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El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 80, 1–33.

ISSN:1083-6489 DOI:10.1214/EJP.v19-2449

Height representation of XOR-Ising loops via bipartite dimers

Cédric Boutillier

Béatrice de Tilière

Abstract

The XOR-Ising model on a graph consists of random spin configurations on vertices of the graph obtained by taking the product at each vertex of the spins of two inde- pendent Ising models. In this paper, we explicitly relate loop configurations of the XOR-Ising model and those of a dimer model living on a decorated, bipartite version of the Ising graph. This result is proved for graphs embedded in compact surfaces of genusg. Using this fact, we then prove that XOR-Ising loops have the same law as level lines of the height function of this bipartite dimer model. At criticality, the height function is known to converge weakly in distribution to 1π a Gaussian free field [dT07b]. As a consequence, results of this paper shed a light on the occurrence of the Gaussian free field in the XOR-Ising model. In particular, they prove a discrete analogue of Wilson’s conjecture [Wil11], stating that the scaling limit of XOR-Ising loops are “contour lines” of the Gaussian free field.

Keywords:Ising model ; XOR-Ising model ; dimers ; 6-vertex model ; height function ; Gaussian free field.

AMS MSC 2010:82B20;82B23.

Submitted to EJP on November 19, 2012, final version accepted on August 10, 2014.

SupersedesarXiv:1211.4825.

SupersedesHAL:hal-00755394.

1 Introduction

Thedouble Ising model consists of two Ising models, living on the same graph. It is related [KW71, Wu71, Fan72, Weg72] to other models of statistical mechanics, as the 8-vertex model [Sut70, FW70] and the Ashkin–Teller model [AT43]. In general, the two models may be interacting. However, in this paper, we consider the case of two non-interacting Ising models, defined on the dualG= (V, E)of a graphG= (V, E), having the same coupling constants(Je)e∈E, where the graphGis embedded either in a compact, orientable, boundaryless surfaceΣof genusg≥0, or in the plane.

We are interested in the polarization of the model [KB79], also referred to as the XOR-Ising model [Wil11] by Wilson. It is defined as follows: given a pair of spin config- urations(σ, σ0)∈ {−1,1}V×{−1,1}V, the XOR-spin configurationbelongs to{−1,1}V

Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris. E-mail:cedric.boutillier@upmc.fr, E-mail: beatrice.de_tiliere@upmc.fr

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and is obtained by taking, at every vertex, the product of the spins. The interface be- tween±1 spin configurations of the XOR-configuration is a loop configuration of the graphG. Using extensive simulations, Wilson [Wil11] finds that, whenG is a specific simply connected domain of the plane, and when both Ising models are critical, XOR loop configurations seem to have the same limiting behavior as “contour lines” of the Gaussian free field, with heights of the contours spaced√

2times as far apart as they should be for the double dimer model on the square lattice. Similar conjectures involv- ing SLE rather than the Gaussian free field, are obtained through conformal field theory [IR11, PS11]. Results of this paper explain the occurrence of the Gaussian free field in the XOR-Ising model and prove a discrete analogue of Wilson’s conjecture.

The first part of this paper concentrates on finite graphs embedded in surfaces.

We explicitly relate XOR loop configurations to loop configurations in a bipartite dimer model, implying in particular that both loop configurations have the same probability distribution. In the second part, we prove that this correspondence still holds for a large class of infinite planar graphs, the so-called isoradial graphs [Ken02, KS05], at criticality, and make the connection with Wilson’s conjecture. Here is an outline.

Outline

Section 2. One of the tools required is a version of Kramers and Wannier’s low/high- temperature duality [KW41a, KW41b] in the case of graphs embedded in surfaces of genusg,with boundary. In the literature, we did find versions of this duality for graphs embedded in surfaces of genus g [LG94], but we could not find versions taking into account boundaries. This is the subject of Propositions 2.1 and 2.2, it involves relative homology theory and the Poincaré–Lefschetz duality.

Sections 3 and 4consist of the extension to general graphs embedded in a surface of genusgof an expansion due to Nienhuis [Nie84], which can be summarized as follows.

Consider the low-temperature expansion of the double Ising model,i.e., consider pairs of polygon configurations separating clusters of ±1 spins of each spin configuration.

Drawing both polygon configurations onG yields an edge configuration consisting of monochromatic edges, that is edges covered by exactly one of the two polygon config- urations, andbichromatic edges, that is edges covered by both polygon configurations.

Monochromatic edge configurations exactly correspond to XOR loop configurations, and separate the surfaceΣinto connected components Σ1, . . . ,ΣN. Inside each connected component, the law of bichromatic edge configurations is that of the low-temperature expansion of an Ising model with coupling constants that are doubled. As a conse- quence, the partition function of the double Ising model can be rewritten using XOR loop configurations and bichromatic edge configurations, see Proposition 3.3.

Fixing a monochromatic edge configuration, and applying low/high-temperature du- ality to the single Ising model corresponding to bichromatic edges, yields a rewriting of the double Ising partition function, as a sum over pairs of non-intersecting polygon con- figurations of the primal and dual graph, where primal polygon configurations exactly correspond to XOR loop configurations, see Proposition 4.2 and Corollary 4.3. Note that there are quite a few difficulties in the proofs, due to the fact that we work on a surface of genusg.

Proposition 1.1.

The double Ising partition function for a graph embedded on a surface of genusg can be rewritten as:

Zd-Ising(G, J) =CI

X

{(P,P)∈P0(G)×P0(G):

P∩P=∅}

Y

e∈P

2e−2Je 1 +e−4Je

! Y

e∈P

1−e−4Je 1 +e−4Je

! ,

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where primal polygon configurations of P0(G) are the XOR loop configurations, and CI= 2|V|+2g+1 Q

e∈Ecosh(2Je) .

Section 5. In Section 5.1, we define the 6-vertex model on the medial graphGMcon- structed fromG. Reformulating an argument of Nienhuis [Nie84], we prove that the 6-vertex partition function can be written as a sum over non-intersecting pairs of poly- gon configurations of the primal and dual graphs.

In Section 5.2, we define the dimer model on the decorated, bipartite graph GQ constructed fromG. Then, we present the mapping between dimer configurations of GQandfree-fermionic6-vertex configurations ofGM[WL75, Dub11b]. Using both map- pings, one assigns to every dimer configurationMa pairPoly(M) = (Poly1(M),Poly2(M)) of non-intersecting primal and dual polygon configurations. The weights of the 6-vertex model chosen to match those of edges in the mixed contour expansion of the double Ising model satisfy thefree-fermioniccondition. As a consequence, we then obtain, see also Proposition 5.4:

Proposition 1.2. The dimer model partition function Zdimer0 (GQ, J) can be rewritten as:

Zdimer0 (GQ, J) = 2 X

{(P,P)∈P0(G)×P0(G):

P∩P=∅}

Y

e∈P

2e−2Je 1 +e−4Je

! Y

e∈P

1−e−4Je 1 +e−4Je

! ,

where primal and dual polygon configurations ofP0(G)× P0(G)are thePolyconfigu- rations.

Combining Proposition 1.1 and Proposition 1.2 yields the following, see also Theo- rem 5.5:

Theorem 1.3. XOR loop configurations of the double Ising model onGhave the same law as Poly1 configurations of the corresponding dimer model on the bipartite graph GQ:

∀P ∈ P0(G), Pd-Ising[XOR =P] =P0Q[Poly1=P].

Remark 1.4. In the paper [Dub11b], Dubédat relates a version of the double Ising model and the same bipartite dimer model in two ways. The first approach uses explicit mappings, most of which are present in the physics literature, and goes as follows.

Consider a slightly different version of the double Ising model, with one model living on the primal graphGand the second on the dual graphG. This double Ising model can be mapped to an 8-vertex model [KW71, Wu71] on the medial graph. Using Fan and Wu’s abelian duality, this 8-vertex model [FW70] can be mapped to a second 8-vertex on the same graph. When coupling constants of the two Ising models satisfy Kramers and Wannier’s duality, the second 8-vertex model is in fact a free-fermionic 6-vertex model.

The free-fermionic 6-vertex model can in turn be mapped to a bipartite dimer model, a result due to [WL75] in the case of the square lattice, and extended by [Dub11b] in the general lattice case. It can also be seen as a specific case of the mapping of the free-fermionic 8-vertex model to a non-bipartite dimer model of [FW70]. Note that this bipartite dimer model is the model of quadri-tilingsstudied by the second author in [dT07a] and [dT07b].

When performing the different steps of the mapping, Dubédat keeps track of or- der/disorder variables, in the vein of [KC71]. Using results of a previous paper of his [Dub11a], this allows him to compute critical correlators in the plane. For simply con- nected regions, this result has independently been obtained by Chelkak, Hongler and Izyurov [CHI12].

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Our goal here is different, since we aim at keeping track of XOR-configurations. This information is not directly available in the above approach. Indeed Fan and Wu’s abelian duality for the 8-vertex model can be compared to a high-temperature expansion, where configurations cannot be interpreted using the initial model. Note that expanding and recombining the identities of [Dub11b] involving correlators, one can recover the iden- tity in law of Theorem 1.3; this proves the existence of a coupling between the two models, which we explicitly provide in this paper.

The second approach uses transformations on matrices. The partition function of the double Ising model can be expressed using the determinant of the Kasteleyn matrix of the Fisher graph [Fis66]; whereas the partition function of the bipartite dimer model can be expressed using the Kasteleyn matrix of the graph GQ. Dubédat shows that the two matrices are related through transformations not affecting the determinant.

Using the fact that the partition function of the double Ising model is also related to the determinant of the Kac–Ward matrix [KW52], Cimasoni and Duminil-Copin use the same approach to relate the Kac–Ward matrix to the matrix of the same bipartite dimer model [CDC13]. Their purpose is to identify the critical point of general bi-periodic Ising models, see also Li [Li10, Li12] for the case of the square lattice with arbitrary fundamental domain.

However, the above transformations on matrices are not easily interpreted in terms of transformations on configurations, and the relation to XOR-configurations is not straightforward.

Using Nienhuis’ mapping [Nie84], the main contribution of this paper is to pro- vide a coupling between the double Ising model and the bipartite dimer model, which keeps trackof XOR loop configurations, and is valid for graphs embedded in surfaces of genusg.

Section 6. Suppose now that the two Ising models are critical and defined on the dual of an infinite isoradial graphGfilling the whole plane, see Section 6.1 for definitions.

Then, the dimer model on the corresponding graphGQis also critical in the dimer sense.

Using the locality property of both probability measures on Ising [BdT11], and dimer configurations [dT07b] on isoradial graphs at criticality, we prove that the equality in law stated in Theorem 1.3 still holds in this infinite context. See Theorem 6.2.

Section 7. The graph GQ being bipartite, using a height function denoted h, dimer configurations can naturally be interpreted as discrete random interfaces. Our second theorem, see also Theorem 7.5, proves the following

Theorem 1.5. XOR loop configurations of the double Ising model defined onG have the same law as level lines of the restriction of the height functionhto vertices of the dual graphG.

Theorem 1.5 can be interpreted as a proof of Wilson’s conjecture mentioned above (see Section 7.2 for a precise statement) in the discrete setting, since it is known that the height function h, seen as a random distribution, converges in law in the scaling limit to 1π times the Gaussian free field in the plane [dT07b]. In particular, we explain the special value of the spacing. It would yield a complete proof of the conjecture if we could overcome the same technical obstacles as those of the proof of the convergence of double dimer loops to CLE4.

Acknowledgments: We would like to warmly thank Thierry Lévy for very helpful discussions on relative homology. We are also grateful to both referees for their useful comments.

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2 Ising model on graphs embedded in surfaces

In this section, we letGbe a graph embedded in a compact, orientable, boundaryless surface of genusg (g ≥ 0), and G be its dual graph. The embedding of G is chosen such that dual vertices are in the interior of the corresponding faces.

Fix some integerp≥0, and suppose first thatp≥1. For everyi∈ {0, . . . , p−1}, let Bibe a union of closed faces ofGhomeomorphic to a disc, such that∀i6=j,Bi∩Bj =∅. Denote by Σ the surface of genus g from which the union of the interiors of Bi’s is removed. Then Σ is a compact, orientable surface of genus g, with boundary ∂Σ =

∂B0∪ · · ·∂Bp−1. Whenp= 0, thenΣis the compact, orientable, boundaryless surface of genusgin which the graphGis embedded.

LetGΣ = (VΣ, EΣ)be the subgraph ofGdefined as follows: VΣconsists of vertices ofV ∩Σ; andEΣconsists of edges ofEjoining vertices ofVΣ, from which edges on the boundary∂Σare removed. LetGΣ= (VΣ, EΣ)be the subgraph ofGwhose vertices are vertices ofV∩Σ, and whose edges are edges ofGjoining vertices ofVΣ; see Figure 1 for an example. Note that the graphGΣ contains all edges dual to edges ofGΣ, i.e., there is a bijection between primal edges ofGΣand dual edges ofGΣ. Note that when p= 0,GΣ=GandGΣ=G.

B

1

B

0

B

2

γ

γ γ

3

1

2

γ

4

Figure 1: The graph G is a piece of Z2 embedded in the torus. The union of faces (Bi)i∈{0,1,2}is pictured in light grey. The graphGΣconsists of black plain lines, and the dual graphGΣof black dotted lines. The paths(γi)4i=1 definingdefects of Section 2.1 are drawn in thick black lines.

Fix a collection of positive constants (Je)e∈E attached to edges of G, referred to as coupling constants. The Ising model on GΣ with free boundary conditions and coupling constants(Je)is defined as follows. Aspin configurationσofGΣis a function of the vertices ofVΣ with values in {−1,+1}. The probability of occurrence of a spin configuration σis given by the Ising Boltzmann measure, denoted PIsing, and defined by:

∀σ∈ {−1,1}VΣ, PIsing(σ) = 1

ZIsing(GΣ, J)exp

 X

e=uv∈EΣ

Jeσuσv

,

whereZIsing(GΣ, J) = X

σ∈{−1,+1}VΣ

exp

 X

e=uv∈EΣ

Jeσuσv

is theIsing partition func-

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tion. Note that to simplify notation, the inverse temperature is included in the coupling constants.

2.1 Ising model with defect lines

When p ≥1, letN = 2g+p−1, and whenp = 0, letN = 2g. We now define2N versions of the original Ising model. Letγ

1,· · · , γ

N beNunoriented paths consisting of edges of the primal graphGΣ; see Figure 1 for an example, where

• for every i ∈ {1, . . . , g}, the paths γ

2i−1, γ

2i wind around thei-th handle in two transverse directions,

• whenp≥2, for everyi∈ {1, . . . , p−1}, the pathγ2g+i joins∂B0and∂Bi.

The pathsγi’s are thought as sets of edges. Let be one of the2N possible “unions”

of paths∪bi∈Iγ

i, where I ⊂ {1, . . . , N}and ∪b means that an edge with multiplicitykis present iffk≡1mod 2. Then, we change the sign of coupling constants of dual edges intersecting with. Spin configurations are defined as above, and so is the probability measure on spin configurations. This defines theIsing model with coupling constants (Je)and defect condition.

In fact, the appropriate framework for defining the Ising model with defects, is rel- ative homology theory, see Appendices A.2, A.3, and A.4. Thefirst homology group of Σrelative to its boundary ∂Σ is denoted byH1(Σ, ∂Σ;Z/2Z). The collection of paths (γ1, . . . , γ

N)defined above, is a representative of a basisΓ = (γ1, . . . , γN)of the first rel- ative homology groupH1(Σ, ∂Σ;Z/2Z)seen as aZ/2Z-vector space. In the case where p= 0,∂Σ =∅ and the first homology group ofΣrelative to its boundary is simply the first homology group.

Let denote the relative homology class of in H1(Σ, ∂Σ;Z/2Z). Then, it will be clear from the low-temperature expansion of Section 2.2 that the partition function is independent of the choice of basis and of the choice of representative of . As a consequence, we refer to this model as theIsing model with coupling constants (Je) and defect condition , and denote by ZIsing (GΣ, J) the corresponding partition func- tion. Nevertheless, since we want the change of signs of coupling constants to be well defined throughout the paper, we fix representatives of relative homology classes in H1(Σ, ∂Σ;Z/2Z), using the collection of pathsγ

1, . . . , γ

N defined above. Note that the original Ising model introduced has defect condition = 0 and is empty. Note also that this treatment is completely equivalent to considering the connected components of the boundary as the boundary of marked faces, and allowing insertion of disorder op- erators on these marked faces. However, the formulation in terms of defect conditions is natural in our context: the graphs on which the Ising model with defect conditions live, arise from the surgery of a larger graph embedded in a surface, and as such, their boundary have a real geometric meaning.

2.2 Low- and high-temperature expansion

Proposition 2.1 below extends thelow-temperature expansionof Kramers and Wan- nier [KW41a, KW41b] to the case of graphs embedded on a compact, orientable surface with boundary. It consists of rewriting the Ising partition function as a sum over poly- gon configurations of the graph GΣ, “separating” clusters of ±1 spins; see Figure 2 (left) for an example.

Apolygon configuration ofGΣis a subset of edges ofGΣ, such that vertices not on the boundary∂Σare incident to an even number of edges. There is no restriction for vertices on the boundary∂Σ. Let us denote byP(GΣ)the set of polygon configurations ofGΣ.

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γ4

+

+ + + + + +

+ + +

+ +

+ + + + + + + + + + + +

+ + + +

+

− −

− −

− − − −

− − − − + +

+ +

+ +

+ + +

+ + + + +

+ + + + + +

+

+ + + + + + + + + + + + + + + + +

− − −

Figure 2: Left: polygon configuration ofGΣ corresponding to a spin configuration of the Ising model with defect condition=γ4. Right: polygon configuration ofGΣ.

Let be an element ofH1(Σ, ∂Σ;Z/2Z), and letP(GΣ) denote the set of polygon configurations ofGΣwhose relative homology class inH1(Σ, ∂Σ;Z/2Z)is, meaning in particular that, for everyi, the number of edges on∂Bihas the same parity as2g+i.

This defines a partition ofP(GΣ):

P(GΣ) = [

∈H1(Σ,∂Σ;Z/2Z)

P(GΣ).

Proposition 2.1(Low-temperature expansion).

For every relative homology class∈H1(Σ, ∂Σ;Z/2Z), ZIsing (GΣ, J) = 2 Y

e∈EΣ

eJe X

P∈P(GΣ)

Y

e∈P

e−2Je. (2.1)

Proof. Suppose for the moment that is the class 0 ∈ H1(Σ, ∂Σ;Z/2Z), so that we deal with the usual Ising model. Using the identity (2.2) below, one can rewrite the partition function as a statistical sum over polygon configurations separating clusters of±1spins: ifσuandσvare two neighboring spins of an edgee=uv, then

eJeσuσv =eJe δuv}+e−2Jeδu6=σv}

. (2.2)

When injecting the right hand side in the expression of the Ising partition function, the product over dual edges e of eJe can be factored out. Since primal and dual edges are in bijection, this can also be written as a product over primal edges. Then, expanding the product, we get a product of contributions for all edges separating two neighboring spins with opposite signs. These edges form a polygon configurationP0of GΣseparating clusters of±1spins. As a consequenceP0has homology class0,i.e.,P0 belongs toP0(GΣ).

Conversely, any polygon configuration ofP0(GΣ)is the boundary of exactly two spin configurations, one obtained from the other by negating all spins, which explains the factor 2 on the right hand side of (2.1).

Suppose now that 6= 0. In the Ising model with defect condition, coupling con- stants of edges crossing paths of the representativeare negated. For these edges, the relation (2.2) should be replaced by the following:

e−Jeσuσv =eJe δu6=σv}+e−2Jeδuv}

.

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Note that, when comparing to (2.2), the two Kronecker symbols have been ex- changed. As a consequence, the construction of polygon configurations as above is slightly modified: the edge configuration, denoted byP, constructed from a spin con- figuration is obtained fromP0by switching the state of every edgeein; see Figure 2 (left). Then, the relative homology class ofP inH1(Σ, ∂Σ;Z/2Z)is:

[P] = [P0] + [] = 0 +=.

As a consequenceP belongs toP(GΣ)and this, independently of the choice of repre- sentative of. Conversely, any element ofP(GΣ)is obtained twice in this way.

For the sequel, it is useful to introduce a symbol for the sum over polygon configu- rations of the low-temperature expansion. For∈H1(Σ, ∂Σ;Z/2Z), define

ZLT (GΣ, J) = X

P∈P(GΣ)

Y

e∈P

e−2Je .

The partition function of the Ising model with defect conditioncan thus be rewritten as:

ZIsing (GΣ, J) = 2 Y

e∈EΣ

eJe

ZLT (GΣ, J).

Proposition 2.2 below extends the high-temperature expansion [KW41a, KW41b, Wan45] to the case of graphs embedded in a compact, orientable surface with boundary.

It consists of rewriting the Ising partition function as a sum over polygon configurations of the graphGΣ, this time. In this case, polygon configurations do not have a simple interpretation in terms of spin configurations.

Apolygon configurationofGΣ(or simplydual polygon configuration) is a subset of edges such that each vertex ofGΣis incident to an even number of edges, see Figure 2 (right) for an example. It is thus a union of closed cycles onGΣ. Let us denote byP(GΣ) the set of polygon configurations ofGΣ.

LetH1(Σ;Z/2Z)be the first homology group ofΣ, see Appendices A.1, A.3 and A.4.

Then, to each dual polygon configuration is assigned its homology class inH1(Σ;Z/2Z). For everyτ∈H1(Σ;Z/2Z), we letPτ(GΣ)denote the set of dual polygon configurations restricted to having homology classτinH1(Σ;Z/2Z). This defines a partition ofP(GΣ):

P(GΣ) = [

τ∈H1(Σ;Z/2Z)

Pτ(GΣ).

Proposition 2.2(High-temperature expansion).

For every relative homology class∈H1(Σ, ∂Σ;Z/2Z), ZIsing (GΣ, J) = 2|VΣ| Y

e∈EΣ

cosh(Je)

· X

τ∈H1(Σ;Z/2Z)

h(−1)(τ|) X

P∈Pτ(GΣ)

Y

e∈P

tanh(Je)i ,

(2.3) where(τ|)is theintersection formevaluated atτ and: it is the parity of the number of intersections of any representative ofτ and any representative of.

For details on the intersection form, see Appendix A.5.

Proof. This result is based on yet another way of rewriting the quantitye±Jeσuσv for a dual edgee=uvofEΣ.

e±Jeσuσv = coshJe±σuσvsinhJe

= coshJe(1±σuσvtanhJe). (2.4)

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The partition function is expanded into a sum of monomials in(σu)u∈V

Σ. In the ex- pansion, the spin variables come by pairs of neighborsσuσv and thus can be formally identified with the dual edge connectinguandv, associated with a weight±tanhJe. Each monomial is then interpreted as a subgraph ofGΣ, the degree ofσubeing the de- gree ofu in the corresponding edge configuration. Because of the symmetryσ↔ −σ, when re-summing over spin configurationsσ, only terms having even degree in each variable remain, giving a factor 2 per dual vertex, and other contributions cancel. As a consequence, the contributing monomials correspond to even subgraphsi.e., polygon configurations ofP(GΣ).

We now determine the sign of dual polygon configurations. Fix τ ∈ H1(Σ;Z/2Z) and a dual polygon configurationP ∈ Pτ(GΣ). Then, edges ofP carrying a negative weight are exactly those crossing edges of. As a consequence, the sign of the contri- bution ofPcorresponds to(−1)to the parity of the number of edges ofPintersecting with, this is exactly given by(τ|). The dual polygon configurationP thus has sign (−1)(τ|).

As in the case of the low-temperature expansion, it is useful to introduce a notation for the sum over dual polygon configurations of the high-temperature expansion. For τ∈H1(Σ;Z/2Z), define:

ZHTτ (GΣ, J) = X

P∈Pτ(GΣ)

Y

e∈P

tanh(Je) .

The relation between (2.1) and (2.3) can then be rewritten in the following compact form. For every relative homology class∈H1(Σ, ∂Σ;Z/2Z):

ZLT (GΣ, J) = 2|VΣ|−1 Y

e∈EΣ

cosh(Je) eJe

!

X

τ∈H1(Σ;Z/2Z)

h(−1)(τ|)ZHTτ (GΣ, J)i

. (2.5)

Remark 2.3. Relation(2.5)can be inverted using the orthogonality identity:

X

∈H1(Σ,∂Σ;Z/2Z)

(−1)(τ|)(−1)0|)= 2Nδτ,τ0, (2.6)

whereN = 2g+p−1whenp≥1, andN = 2gwhenp= 0. This orthogonality relation is proved as follows. The summand can be rewritten as(−1)(τ−τ0|). The application 7→(−1)(τ−τ0|)is a group homomorphism fromH1(Σ, ∂Σ;Z/2Z)toZ/2Z. Whenτ =τ0, this application is constant, equal to 1, all terms in the sum (2.6) equal 1, and the total sum equals2N. Otherwise, since the intersection pairing is non-degenerate (see Appendix A.5), the application 7→ (−1)(τ−τ0|) takes the values 1 and -1 the same number of times, and the sum (2.6)is zero. Using this identity, we obtain the inverted version of relation(2.5):

ZHTτ (GΣ, J) = 2−N−|VΣ|+1 Y

e∈EΣ

eJe cosh(Je)

!

X

∈H1(Σ,∂Σ;Z/2Z)

h(−1)(τ|)ZLT (GΣ, J)i .

3 Double Ising model on a boundaryless surface of genus g

In this section, we letGbe a graph embedded in a compact, orientable, boundaryless surfaceΣof genusg, andGdenote its dual graph. SinceΣhas no boundary, the first homology groupH1(Σ, ∂Σ;Z/2Z)ofΣrelative to its boundary is identified with the first homology groupH1(Σ,Z/2Z).

Instead of one Ising model on G, we now consider two copies of the Ising model, say a red one and a blue one, with the same coupling constants(Je). These two models

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are not taken to be completely independent: we require that they have the same defect conditions,i.e., we ask that polygon configurations coming from the low-temperature expansion of both spin configurations have the same homology class.

More precisely, from the point of view of the low-temperature expansion, we are interested in the probability measure Pd-Ising, on P := S

∈H1(Σ;Z/2Z)P(G)× P(G), defined by, for every(Pred, Pblue)∈P:

Pd-Ising(Pred, Pblue) = C

Q

e∈Prede−2Je Q

e∈Pbluee−2Je Zd-Ising(G, J) ,

whereC= 2Q

e∈EeJe2

, and the partition functionZd-Ising(G, J)is given by:

Zd-Ising(G, J) = X

∈H1(Σ;Z/2Z)

X

(Pred,Pblue)∈P(G)×P(G)

C Y

e∈Pred

e−2Je Y

e∈Pblue

e−2Je

= X

∈H1(Σ;Z/2Z)

(ZIsing (J))2.

Given a pair(Pred, Pblue)∈ P, and looking at the superimpositionPred∪PblueonG, one defines two new edge configurations:

• Mono(Pred, Pblue): consisting of monochromatic edges of the superimpositionPred∪ Pblue,i.e., edges covered by exactly one of the polygon configuration;

• Bi(Pred, Pblue): consisting of bichromatic edges of the superimposition,i.e., edges covered by both polygon configurations.

Edges which are not in the two configurations above are covered neither byPblue nor byPred. In Sections 3.1 and 3.2 below, we characterize these two sets of edges.

3.1 Monochromatic edges

Let ∈ H1(Σ;Z/2Z), and consider a pair of polygon configurations (Pred, Pblue)in P(G)× P(G). Then, it can be realized as four pairs of Ising spin configurations (±σ,±σ0), each with defect type , where coupling constants are negated along the representativeof, chosen in Section 2.1.

Following Wilson [Wil11], to each of the four pairs of spin configurations, one as- signs an XOR-spin configurationdefined as follows: at every vertex, the XOR-spin is the product of the Ising-spins at that same vertex.

Note that the four pairs of spin configurations yield two distinct XOR-spin configura- tions, one being obtained from the other by negating all spins. As a consequence, both XOR-spin configurations have the same polygon configuration separating clusters of±1 spins, meaning that this polygon configuration is independent of the choice of(±σ,±σ0) realizing(Pred, Pblue), let us denote it byXOR(Pred, Pblue). Note also, that although the definition of σ and σ0 depends on the particular choice of representative , the XOR polygon configuration does not: it is defined intrinsically from(Pred, Pblue).

Lemma 3.1. For every pair of polygon configurations(Pred, Pblue)∈P, the monochro- matic edge configurationMono(Pred, Pblue)is exactly the XOR loop configuration XOR(Pred, Pblue). In particular, it is a polygon configuration ofP0(G).

Proof. Fix a pair of red and blue polygon configurations(Pred, Pblue)∈P(G)× P(G)for some∈H1(Σ;Z/2Z). Let(σ, σ0)be one of the four pairs of spin configurations whose low-temperature expansion is(Pred, Pblue). We need to show that, for every edgeeofG,

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eis monochromatic, if and only if XOR-spins at verticesu, v of the dual edge e are distinct. Suppose thatedoes not belong to. Then,

- the edgeeis red only⇔ σu6=σv and σu00v, - the edgeeis blue only⇔ σuv and σ0u6=σv0.

Ifebelongs to, the two above conditions hold with colors exchanged. In all cases,eis monochromatic if and only if XOR spins at verticesuandvare distinct.

Being the boundary of some domain, the set of monochromatic edges must be a polygon configuration ofG, with homology class0.

3.2 Bichromatic edge configurations

Before describing features of bichromatic edge configurations, we recall some gen- eral facts. A polygon configurationP of the graph Gseparates the surfaceΣinto nP connected componentsΣ1, . . . ,ΣnP, where nP ≥ 1. For every i ∈ {1, . . . , nP}, Σi is a surface of genus gi with boundary ∂Σi. The boundary is either empty or consists of cycles ofΣ.

As in Section 2,GΣidenotes the subgraph ofG, whose vertex setVΣi isV ∩Σi, and whose edge setEΣi consists of edges ofE joining vertices ofVΣi, from which edges on the boundary∂Σiare removed. The dual graph is denoted byGΣ

i.

Recall thatH1i, ∂Σi;Z/2Z)denotes the first homology group ofΣi relative to its boundary. Consider the morphismΠi = ΠΣ,Σi, fromH1(Σ;Z/2Z)toH1i, ∂Σi;Z/2Z) defined as follows: for every∈H1(Σ;Z/2Z),Πi()is the homology class in

H1i, ∂Σi;Z/2Z)of the restriction of any representativeoftoΣi, see Appendix A.6 for details.

The following lemma characterizes bichromatic edge configurations.

Lemma 3.2. Fix ∈ H1(Σ;Z/2Z), and let P ∈ P0(G) be a polygon configuration, separating the surfaceΣinto connected componentsΣ1, . . . ,ΣnP.

• If there exists a pair of polygon configurations(Pred, Pblue)∈ P(G)× P(G)such thatMono(Pred, Pblue) =P; then, for everyi∈ {1, . . . , nP}, the restriction of bichro- matic edges toGΣi is the low-temperature expansion of an Ising configuration on GΣ

i, with coupling constants(2Je)and defect conditionΠi(). As a consequence, it is a polygon configuration inPΠi()(GΣi).

• Given, for everyi ∈ {1, . . . , nP}, a polygon configurationPi ∈ PΠi()(GΣi), there are2nP−1 pairs(Pred, Pblue)∈ P(G)× P(G)such thatMono(Pred, Pblue) =P and such that, for every i ∈ {1, . . . , nP}, the restriction of bichromatic edges to GΣi isPi.

Proof.

• Suppose that there exists a pair of polygon configurations(Pred, Pblue)ofP(G)× P(G)such thatMono(Pred, Pblue) =P. Then, for everyi∈ {1, . . . , nP}, the restric- tion of bichromatic edges toGΣi exactly consists of the restriction toGΣi of one of two original polygon configurations. Since this polygon configuration has homol- ogyinΣ, the homology class inH1i, ∂Σi;Z/2Z)of the restriction toΣiisΠi() by definition. As a consequence, the bichromatic edge configuration on Σi is a polygon configuration ofPΠi()(GΣi). Moreover, since all edges in the bichromatic configuration are present twice, and since the weight of pairs of polygon configu- rations is the product of the edge-weights contained in the pair of configurations, the effective weight of a bichromatic edgeeis squared and becomes:

e−2Je2

=e−2(2Je),

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which corresponds to a doubling of the coupling constants.

• There are two spin configurations, denoted by±ξ, whose low-temperature expan- sion is P. Suppose that there exists a pair of spin configurations (σ, σ0) whose low-temperature expansion hasP as monochromatic edges, thenσσ0=±ξ. Let us assumeσσ0 =ξ, the argument being similar in the other case, this has the effect of adding a global factor 2 when speaking of spin configurations. The relation σσ0 =ξimplies that there is freedom of choice for exactly one spin configuration, sayσ, the other being determined by their productξ.

Consider a connected componentΣi, and a polygon configurationPi ∈ PΠi()(GΣi). We wantPi to consist of doubled edges, so that in particular, it must contain all red edges. There are thus two choices for the first spin configuration of GΣ

i, denoted by ±σi. This holds for every i ∈ {1, . . . , nP} and thus defines 2nP spin configurations (±σ1, . . . ,±σnP) of G. Recall that in each of the 2nP cases, the second spin configuration is determined by the conditionσσ0 =ξ. Since on each connected componentΣi,ξis identically equal to±1, we deduce that(σ0)i=±σi. As a consequence, the low-temperature expansion ofσ0 exactly consists of edges of Pi, i.e., Pi consists of red and blue edges. Summarizing, there are 2 ·2nP pairs of spin configurations, or2nP−1pairs of polygon configurations(Pred, Pblue), such that monochromatic edges are those ofP and bichromatic edges those ofPi, i ∈ {1, . . . , nP}. Note that by construction (choice ofσi’s), each polygon configu- rationPred,Pblueis inP(G).

Consider a polygon configuration P ∈ P0(G), and let ∈ H1(Σ,Z/2Z). Denote by Wd-Ising[Mono =P]the contribution of the set

{(Pred, Pblue)∈ P(G)× P(G) : Mono(Pred, Pblue) =P},

to the partition function(ZIsing (J))2, and by Wd-Ising[Mono =P] = X

∈H1(Σ;Z/2Z)

Wd-Ising[Mono =P].

By the low-temperature expansion of the Ising partition function, the weight of each polygon configurationPred, Pblue is the product of edge-weights contained in the con- figuration. As a consequence, the contribution of(Pred, Pblue)can be decomposed as a product over monochromatic edges, and bichromatic edges of each of the components.

Using Lemmas 3.1 and 3.2, this yields

Proposition 3.3. For every polygon configurationP ∈ P0(G)and every∈H1(Σ,Z/2Z),

Wd-Ising[Mono =P] = 2−1C Y

e∈P

e−2JeYnP

i=1

2ZLTΠi()(GΣi,2J)

, (3.1)

whereC= 2Q

e∈EeJe2

. Moreover, the double Ising partition function can be rewrit- ten as:

Zd-Ising(J) = X

P∈P0(G)

Wd-Ising[Mono =P],

and the probability measurePd-Isinginduces a probability measure on polygon configu- rations ofP0(G), given by:

∀P ∈ P0(G), Pd-Ising[Mono =P] = Wd-Ising[Mono =P]

Zd-Ising(J) . (3.2)

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4 Mixed contour expansion

In [Nie84], Nienhuis rewrites the partition function of the Ashkin–Teller model on the square lattice as a statistical sum over polygon families onGandG which do not intersect. We apply the same approach to the double Ising model onΣbut some care is required to keep track of the homology class of the polygon configurations involved.

We fix∈H1(Σ;Z/2Z)and a polygon configurationP ∈ P0(G). In Proposition 4.1, we apply the low/high-temperature duality to each of the termsZLTΠi()(GΣi,2J)involved in the expression ofWd-Ising[Mono =P]of Equation (3.1). This has the effect of trans- forming bichromatic polygon configurations ofGΣi into dual polygon configurations of GΣ

i. Then, in Proposition 4.3, we sum over ∈ H1(Σ;Z/2Z), and show that the out- come simplifies to a sum over dual polygon configurations of the dual graph, having 0 homology class inH1(Σ;Z/2Z), and not intersectingP.

Proposition 4.1. For every polygon configurationP ∈ P0(G)and every∈H1(Σ,Z/2Z),

Wd-Ising[Mono =P] =C0 Y

e∈P

2e−2Je 1 +e−4Je

×

×

nP

Y

i=1

h X

τi∈H1i,Z/2Z)

(−1)ii()) X

Pi∈Pτ i(GΣ

i)

Y

e∈Pi

1−e−4Je 1 +e−4Je

i ,

where,C0= 2|V|+1 Q

e∈Ecosh(2Je) .

Proof. The expression forWd-Ising[Mono =P]of Equation (3.1) can be rewritten as:

Wd-Ising[Mono =P] = 2nP−1C Y

e∈P

e−2JeYnP

i=1

ZLTΠi()(GΣi,2J) .

For every i ∈ {1, . . . , nP}, the contribution, ZLTΠi()(GΣi,2J)is the low-temperature expansion of an Ising model on vertices of VΣ

i with coupling constants 2Je and de- fect conditionΠi(). Using the relation between Kramers and Wannier’s low and high- temperature expansions of (2.5), it can be expressed as:

ZLTΠi()(GΣi,2J) =Ai× X

τi∈H1i;Z/2Z)

(−1)ii())ZHTτi (GΣi,2J),

where

Ai= 2|VΣi|−1 Y

e∈EΣi

cosh(2Je) e2Je .

Let us first compute the part which is independent of. Observing that the collection of sets of dual vertices(VΣ

i)ni=1P is a partition ofV, one writes:

2nP−1C Y

e∈P

e−2Je

nP

Y

i=1

Ai

= 2nP−122 Y

e∈E

e2Je Y

e∈P

e−2Je

2|V|−nP Y

e∈EΣi

cosh(2Je) e2Je

.

Noticing that the collection of edges in theΣi’s is exactly the set of edges ofGnot inP, we have:

2nP−1C Y

e∈P

e−2Je

nP

Y

i=1

Ai

= 2|V|+1 Y

e∈E

cosh(2Je) Y

e∈P

cosh(2Je)−1 .

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