graphs.

• Section 4: definition of the Gaussian free field in the plane.

• Section 5 and Section 6: proof of Theorem 1 and Corollary 2.

2. Minimal free energy Gibbs measure for isoradial dimer models

In the whole of this section, we letGbe an isoradial graph, whose dual graphG^∗is bipartite;Bdenotes the set of black vertices, andW the set of white ones. In the proof of Theorem 1, we use the explicit expression of [3] for the minimal free energy per fundamental domain Gibbs measureμon 2-tilings ofG, and in the proof of Corollary 2, we use the explicit expression of [3] for the Gibbs measurePon triangular quadri-tilings. The goal of this section is to state the expressions forμandP. In order to do so, we first define the critical weight function and the Dirac operator, introduced in [13].

2.1. Critical weight function 2.1.1. Definition

The following definition is taken from [13]. To each edgeeofG^∗, we associate a unit side-length rhombusR(e) whose vertices are the vertices ofeand of its dual edge e^∗ (R(e)may be degenerate). Let R= _e_∈_G∗R(e). The critical weight functionνat the edgeeis defined byν(e)=2 sinθ, where 2θis the angle of the rhombusR(e)at the vertex it has in common withe;θis called therhombus angleof the edgee. Note thatν(e)is the length ofe^∗. 2.1.2. Example: critical weights for triangular quadri-tilings

Recall that triangular quadri-tilings of Q correspond to two superimposed isoradial dimer models, the first on lozenge-with-diagonals tilings and the second on the equilateral triangular latticeT. Let us note that the dual graphs of lozenge-with-diagonals tilings and ofTare bipartite. We now compute the critical weights in the above two cases.

Consider the equilateral triangular latticeT, then edges of its dual graphT^∗, known as the honeycomb lattice, all have the same rhombus angle, equal toπ/3, and the same critical weight, equal to√

3.

Consider a lozenge-with-diagonals tilingL∈L. Observe that the circumcenters of the faces ofLare on the boundary of the faces, so that in the isoradial embedding of the dual graphL^∗some edges have length 0, and the rhombi associated to these edges are degenerate, see Fig. 2. Since edges ofL^∗ correspond to quadri-tiles ofL, we classify them as being of type I, II, III, IV. Fig. 2 below gives the rhombus angles and the critical weights associated to edges of type I, II, III and IV, denoted bye_I, e_II, e_III, e_IVrespectively.

(6)

Fig. 2. Dual graph of a lozenge-with-diagonals tiling (left). Critical weights for quadri-tiles (right).

2.2. Dirac and inverse Dirac operator

Results in this section are due to Kenyon [13], see also Mercat [16]. Define the Hermitian matrixK indexed by vertices ofG^∗as follows. Ifv1andv2are not adjacent,K(v1, v2)=0. Ifw∈Wandb∈Bare adjacent vertices, then K(w, b)=K(b, w)is the complex number of modulusν(wb)and direction pointing fromwtob. Another useful way to say this is as follows. LetR(wb)be the rhombus associated to the edgewb, and denote byw, x, b, yits vertices in counterclockwise order, thenK(w, b)isitimes the complex vectorx−y. Ifwandbhave the same image in the plane, then|K(w, b)| =2, and the direction ofK(w, b)is that which is perpendicular to the corresponding dual edge, and has sign determined by the local orientation. The infinite matrixKdefines theDirac operatorK:C^{V (G}^∗⁾→C^{V (G}^∗⁾, by

(Kf )(v)=

u∈V (G∗)

K(v, u)f (u),

whereV (G^∗)denotes the set of vertices of the graphG^∗.

Theinverse Dirac operatorK⁻¹is defined to be the operator which satisfies 1. KK⁻¹=Id,

2. K⁻¹(b, w)→0, when|b−w| → ∞.

In [13], Kenyon proves uniqueness ofK⁻¹, and existence by giving an explicit expression forK⁻¹(b, w)as a function of thelocalgeometry of the graph.

2.3. Minimal free energy Gibbs measure for isoradial graphs

Ife₁=w₁b₁, . . . , e_k=w_kb_k is a subset of edges ofG^∗, define thecylinder set{e₁, . . . , e_k}ofG^∗to be the set of dimer configurations ofG^∗which contain the edgese₁, . . . , e_k. LetAbe the field consisting of the empty set and of the finite disjoint unions of cylinders. Denote byσ (A)theσ-field generated byA.

Theorem 3.[3]AssumeG^∗is doubly periodic. Then, there is a probability measureμon(M(G^∗), σ (A))such that for every cylinder{e₁, . . . , e_k}ofG^∗,

μ(e₁, . . . , e_k)= _k

i=1

K(w_i, b_i)

det

1i,jk

K⁻¹(b_i, w_j)

. (1)

Moreoverμis a Gibbs measure onM(G^∗), and it is the unique Gibbs measure which has minimal free energy per fundamental domain among the two-parameter family of translation invariant, ergodic Gibbs measures of[17].

(7)

Remark 4.

• Refer to [15] for the definition of the free energy per fundamental domain.

• In [3], we prove that the periodicity assumption can be released in the case of lozenge-with-diagonals tilings.

That is, given any lozenge-with-diagonals tiling ofL, Eq. (1) defines a Gibbs measure on dimer configurations of its dual graph. Although fundamental domains make no sense in case of non-periodic graphs, the minimal free energy property can still be interpreted in some wider sense.

2.4. Gibbs measure on triangular quadri-tilings

The construction of this section is taken from [3]. Consider the setQof all triangular quadri-tilings of the plane up to isometry, and assume that quadri-tiles are assigned a positive weight function. Then the notion ofGibbs measure on Qis a natural extension of the one used in the case of dimer configurations of fixed graphs. It is a probability measure that satisfies the following: if a triangular quadri-tiling is fixed in an annular region, then triangular quadri-tilings inside and outside of the annulus are independent; moreover, the probability of any interior triangular quadri-tiling is proportional to the product of the weights of the quadri-tiles. Denoting by M the set of dimer configurations corresponding to triangular quadri-tilings ofQ, and using the bijection betweenQandM, we obtain the definition of a Gibbs measure onM.

DefineL^∗to be set of dual graphsL^∗of lozenge-with-diagonals tilingsL∈L. Although some edges ofL^∗have length 0, we think of them as edges of the one skeleton of the graphs, so that to every edge ofL^∗, there corresponds a unique quadri-tile. Letebe an edge ofL^∗, and letq_ebe the corresponding quadri-tile, thenq_eis made of two adjacent right triangles. If the two triangles share the hypotenuse edge, they belong to two adjacent lozenges; else if they share a leg, they belong to the same lozenge. Let us call these lozenge(s) thelozenge(s) associated to the edge e, and denote it/them byl_e (that isl_e consists of either one or two lozenges). Letk_e be the edge(s) ofT^∗corresponding to the lozenge(s)l_e. Let us introduce one more definition, if{e₁, . . . , e_k}is a subset of edges ofL^∗, then thecylinder set {e₁, . . . , e_k}is the set of dimer configurations ofMwhich contain these edges. Denote byCthe field consisting of the empty set and of the finite disjoint unions of cylinders. Denote byσ (C)theσ-field generated byC.

Consider a lozenge-with-diagonals tilingL∈L, and denote byμ^Lthe minimal free energy per fundamental domain Gibbs measure on(M(L^∗), σ (A))given by Theorem 3, whereσ (A)is theσ-field of cylinders ofM(L^∗). Similarly, denote byμ^Tthe minimal free energy per fundamental domain Gibbs measure on(M(T^∗), σ (B)), whereσ (B)is the σ-field of cylinders ofM(T^∗).

Let us defineμ˜^Lon(M, σ (C))by:

˜

μ^L(e₁, . . . , e_k)=

μ^L(e1, . . . , ek) if the lozenges^le₁, . . . ,l_e_k belong to^L,

0 else,

where we recall thatLis the lozenge tiling obtained from the lozenge-with-diagonals tilingLby removing the diagonals. Then, it is easy to check thatμ˜^Lis a probability measure on(M, σ (C)). In order to simplify notations, we write μ^Lforμ˜^Lwhenever no confusion occurs.

Now, on(M×M(T^∗),B×C), define:

P

(e₁, . . . , e_k)×(k₁, . . . , k_m)

=

{^L^∗∈M(T^∗):k1,...,km∈^L^∗}

μ^L(e₁, . . . , e_k)dμ^T(L^∗).

Using Kolmogorov’s extension theorem,P extends to a probability measure on(M×T^∗, σ (B×C)). Let us also denote byPthe marginal ofPonM, then in [3],Pis shown to be a Gibbs measure onM, and conjectured to be of minimal free energy per fundamental domain among afour-parameter family of translation invariant, ergodic Gibbs measures.

3. Height functions

In the whole of this section, we letGbe an isoradial graph whose dual graphG^∗is bipartite; as before,Bdenotes the set of black vertices,W the set of white ones. We define theheight functionhon vertices of 2-tilings ofG, whose fluctuations are described in Theorem 1. As in [15], see also [2],his defined using flows.

(8)

The bipartite coloring of the vertices ofG^∗ induces an orientation of the faces ofG: color the dual faces of the black (white) vertices black (white); orient the boundary edges of the black faces counterclockwise, the boundary edges of the white faces are then oriented clockwise.

3.1. Definition

Let us first define a flowω₀on the edges ofG^∗. Consider an edgewbofG^∗, thenR(wb)is the rhombus associated towb, andθ_wb is the corresponding rhombus angle. Defineω₀to be the white-to-black flow, which flows byθ_wb/π along every edgewbofG^∗.

Lemma 5.The flowω0has divergence1at every white vertex, and−1at every black vertex ofG^∗. Proof. By definition of the rhombus angle, we have

∀w∈W,

b:b∼w

2θwb=2π; ∀b ∈ B,

w:w∼b

2θwb=2π. 2

Now, consider a 2-tilingT ofG, and letMbe the corresponding perfect matching ofG^∗. ThenMdefines a white-to- black unit flowωon the edges ofG^∗: flow by 1 along every edge ofM, from the white vertex to the black one. The differenceω0−ωis a divergence free flow, which means that the quantity of flow that enters any vertex ofG^∗equals the quantity of flow which exits that same vertex.

We are ready for the definition of theheight functionh. Choose a vertexv0ofG, and fixh(v0)=0. For every other vertexv of T, take an edge-pathγ of Gfromv0 tov. If an edgeuvof γ is oriented in the direction of the path, and if we denote byeits dual edge, thenh increases byω0(e)−ω(e)alonguv; if an edgeuvis oriented in the opposite direction, thenhdecreases by the same quantity alonguv. As a consequence of the fact thatω0−ωis a divergence free flow, the height functionhis well defined.

The following lemma gives a correspondence between height functions defined on vertices ofG, and 2-tilings ofG.

Lemma 6.Leth˜be anR-valued function on vertices ofGsatisfying

• ˜h(v0)=0,

• ˜h(v)− ˜h(u)=ω0(e)orω0(e)−1for any edgeuvoriented fromutov, whereedenotes the dual edge ofuv.

Then, there is a bijection between functionsh˜ satisfying these two conditions, and2-tilings ofG.

Proof. The idea of the proof closely follows [5]. LetT be a 2-tiling ofG,Mbe the corresponding matching, and ωbe the unit white-to-black flow defined byM. Then, the height functionhsatisfies the conditions of the lemma:

consider an edgeuvofGoriented fromutovand denote byeits dual edge, thenh(v)−h(u)=ω₀(e)−ω(e), and by definitionω(e)=0 or 1.

Conversely, consider anR-valued functionh˜as in the lemma. Let us construct a 2-tilingT whose height function ish. Consider a black face˜ F ofG, and lete₁, . . . , e_mbe the dual edges of its boundary edges. Then_m

i=1ω₀(e_i)=1, so that there is exactly one boundary edgeuvalong whichh(v)˜ − ˜h(u)isω₀(e_i)−1 (wheree_iis the dual edge ofuv).

To the faceF, we associate the 2-tile ofGwhich is crossed by the edgeuv. Repeating this procedure for all black faces ofG, we obtainT. 2

3.2. Interpretation

The discrete interface interpretation of 2-tilings was first given by Thurston in the case of lozenges [19]. Following him, a 2-tiling ofGcan be seen as a discrete 2-dimensional surfaceSin a 3-dimensional space that has been projected orthogonally to the plane; the “height” ofS is given by the functionh. Stated in physics terminology, 2-tilings ofG are random interfaces in dimension 2+1.

(9)

Fig. 3. From left to right: triangular quadri-tiling T and first height functionh₁, underlying lozenge tilingL(T )and second height functionh₂, T with height functionsh₁(above) andh₂(below).

3.3. Example: the two height functions of triangular quadri-tilings

Consider a triangular quadri-tilingT ∈Q. Then, recall thath₁assigns to vertices ofT the “height” ofT interpreted as a 2-tiling of its underlying lozenge-with-diagonals tilingL(T ), andh₂assigns to vertices ofT the “height” ofL(T ) interpreted as a 2-tiling ofT. Let us now explicitly computeh₁andh₂.

The definition of the flowω0(L(T ))on edges ofL(T )^∗uses the rhombus angles of the edges ofL(T )^∗. These have been computed in Section 2.1.2 and are equal to^π₆,^π₃,^π₂ for edges of type I, II, III and IV respectively. Hence if uvis the boundary edge of a quadri-tile ofT, oriented fromutov, then the height change ofh₁alonguvis ¹₆,¹₃,¹₂ depending on whether the dual edgeeof the edgeuvis of type I, II, III or IV respectively, see Fig. 3.

In a similar way, the flowω₀(T^∗)on edges ofT^∗flows by ¹₃along every edge, so that ifuvbe a boundary edge of a lozenge of^L(T )oriented fromutov, then the height change ofh2alonguvis equal to ¹₃. Thinking of a lozenge as the projection to the plane of the face of a cube [19], there is a natural way to assign a value forh2at the vertex at the crossing of the diagonals of the lozenges, see Fig. 3.

By means ofh₁andh₂, a triangular quadri-tilingT ofQis interpreted in [3] as a 2-dimensional discrete surface S₁in a 4-dimensional space that has been projected orthogonally to the plane.S₁can also be projected to¹₃Z³(Z³is the spaceZ³where the cubes are drawn with diagonals on their faces), and one obtains a surfaceS₂. When projected to the planeS₂is the underlying lozenge-with-diagonals tilingL(T ). This can be restated by saying that triangular quadri-tilings ofQare discrete interfaces in dimension 2+2.

4. Gaussian free field in the plane

Theorem 1 and Corollary 2 prove convergence of the height functionhto a Gaussian free field. The goal of this section is to define theGaussian free field in the plane. We refer to [6,18] for other ways of defining the Gaussian free field.

4.1. Green function of the plane, and Dirichlet energy

The Green function of the plane, denoted by g, is the kernel of the Laplace equation in the plane, it satisfies _xg(x, y)=δ_x(y), whereδ_xis the Dirac distribution atx. Up to an additive constant,gis given by

g(x, y)= − 1

2π log|x−y|. Define the following bilinear form

G:C_c,0^∞ R²

×C_c,0^∞ R²

−→R,

(ϕ₁, ϕ₂)−→G(ϕ₁, ϕ₂)=

R²

g(x, y)ϕ₁(x)ϕ₂(y)dxdy.

(10)

G(ϕ, ϕ)is called theDirichlet energyofϕ. Let us consider the topology induced by theL^∞norm onC_c,0^∞(R²). Then, Gis a continuous, positive definite, bilinear form.

4.2. Random distributions

The following definitions are taken from [7]. Arandom functionF associates to every functionϕ∈C_c,0^∞(R²)a real random variableF ϕ. Forϕ₁, . . . , ϕ_k∈C_c,0^∞(R²), we suppose that the joint probabilitiesa_nF ϕ_n< b_n, 1nk are given, and we ask that they satisfy the compatibility relation. A random functionFislinearif∀ϕ₁, ϕ₂∈C_c,0^∞(R²),

F (αϕ₁+βϕ₂)=αF ϕ₁+βF ϕ₂.

It iscontinuousif convergence of the functionsϕ_n_j toϕ_j (1jk)implies

nlim→∞(F ϕ_n₁, . . . , F ϕ_n_k)=(F ϕ₁, . . . , F ϕ_k),

that is, ifP (x)(resp.P_n(x)) is the probability measure corresponding to the random variable(F ϕ₁, . . . , F ϕ_k)(resp.

(F ϕn₁, . . . , F ϕn_k)), then for any bounded continuous functionf

nlim→∞

f (x₁, . . . , x_k)dPn(x)=

f (x₁, . . . , x_k)dP (x).

Arandom distributionF is a random function which is linear and continuous. It is said to beGaussianif for every linearly independent functionsϕ₁, . . . , ϕ_k∈C_c,0^∞(R²), the random vector(F ϕ₁, . . . , F ϕ_k)is Gaussian.

Two random distributionsF andG are said to beindependentif for any functionsϕ1, . . . , ϕk∈C_c,0^∞(R²), the random vectors(F ϕ₁, . . . , F ϕ_k)and(Gϕ₁, . . . , Gϕ_k)are independent.

4.3. Gaussian free field in the plane

Lemma 7.[1]If G:C_c,0^∞(R²)×C_c,0^∞(R²)→Ris a bilinear, continuous, positive definite form, then there exists a Gaussian random distributionF, whose covariance function is given by

E(F ϕ₁F ϕ₂)=G(ϕ₁, ϕ₂).

Using Lemma 7, we define aGaussian free field in the planeto be a Gaussian random distribution whose covariance function is

E(F ϕ₁F ϕ₂)= − 1 2π

R²

log|x−y|ϕ₁(x)ϕ₂(y)dxdy.

5. Proof of Theorem 1

We place ourselves in the context of Theorem 1:Gis an isoradial graph whose dual graphG^∗is doubly periodic and bipartite; edges ofG^∗are assigned the critical weight function, andG^ε is the graphGwhose edge-lengths have been multiplied byε. Recall the following notations: H^εϕ=ε²

v∈V (G^ε)a(v^∗)ϕ(v)h^ε(v), μis the minimal free energy per fundamental domain Gibbs measure for the dimer model onG^∗of Section 2.3, andF is a Gaussian free field in the plane.

Since the random vector(F ϕ1, . . . , F ϕk)is Gaussian, to prove convergence of(H^εϕ1, . . . , H^εϕk)to(F ϕ1, . . . , F ϕ_k), it suffices to prove convergence of the moments of(H^εϕ₁, . . . , H^εϕ_k)to those of(F ϕ₁, . . . , F ϕ_k); that is we need to show that for everyk-tuple of positive integers(m₁, . . . , m_k), we have

εlim→0E

H^εϕ₁m₁

. . .

H^εϕ_km_k

=E F ϕ₁m₁

. . . (F ϕ_k)^m^k

. (2)

In Section 5.1, we prove two properties of the height functionh, and in Section 5.2, we give the asymptotic formula of [13] for the inverse Dirac operatorK⁻¹. Using these results in Section 5.3, we prove a formula for the limit (as ε→0) of thekth moment ofh. This allows us to show convergence ofE[(H^εϕ)^k]toE[(F ϕ)^k]in Section 5.4. One then obtains Eq. (2) by choosingϕto be a suitable linear combination of theϕi’s.

As before,Bdenotes the set of black vertices ofG^∗, andWthe set of white ones. Moreover, we suppose that faces ofGhave the orientation induced by the bipartite coloring of the vertices ofG^∗.

(11)

5.1. Properties of the height function

Letu, vbe two vertices ofG, and letγ be an edge-path ofGfromutov. First, consider edges ofγ which are oriented in the direction of the path, that is edges which have a black face ofGon the left, and denote byf1, . . . , fn

their dual edges. Hence an edgefjconsists of a black vertex on the left ofγ, and of a white one on the right. Similarly, consider edges ofγ which are oriented in the opposite direction, and denote bye1, . . . , emtheir dual edges, hence an edgee_i consists of a white vertex on the left ofγ, and of a black one on the right. LetIebe the indicator function of M(G^∗):Ie(M)=1, if the edgeebelongs to the dimer configurationMofG^∗, and 0 otherwise.

Lemma 8.

h(v)−h(u)= m

j=1

Iej−μ(e_j) +

n

j=1

−Ifj+μ(f_j) .

Proof. Lete_j be the dual edge of an edgeu_jv_j ofγ oriented fromv_j tou_j. Denote byθ_j the rhombus angle of the edgee_j, then by Lemma 6,

h(vj)−h(uj)=

−^θ_π^j +1 if the edgee_j belongs to the dimer configuration ofG^∗,

−^θ_π^j else.

Hence

h(v_j)−h(u_j)=

−θ_j π +1

Iej −θ_j

π

1−Iej

=Iej −θ_j π. Moreover, a direct computation using formula (1) yieldsμ(e_j)=θ_j/π, so

h(v_j)−h(u_j)=Iej−μ(e_j).

Similarly, whenf_j is the dual edge of an edgeu_jv_jofγ oriented fromu_j tov_j, we obtainh(v_j)−h(u_j)= −Ifj+ μ(f_j), and we conclude

h(v)−h(u)= m

j=1

h(v_j)−h(u_j)+ n

j=1

h(v_j)−h(u_j)= m

j=1

Iej−μ(e_j) +

n

j=1

−Ifj+μ(f_j)

. 2

Lemma 9.

Eμ

h(v)−h(u)

=0.

Proof. By Lemma 8 we have, Eμ

h(v)−h(u)

= m

j=1

Eμ

Iej −μ(e_j) +

n

j=1

Eμ

−Ifj+μ(f_j)

=0. 2

5.2. Asymptotics of the inverse Dirac operatorK⁻¹

In order to state the asymptotic formula of [13] for the inverse of the Dirac operatorKindexed by vertices ofG^∗, we letw∈Wbe a white vertex ofG^∗,xany other vertex ofG^∗, and define the rational functionf_wx(z)of [13]. Recall thatRis the set of rhombi associated to edges ofG^∗, and considerw=x₀, x₁, x₂, . . . , x_k=xan edge-path ofRfrom wtox. Each edgex_jx_j₊₁has exactly one vertex ofG^∗(the other is a vertex ofG). Direct the edge away from this vertex if it is white, and towards this vertex if it is black. Let e^iα^j be the corresponding vector inR(which may point contrary to the direction of the path). Then,f_wxis defined inductively along the path, starting from

f_ww(z)=1.

(12)

If the edge leads away from a white vertex, or towards a black vertex, then fwx_j₊₁(z)=fwx_j(z)

1−e^iα^j,

else, if it leads towards a white vertex, or away from a black vertex, then f_wx_j+1(z)=f_wx_j(z)·

z−e^iα^j .

The functionf_wx(z)is well defined (i.e. independent of the edge-path ofRfromwtox). Then, Kenyon gives the following asymptotics for the inverse Dirac operatorK⁻¹:

Theorem 10.[13]Asymptotically, as|b−w| → ∞, K⁻¹(b, w)= 1

2π 1

b−w+fwb(0) b−w

+O

1

|b−w|²

.

5.3. Moment formula

Letu₁, . . . , u_k, v₁, . . . , v_k be distinct points ofR², and letγ₁, . . . , γ_k be pairwise disjoint, simple, smooth paths such thatγ_j runs fromu_j tov_j. Letu^ε_j, v_j^εbe vertices ofG^εlying within O(ε)ofu_j andv_j respectively. In order to simplify notations, we writehfor the unnormalized height functionh^εon 2-tilings ofG^ε. Then, we have

Proposition 11.For everyk∈N,k2

εlim→0E

h(v₁^ε)−h(u^ε₁)

· · ·

h(v_k^ε)−h(u^ε_k)

=(−i)^k (2π )^k

ρ=0,1

(−1)^kρ

γ1

. . .

γk

i,jdet∈[1,k] i=j

1 z^ρ_i −z^ρ_j

dz₁^ρ· · ·dz^ρ_k

,

(3) wherez⁰_i =zi andz¹_i = ¯zi.

Proof. The structure of the proof of Proposition 11 is essentially that of [11], but since we work in a much more general setting, each step is adapted in a non-trivial way. The proof being quite long, here is a short outline: after having initiated the proof, we derive and show four lemmas (Lemmas 12–15); then, after the last of the four lemmas has been shown, we conclude the proof of Proposition 11.

Letγ₁^ε, . . . , γ_k^εbe pairwise disjoint paths ofG^ε, such thatγ_j^εruns fromu^ε_j tov^ε_jand approximatesγjwithin O(ε).

For everyj, denote byfj sthe dual edge of thesth edge of the pathγ_j^ε, which is oriented in the direction of the path:

fj sconsists of a black vertex on the left ofγ_j^ε, and of a white one on the right. Denote byej t the dual edge of thetth edge of the pathγ_j^ε, which is oriented in the opposite direction:ej t consists of a black vertex on the right ofγ_j^ε, and of a white one on the left. Using Lemma 8, we obtain

E

h(v^ε₁)−h(u^ε₁)

· · ·

h(v^ε_k)−h(u^ε_k)

=E

t1

Ie1t1 −μ(e_1t₁)

−

s1

If1s1 −μ(f_1s₁) . . .

tk

Ie_ktk −μ(e_kt_k)

−

sk

If_ksk−μ(f_ks_k)

=

t₁,...,t_k

E

Ie_1t₁−μ(e_1t₁)

· · ·

Ie_ktk −μ(e_kt_k)

− · · · +(−1)^k

s₁,...,s_k

E

If_1s₁ −μ(f_1s₁)

· · ·

If_ksk −μ(f_ks_k)

=

δ1,...,δk∈{0,1}

t₁^δ¹,...,t_k^δk

(−1)^δ¹^+···+^δ^kE Ie

1tδ1 1

−μ(e

1t₁^δ¹)

· · · Ie

ktδk k

−μ(e

kt_k^δk)

, (4)

where t_j^δ^j=

tj ifδj=0, s_j ifδ_j=1, e

j t_j^δj =e^δ^j

j t_j^δj =

e_{j t}_j ifδ_j=0, f_{j s}_j ifδ_j=1.