The energy density in the planar Ising model

c

2013 by Institut Mittag-Leffler. All rights reserved

The energy density in the planar Ising model

by

Cl´ement Hongler

Columbia University New York, NY, U.S.A.

Stanislav Smirnov

Universit´e de Gen`eve Gen`eve, Switzerland

Contents

1. Introduction . . . 191

1.1. The model . . . 191

1.2. The energy density . . . 192

1.3. Graph notation . . . 194

1.4. The energy density . . . 196

1.5. Contour statistics and discrete fermionic correlators . . . 198

1.6. Convergence results and proof of Theorem 1.1 . . . 202

1.7. Proof of convergence . . . 204

2. Analysis of the discrete fermionic correlators . . . 204

2.1. Discrete holomorphicity . . . 205

2.2. Singularity . . . 209

2.3. Boundary values . . . 210

2.4. Discrete integration . . . 210

3. Convergence of the discrete fermionic correlators . . . 212

3.1. Precompactness . . . 213

3.2. Identification of the limit . . . 215

Appendix A. Proof of Lemma 1.4 . . . 220

Appendix B. The full-plane fermionic correlator . . . 221

References . . . 224

1. Introduction 1.1. The model

The Lenz–Ising model in two dimensions is probably one of the most studied models for an order-disorder phase transition, exhibiting very rich and interesting behavior, yet well understood both from the mathematical and physical viewpoints [B], [MW2], [P].

After Kramers and Wannier [KW] derived the value of the critical temperature and Onsager [O] analyzed the behavior of the partition function for the Ising model on the

two-dimensional square lattice, a number of exact derivations were obtained by a variety of methods. Thus it is often said that the 2D Ising model isexactly solvableorintegrable.

Moreover, it has a conformally invariant scaling limit at criticality, which allows one to use conformal field theory (CFT) orSchramm’s SLE techniques. CFT provides predictions for quantities like the correlation functions of the spin or the energy fields, which in principle can then be related to SLE.

In this paper, we obtain a rigorous exact derivation of the one-point function of the energy density, matching the CFT predictions [DMS], [C], [BG]. We exploit the integrable structure of the 2D Ising model, but in a different way from the one employed in the classical literature. Our approach is rather similar to Kenyon’s approach to the dimer model [Ken].

We write the energy density in terms of discrete fermionic correlators, of the form introduced in [S1]. These correlators solve a discrete version of a Riemann boundary value problem, which identifies them uniquely. In principle, this could be used to give an exact, albeit very complicated, formula, that one could try to simplify—a strategy similar to most of the earlier approaches. Instead we pass to the scaling limit, showing that the solution to the discrete boundary value problem approximates well its continuous counterpart, which can be easily written using conformal maps. Thus we obtain a short expression, approximating the energy density to the first order. Moreover, our method works in any simply connected planar domain, and the answer is, as expected, conformally covariant.

The fermionic approach to the Ising model was introduced by Kaufman [Ka]. The fermionic correlators were in particular studied by Kadanoff and Ceva [KC] and later in Mercat [M], but their scaling limits with boundary conditions were not discussed before [S1]. Our results have been generalized in [Ho], where the limits ofn-point correlation functions of the energy density are obtained, via the introduction of multipoint fermionic correlators. Scaling limits of then-pointspincorrelations are obtained in [CHI], via the introduction of spinors and the analysis of their scaling limits.

Recall that the Ising model on a graphGis defined by a Gibbs probability measure on configurations of ±1 (or up/down) spins located at the vertices: it is a random assignment (σx)_x∈V of±1 spins to the vertices V of G and the probability of a state is proportional to its Boltzmann weighte^−βH, whereβ >0 is the inverse temperature of the model andH is the Hamiltonian, or energy, of the state σ. In the Ising model with no external magnetic field, we haveH:=−P

i∼jσiσj, where the sum is over all the pairs of adjacent vertices ofG.

1.2. The energy density

Let Ω be a Jordan domain and let Ωδ be a discretization of it by a subgraph of the square grid of mesh sizeδ >0. We consider the Ising model on the graph Ωδ at the critical inverse temperatureβc=¹₂log(√

2+1); on the boundary of Ωδ, we may impose the value +1 to the spins or let them free (we call these + andfree boundary conditions respectively).

Our main result about the energy density is the following.

Theorem. Let a∈Ωand for each δ >0,let hxδ, y_δi be the closest edge to a in Ω_δ. Then, as δ!0,we have

E⁺

σ_xδσ_yδ−

√2 2

=l_Ω(a)

2π δ+o(δ) and Efree

σ_xδσ_yδ−

√2 2

=−l_Ω(a)

2π δ+o(δ), where the subscripts +and freedenote the boundary conditions and l_Ωis the element of the hyperbolic metric of Ω.

A precise version of this theorem in terms of the energy density field is given in§1.4.

This result has been predicted for a long time by CFT methods (see [DMS] and [BG]

for instance), notably using Cardy’s celebrated mirror image technique [C]. However, the CFT approach does not allow one to determine the lattice-specific constant 1/2π appearing in front of the hyperbolic metric element.

This is one of the first results where full conformal invariance (i.e. not only M¨obius invariance) of a correlation function for the Ising model is actually shown. The proof does not appeal to the SLE machinery, although the fermionic correlator that we use is very similar to the one employed to prove convergence of Ising interfaces to SLE(3) [CS2]. A generalization of our result with mixed boundary conditions could also be used to deduce convergence to SLE.

In the case of the full plane, the energy density correlations have been first computed by Hecht [He] using transfer matrix techniques. These results were later generalized by Boutillier and De Tili´ere, using dimer model techniques [BT1], [BT2]. However, their approach works only in the infinite-volume limit or in periodic domains and does not directly apply to arbitrary bounded domains.

In the case of the half-plane, the energy density one-point function has been recently obtained by Assis and McCoy [AM] (passing to the limit the finite-scale results of [MW1]), using transfer matrix techniques.

The strategy for the proof of our theorem relies mainly on:

• The introduction of a discrete fermionic correlator, which is a complex deformation of a certain partition function, and of an infinite-volume version of this correlator.

• The expression of the energy density in terms of discrete fermionic correlators.

• The proof of the convergence of the discrete fermionic correlators to continuous ones, which are holomorphic functions.

1.3. Graph notation

Let us first give some general graph notation. LetGbe a graph embedded in the complex planeC.

• We denote by V_G the set of the vertices ofG, by E_G the set of its (unoriented) edges, and byE~_G the set of its oriented edges.

• We identify the vertices V_G with the corresponding points in the complex plane (since G is embedded). An oriented edge is identified with the difference of the final vertex minus the initial one.

• Two vertices v1, v2∈V_G are said to be adjacent if they are the endpoints of an edge, denotedhv1, v2i, and two distinct edges e1, e2∈E_G are said to beincident if they share an endvertex.

1.3.1. Discrete domains

• We denote by Cδ the square grid of mesh size δ >0. Its vertices and edges are defined by

V_Cδ:={δ(j+ik) :j, k∈Z} and E_Cδ:={hv1, v2i:v1, v2∈ V_Cδ and|v1−v2|=δ}.

• In order to keep the notation as simple as possible, we will only look at finite induced subgraphs Ωδ of Cδ (two vertices of Ωδ are linked by an edge in Ωδ whenever they are linked inCδ), that we will also calldiscrete domains.

• For a discrete domain Ωδ, we denote by Ω^∗_δ thedual graph of Ωδ: its verticesVΩ^∗_δ

are the centers of the bounded faces of Ωδ and two vertices ofVΩ^∗_δ are linked by an edge ofEΩ^∗_δ if the corresponding faces of Ωδ share an edge.

• We denote by∂VΩ_δ the set of vertices of V_Cδ\VΩ_δ that are at distance δ from a vertex ofVΩ_δ (i.e. that are adjacent inCδ to a vertex ofVΩ_δ) and by∂EΩ_δ⊂E_Cδ the set of edges between a vertex of VΩ_δ and a vertex of ∂VΩ_δ. The vertices in ∂VΩ_δ appear with multiplicity: if a vertex ofV_Cδ\VΩ_δ is at distanceδ to several vertices ofVΩ_δ, then it appears as so many distinct elements of∂VΩ_δ. In other words, there is a one-to-one correspondence between∂VΩ_δ and∂EΩ_δ.

• We denote by∂VΩ^∗_δ the centers of the faces of Cδ that are adjacent to a face of Ω_δ. We denote by∂EΩ^∗_δ⊂E_C^∗_δ the set of dual edges between a vertex ofVΩ^∗_δ and a vertex of∂VΩ^∗_δ. For an edge e∈EΩδ we denote bye^∗∈EΩ^∗_δ its dual (eand e^∗ intersect at their midpoint).

VΩ_δ

∂VΩ_δ

VΩ^∗_δ

∂VΩ^∗_δ

V_ΩM δ

EΩ_δ

E_ΩM δ

∂EΩ_δ

δ

Figure 1.1. Notation for discrete domains

• We writeVΩ_δ forVΩ_δ∪∂VΩ_δ andVΩ^∗_δ forVΩ^∗_δ∪∂VΩ^∗_δ.

• We denote by E_Ω^h

δ⊂EΩ_δ the set of the horizontal (i.e. parallel to the real axis) edges of Ωδ and byE_Ω^v

δ:=EΩ_δ\E_Ω^h

δ the set of the vertical ones.

• We denote by Ω^M_δ the medial graph of Ωδ: its vertices V_ΩM

δ are the midpoints of the edges ofEΩ_δ∪∂EΩ_δ and the medial edges E_ΩM

δ link midpoints of incident edges of EΩ_δ∪∂EΩ_δ.

• We say that a family (Ωδ)δ>0 of discrete domains (with Ωδ⊂Cδ for each δ >0) approximates or discretizes a continuous domain Ω if for each δ >0, Ωδ is the largest connected induced subgraph ofCδ contained in Ω.

1.3.2. The Ising model with boundary conditions

The Ising model (withfree boundary conditions) on a finite graphG(in this paper,Gwill be a discrete domain Ωδ or its dual Ω^∗_δ) atinverse temperature β >0 is a model whose state space Ξ_G is given by Ξ_G:={(σx)_x∈VG:σx∈{±1}}: a state assigns to every vertexx ofGa spin σx∈{±1}. The probability of a configurationσ∈Ξ_G is

P^β,freeG {σ}:= 1

Z_G^β,freee^{−βHGβ,free(σ)},

with theenergy (or Hamiltonian)H_G^β,free of a configurationσgiven by H_G^β,free(σ) :=− X

hx,yi∈EG

σ_xσ_y,

and thepartition function Z_G^β,free by

Z_G^β,free:= X

σ∈ΞG

e^−βH(σ).

Given a graph G with boundary vertices∂V_G (like Ω^∗_δ with∂V_Ω^∗

δ) the Ising model on G with + boundary condition is defined as the Ising model on G, with extra spins located at the vertices of∂V_G that are set to +1 and with energy

H_G^β,+(σ) :=− X

hx,yi∈E¯G

σxσy,

where ¯E_G is the set of edges linking vertices ofV_G∪∂V_G.

In this paper, we will be interested in the Ising model with free and + boundary conditions on discrete square grid domains Ωδ at the critical inverse temperature

βc:=1 2log(√

2+1), when the mesh sizeδis small.

We will from now on omit the inverse temperature parameterβ in the notation and will denote byP^freeG andP⁺G the probability measures of the Ising model onG at β=β_c with free and + boundary conditions and byE^freeG andE⁺Gthe corresponding expectations.

1.4. The energy density

Let Ωδ be a discrete domain and let aδ∈V_ΩM

δ be the midpoint of a horizontal edge of Ωδ. We introduce the two quantities hεδ(aδ)i^free_Ωδ and hεδ(aδ)i⁺_Ω∗

δ, called average energy density (with free and + boundary conditions), defined by

hεδ(aδ)i^free_Ω

δ :=E^freeΩ_δ

σe_δσw_δ−

√2 2

and hεδ(aδ)i⁺_Ω∗ δ:=E⁺Ω^∗_δ

σn_δσs_δ−

√2 2

, whereheδ, wδi∈EΩ_δ andhnδ, sδi∈EΩ^∗_δ are respectively the (horizontal) edge and the dual (vertical) edge, the midpoint of both of which is aδ (see Figures 1.2 and 1.3). The quantity ¹₂√

2 is the infinite-volume limit of the product of two adjacent spins (it can be found in [MW2, Chapter VIII, Formula 4.12], for instance). The energy density field is the fluctuation of the product of adjacent spins around this limit: it measures the distribution of the energyH among the edges, as a function of their locations. We are considering horizontal edges on Ωδ and vertical edges on Ω^∗_δ for concreteness and for making the notation simpler, but our results are rotationally invariant.

We can now state the main result of this paper, which is the conformal covariance of the average energy density.

+

+

+

+

+

+ + + + + +

+ +

+ +

+ +

+

+ + + +

+ +

+ +

+ +

+ + + + +

+ + +

+ +

-

- -

-

- -

- -

- -

- - - -

a

Figure 1.2. The Ising model onV_Ω∗

δ with + boundary condition, with the contours corre- sponding to its low-temperature expansion.

Theorem 1.1. Let Ωbe a C¹ simply connected domain and let a∈Ω. Consider a family (Ω_δ)_δ>0 of discrete domains approximating Ωand for each δ >0,let a_δ∈V_ΩM

δ be the midpoint of the horizontal edge that is the closest to a. Then,as δ!0,uniformly on the compact subsets of Ω,we have

1

δhεδ(aδ)i⁺_Ω∗ δ! 1

2π`Ω(a) and 1

δhεδ(aδ)i^free_Ω∗ δ !− 1

2π`Ω(a),

`Ω(a)being the hyperbolic metric element of Ωat a. Namely, `Ω(a):=2ψ_a⁰(a),where ψa

is the conformal mapping from Ωto the unit disk D:={z∈C:|z|<1}such that ψa(a)=0 and ψ⁰_a(a)>0.

The proof will be given in§1.6.

Corollary 1.2. The conclusions of Theorem 1.1 hold under the assumption that Ωis a Jordan domain.

Proof. We have that hεδ(aδ)i⁺_Ω

δ and hεδ(aδ)i^free_Ω

δ are, respectively, non-increasing and non-decreasing with respect to the discrete domain Ωδ, as follows easily from the Fortuin–Kasteleyn–Ginibre inequality applied to the Fortuin–Kasteleyn representation of the model (see [G, Chapters 1 and 2], for instance): if Ωδ⊂eΩδ, then

hεδ(aδ)i⁺_Ω∗

δ>hεδ(aδ)i⁺

Ωe^∗_δ and hεδ(aδ)i^free_Ωδ 6hεδ(aδ)i^free

Ωe_δ .

+

+

+ + +

+ + +

+ +

+

+ + + +

+

+ + +

+ +

+ + +

- -

- - -

-

- -

- -

- -

-

- -

-

- -

- - - -

a

Figure 1.3. The Ising model onV_Ωδ with free boundary condition.

If Ω is a Jordan domain, we can approximate Ω by increasing and decreasing sequences of smooth domains, for which Theorem1.1applies, and deduce the result for Ω.

The central idea for proving Theorem 1.1is to introduce a discrete fermionic cor- relator which is a two-point function f_Ωδ(a, z); it is defined in the next subsection. We then relatef_Ωδ to the average energy density and prove its convergence to a holomorphic functionf_Ω.

1.5. Contour statistics and discrete fermionic correlators 1.5.1. Contour statistics

Let Ωδ be a discrete domain. We denote byCΩ_δ the set of edge collections ω⊂EΩ_δ such that every vertex v∈VΩ_δ belongs to an even number of edges of ω: in other words, by Euler’s theorem for walks, the edge collections ω∈CΩ_δ are the ones that consist of edges forming (not necessarily simple) closed contours. For an edge e∈EΩ_δ, we denote byC_Ω^{e+}

δ the set of configurations ω∈CΩ_δ thatdo not contain eand byC_Ω^{e−}

δ the set of configurations thatdo contain e.

Setα:=√

2−1. For a collection of edgesω⊂EΩδ, we denote by |ω| its cardinality.

Fore∈E_Ωδ we define Z_Ωδ:= X

ω∈CΩδ

α^|ω|, Z^{e_Ω^+}

δ := X

ω∈C^{e+}_Ω

δ

α^|ω| and Z^{e_Ω^−}

δ := X

ω∈C^{e−}_Ω

δ

α^|ω|.

We now have the following representation of the energy density in terms of contour statistics.

Proposition 1.3. Let e∈E_Ω^h

δ be a horizontal edge and let its midpoint be a∈V_ΩM δ . Then we have

hεδ(a)i⁺_Ω∗

δ=Z^{e_Ω^+}

δ

Z_Ωδ −Z^{e_Ω^−}

δ

Z_Ωδ −

√2

2 = 2Z^{e_Ω^+}

δ

Z_Ωδ −

√2+2

2 and hεδ(a)i^free_Ω

δ =−hεδ(a)i⁺_Ω∗ δ. Proof. From the low-temperature expansion of the Ising model (see [P, Chapter 1], for instance), there is a natural bijection between the configurations of spins σon VΩ^∗_δ

with + boundary condition on∂VΩ^∗_δ, and the edge collections ω∈CΩδ: one puts an edge e∈EΩδ in the edge collectionωif the spins ofσat the endpoints of the dual edgee^∗∈EΩ^∗_δ

are different. It is easy to see that the probability measure on C_Ωδ induced by this bijection gives to each edge collection ω∈C_Ωδ a weight proportional to (e^−2β)^|ω|, and hence to α^|ω|, where α=√

2−1 as above (since β=β_c=¹₂log(√

2+1)). The event that the spins at two adjacent dual verticesx, y∈V_Ω^∗

δ are the same (respectively are different) corresponds through the natural bijection toC^{e_Ω^+}

δ (respectivelyC_Ω^{e−}

δ ), wheree∈EΩ_δ is such thate^∗=hx, yi. Using that Z^{e_Ω^+}

δ +Z^{e_Ω^−}

δ =ZΩ_δ, we deduce the first identity.

From the so-called high-temperature expansion (see [P, Chapter 1]) we have that for the Ising model on Ω_δ with free boundary condition, the correlation of two spins z₁, z₂∈V_Ωδ is equal to

P

ω∈Ce Ωδ(z1,z2)(tanhβ)^|ω|e P

ω∈CΩδ

(tanhβ)^|ω| ,

whereCΩ_δ(z1, z2) is the set of edge collectionsωe such that every vertex inVΩ_δ\{z1, z2} belongs to an even number of edges of ωe and such that z1 and z2 both belong to an odd number of edges ofω. Ate β=βc, we have tanhβ=α(the fact that tanhβc=e^−2βc actually characterizesβc).

Let us now takez₁ andz₂ adjacent, sete:=hz₁, z₂i∈E_Ωδ, and denote byC_Ω⁺

δ(z₁, z₂) and C_Ω⁻

δ(z₁, z₂) the sets of ωe∈C_Ωδ(z₁, z₂) such that e∈eω and e /∈ωe respectively. From eachωe∈C_Ω⁺

δ(z₁, z₂), we can remove e and obtain an edge collection in C^{e_Ω^+}

δ (this map C_Ω⁺

δ(z₁, z₂)!C_Ω^{e+}

δ is bijective) and to each ωe∈C_Ω⁻

δ(z₁, z₂), we can add eand obtain an edge collection inC_Ω^{e−}

δ . Hence we have hε_δ(a)i^free_Ω

δ = P

ω∈Ce _Ω⁺

δ(z1,z2)α^|ω|e Z_Ωδ +

P

ω∈Ce ⁻_Ω

δ(z1,z2)α^|ω|e Z_Ωδ −

√2 2

= αP

ω∈C^{e+}_Ω

δ

α^|ω|

Z_Ωδ + α⁻¹P

ω∈C_Ω^{e−}

δ

α^|ω|

Z_Ωδ −

√2

2 =αZ^{e_Ω^+}

δ +α⁻¹Z^{e_Ω^−}

δ

Z_Ωδ −

√2 2 . Using the relationZ^{e_Ω^+}

δ +Z^{e_Ω^−}

δ =Z_Ωδ, we obtain the second identity.

1.5.2. The discrete fermionic correlator in bounded domains Ifa∈V_ΩM

δ is the midpoint of a horizontal edgee₁∈E_Ω^h_δ andz∈V_ΩM

δ is the midpoint of an arbitrary edgee₂∈EΩδ∪∂EΩδ, we denote by CΩδ(a, z) the set ofγ consisting of edges of E_Ωδ\{e₁, e₂} and of two half-edges (half of an edge between its midpoint and one of its ends) such that

• one of the half-edges has endpointsaanda+¹₂δ;

• the other half-edge is incident toz;

• every vertexv∈V_Ωδ belongs to an even number of edges or half-edges ofγ.

For a configurationγ∈C_Ωδ(a, z), we say that a sequencee₀, e₁, ..., e_nis anadmissible walk along γif

• e0is the half-edge incident to a;

• en is the half-edge incident toz;

• e1, ..., e_n−1∈EΩ_δ are edges;

• ej andej+1 are incident for each j∈{0, ..., n−1};

• each edge appears at most once in the walk;

• when one follows the walk and arrives at a vertex that belongs to four edges or half-edges ofγ (we call this anambiguity), one either turns left or right (going straight in that case is forbidden).

It is easy to see that, for any γ∈CΩ_δ(a, z), such a walk always exists, though in general it is not unique (see Figure1.4).

Given a configuration γ∈CΩ_δ(a, z) and an admissible walk along γ, we define the winding number Wofγ, denotedW(γ)∈R/4πZ, by

W(γ) :=¹₂π(n_l−nr),

where n_l and n_r are the number of left turns and right turns, respectively, that the admissible walk makes fromatoz: it is the total rotation of the walk betweenaandz, measured in radians. More generally, we define the winding number of a rectifiable curve as its total rotation from its initial point to its final point, measured in radians.

The following lemma shows that the winding number (modulo 4π) of a configuration γ∈CΩ_δ(a, z) is actually independent of the choice of the walk onγ.

Lemma 1.4. For any γ∈CΩ_δ(a, z), the winding number W(γ)∈R/4πZ is indepen- dent of the choice of admissible walk along γ.

The proof is given in Appendix A.

Due to Lemma1.4, we can now define the discrete fermionic correlatorf_Ωδ that will be instrumental in our studies of the energy density.

a z

Figure 1.4. An admissible walk in a configuration inCΩ_δ(a, z). There are sixteen different choices of such walks in this configuration.

Definition 1.5. For any midpoint of a horizontal edgea∈V_ΩM

δ , we define the discrete fermionic correlatorfΩ_δ(a,·):V_ΩM

δ !Cby fΩ_δ(a, z) := 1

ZΩ_δ

X

γ∈CΩδ(a,z)

α^|γ|e^{−(i/2)W(γ)} and fΩ_δ(a, a) :=Z^{e_Ω^+}

δ

ZΩ_δ

,

where|γ|denotes the number of edges and half-edges ofγ, with the half-edges contribut- ing ¹₂ each.

In this way, z7!fΩ_δ(a, z) is a function whose value at z=agives (up to an additive constant) the average energy density ata. As we will see, moving the pointz across the domain will allow us to gain information about the effect of the geometry of the domain on the energy density.

1.5.3. The discrete fermionic correlator in the full plane As mentioned above, the ¹₂√

2 appearing in the definition of the average energy density (§1.4) is the infinite volume limit (or full-plane) average product of two adjacent spins, which one has to subtract in order for the effect of the shape of the domain to be studied. We now introduce a full-plane version of the discrete fermionic correlator, whose definition a priori seems quite different from the bounded domain version. It will allow us to represent the energy density (with the correct additive constant) in terms of the difference of two discrete fermionic correlators.

Definition 1.6. For a, z∈V_ΩM

δ with a6=z and abeing the midpoint of a horizontal edge, definef_Cδ by

f_Cδ(a, z) := cosπ 8

e^πi/8

C₀

2 a+¹₂δ δ ,2z

δ

+C₀

2 a−¹₂iδ δ ,2z

δ

+sinπ 8

e^−3πi/8

C₀

2 a−¹₂δ δ ,2z

δ

+C₀

2 a+¹₂iδ δ ,2z

δ

, where C0(z1, z2):=C0(0, z2−z1) is the dimer coupling function of Kenyon (see [Ken]), defined (on{(z1, z2)∈Z²:z1+z2 is odd}) by

C(0, x+iy) := 1 4π²

Z 2π 0

Z 2π 0

e^i(xθ−yφ)

2isinθ+2 sinφdθ dφ.

We setf_Cδ(a, a):=¹₄(2+√

2). This value corresponds to the limit off_Ωδ(a, a)=Z^{e_Ω^+}

δ /Z_Ωδ when Ω_δ!Cδ. We will not use this fact, though, but rather that f_Cδ(a,·) and f_Ωδ(a,·) have the same discrete singularity ata(see Propositions2.6and2.9).

From our definitions up to now and from Proposition1.3, we deduce the following result.

Lemma1.7. Let Ωδ be a discrete domain anda∈V_ΩM

δ be the midpoint of a horizontal edge of Ωδ. Then the average energy density can be represented as

hε_δ(a)i⁺_Ω∗

δ= 2(f_Ωδ−f_Cδ)(a, a) and hε_δ(a)i^free_Ω

δ =−2(f_Ωδ−f_Cδ)(a, a).

1.6. Convergence results and proof of Theorem1.1

The core of this paper is the convergence of the discrete fermionic correlators to continu- ous ones, which are holomorphic functions. Let us define these functions first: fora, z∈Ω witha6=z, we define

f_Ω(a, z) := 1 2π

pψ⁰_a(a)p

ψ_a⁰(z)ψ_a(z)+1

ψa(z) and f_C(a, z) := 1 2π(z−a),

whereψa is the unique conformal mapping from Ω to the unit diskDwithψa(a)=0 and ψ⁰_a(a)>0 (this mapping exists by the Riemann mapping theorem). Note thatz7!fΩ(a, z) andz7!f_C(a, z) both have a simple pole of residue 1/2πatz=aand that (fΩ−f_C)(a, z) hence extends holomorphically toz=a.

We can now state the key theorem of this paper.

Theorem1.8. For each δ >0,identify a∈Cwith the closest midpoint of a horizontal edge of Cδ and z∈Cwith the closest midpoint of an edge of Cδ. Then,as δ!0,we have the following convergence results:

fΩ_δ(a, z)

δ !fΩ(a, z) for all a, z∈Ωwith a6=z, f_Cδ(a, z)

δ !f_C(a, z) for all a, z∈Cwith a6=z, (fΩ_δ−f_Cδ)(a, z)

δ !(f_Ω−f_C)(a, z) for all a, z∈Ω,

where the convergence of fΩ_δ/δ is uniform on the compact subsets of Ω×Ωaway from the diagonal{(w, w):w∈Ω},the convergence of f_Cδ/δis uniform on C×Caway from the diagonal {(w, w):w∈C}and the convergence of (fΩ_δ−f_Cδ)/δis uniform on the compact subsets of Ω×Ω.

From this result, the proof of the main theorem follows readily: since we have (Lemma1.7)

hεδ(a)i⁺_Ω∗

δ= 2(fΩ_δ−f_Cδ)(a, a) and hεδ(a)i^free_Ωδ =−2(fΩ_δ−f_Cδ)(a, a), and since (fΩ_δ−f_Cδ)/δ converges to (fΩ−f_C)(a, a), it suffices to check that

pψ⁰_a(a)p

ψ_a⁰(z)ψa(z)+1 ψa(z) − 1

z−a

!ψ_a⁰(a) as z!a,

which follows readily by verifying that pψ_a⁰(a)p

ψ_a⁰(z) ψa(z) − 1

z−a!0 asz!a.

To prove this, notice that since pψ_a⁰(a)p

ψ_a⁰(z) ψ_a(z) = 1

z−a+A+O(z−a) asz!a,

by squaring this expression, it suffices to check that the residue ofψ⁰_a(z)/ψ_a²(z) at z=a vanishes. By contour integrating on a small circleC aroundaand using the change of variable formula (sinceψa is conformal), we have

4πiA ψ⁰_a(a)=

I

C

ψ_a⁰(z)dz ψ_a(z)² =

I

ψa⁻¹(C)

dw w²= 0.

1.7. Proof of convergence

The rest of this paper is devoted to the proof of the key theorem (Theorem1.8). This proof consists mainly of two parts:

• The analysis of the discrete fermionic correlators fΩ_δ and f_Cδ as functions of their second variable (§2):

– We prove thatfΩ_δ(a,·) andf_Cδ(a,·) are discrete holomorphic (in a specific sense) onV_ΩM

δ \{a} andV_CM

δ \{a}, respectively (Propositions2.5and2.6).

– We show that f_Ωδ(a,·) and f_Cδ(a,·) have the same discrete singularity at a: their difference (f_Ωδ−f_Cδ)(a,·) is hence discrete holomorphic on V_ΩM δ

(Propositions 2.7–2.9).

– We observe that f_Ωδ(a,·) has some specific boundary values on ∂₀V_ΩM δ

(Proposition 2.10).

• The proof of convergence of the functionsf_Ωδ/δ,f_Cδ/δ and (f_Ωδ−f_Cδ)/δ(§3):

– The convergence of f_Cδ/δ follows directly from the convergence result of Kenyon for the dimer coupling function (Theorem3.1).

– We show that the family of functions ((f_Ωδ−f_Cδ)/δ)_δ>0is precompact on the compact subsets of Ω×Ω: it admits subsequences which converge as δ!0 (Proposition 3.2). Hence (f_Ωδ/δ)_δ>0 is also precompact on the compact subsets of Ω×Ω away from the diagonal.

– We identify the δ!0 limits of subsequences of fΩ_δ/δ with the functionfΩ

(Proposition 3.6). This allows us to conclude the proof of Theorem1.8.

Acknowledgements

This research was supported by the Swiss N. S. F., the European Research Council AG CONFRA, by EU RTN CODY, by the Chebyshev Laboratory (Department of Math- ematics and Mechanics, St. Petersburg State University) under RF governement grant 11.G34.31.0026 and by the National Science Foundation under grant DMS-1106588.

The authors would like to thank D. Chelkak for advice and useful remarks, as well as V. Beffara, Y. Velenik, K. Kyt¨ol¨a, K. Izyurov, C. Boutillier and B. de Tili`ere, for useful discussions. The first author would like to thank W. Werner for his invitation to ´Ecole Normale Sup´erieure, during which part of this work was completed.

2. Analysis of the discrete fermionic correlators

In this section, we study the properties of the discrete fermionic correlatorsf_Ωδ andf_Cδ that follow from their constructions. In the next section, we will use these properties to

prove Theorem1.8.

We study both correlators as functionsfΩ_δ(a,·) andf_Cδ(a,·) of their second variable, keeping fixed the medial vertexa∈V_ΩM

δ (which is the midpoint of a horizontal edge of Ωδ).

Let us first introduce the discrete versions of the differential operators ¯∂and ∆ that will be useful in this paper: for aC-valued functionf, we define, wherever it makes sense (i.e. for vertices ofV_Ωδ∪V_Ω^∗

δ),

∂¯_δf(x) :=f x+¹₂δ

−f x−¹₂δ

+i f x+¹₂iδ

−f x−¹₂iδ ,

∆_δf(x) :=f(x+δ)+f(x+iδ)+f(x−δ)+f(x−iδ)−4f(x).

In the case where one has that a vertexy∈{x±δ, x±iδ}belongs to∂V_Ωδ in the definition of ∆_δ, the boundary vertex is the one identified with the edgehx, yi∈∂E_Ωδ.

If e=−xy→∈E~Ω_δ is an oriented edge with x, y∈VΩ_δ and f is a function VΩ_δ!C we denote by∂ef the discrete partial derivative defined by∂ef:=f(y)−f(x).

2.1. Discrete holomorphicity

It turns out that the functionsf_Ωδ(a,·) andf_Cδ(a,·) are discrete holomorphic in a specific sense, which we calls-holomorphicity or spin-holomorphicity.

Let us first define this notion. With any medial edge e∈E_CM

δ , we associate a line

`(e)⊂Cof the complex plane defined by

`(e) := (d−v)^−1/2R={(d−v)^−1/2t:t∈R}, wherev∈V_Cδ is the closest vertex toeandd∈V_C^∗

δ is the closest dual vertex toe. On the square lattice, the four possible lines that we obtain aree^±πi/8R ande^±3πi/8R. When

`:=e<sup>iθ</sup>R is a line in the complex plane passing through the origin, let us denote byP`

the orthogonal projection on`, defined by

P`[z] :=¹₂(z+e^2iθz)¯ for allz∈C.

Definition 2.1. Let Mδ⊂V_CM

δ be a collection of medial vertices. We say that f:Mδ!C is s-holomorphic on Mδ if for any two medial vertices x, y∈Mδ that are adjacent inC^Mδ ,

P_{`(e)</sub>[f(x)] =P<sub>`(e)}[f(y)], wheree=hx, yi∈E_CM

δ .

e^πi/8R

e^−πi/8R

e^3πi/8R

e^−3πi/8R

Figure 2.1. The lines associated with the medial edges of Ωδ.

Remark 2.2. Our definition is the same as the one introduced in [S2], except that the lines that we consider are rotated by a phase ofe^πi/8, and that our lattice is rotated by an angle of ¹₄π, compared to the definitions there. In [CS2] this definition is also used, in the more general context of isoradial graphs.

Remark 2.3. The definition of s-holomorphicity implies that a discrete version of the Cauchy–Riemann equations is satisfied: if f:Mδ⊂V_CM

δ !C is s-holomorphic and v∈V_Cδ∪V_C^∗

δ is such that the four medial verticesv±¹₂δand v±¹₂iδ are inMδ, then we have

∂¯δf(v) = 0.

This can be found in [S2] (it follows by taking a linear combination of the four s- holomorphicity relations between the values f v±¹₂δ

and f v±¹₂iδ

), as well as the fact that satisfying this difference equation is strictly weaker than being s-holomorphic.

Remark 2.4. If λδ={−−−→vivi+1∈E~_ΩM

δ :i∈Z/nZ} is a simple counterclockwise-oriented closed discrete contour of medial edges and Λδ is the collection of points in VΩ_δ∪VΩ^∗_δ

surrounded byλδ, then it is easy to check that for any functionf:V_ΩM

δ !Cwe have X

−−−−→

v_iv_i+1∈λδ

f(vi)+f(vi+1)

2 (vi+1−vi) =iδ X

z∈Λδ

∂¯δf(v).

In particular this sum vanishes iff is discrete holomorphic.

Proposition 2.5. The function fΩ_δ(a,·)is s-holomorphic on V_ΩM δ \{a}.

Proof. Let z, w∈V_ΩM

δ \{a} be two adjacent medial vertices and let e∈E_ΩM δ be the medial edge linking them. Suppose thatzis the midpoint of a horizontal edge and thatw is the midpoint of a vertical edge. We prove the result in the case wherew=z+¹₂(1+i)δ (the other ones are symmetric). Denote by hthe half-edge between z and z+¹₂δ∈VΩδ

and by ˜hthe half-edge betweenz+¹₂δandw. For anyγ∈CΩδ(a, z), defineϕ(γ):=γ⊕h⊕˜h

as the symmetric difference ofγwith{h,h}: if˜ his not inγ, add it, otherwise remove it, and similarly for ˜h. Clearly,ϕis an involution mapping CΩ_δ(a, z) toCΩ_δ(a, w) and vice versa. Moreover, forγ∈CΩ_δ(a, z), we have

P_e−3πi/8

R[α^|γ|e^{−(i/2)W(γ)}] =P_e−3πi/8

R[α^|ϕ(γ)|e−(i/2)W(ϕ(γ))]. (2.1) This identity follows from considering the four possible cases, as shown in Figure2.2:

(1) Ifh /∈γ and ˜h /∈γ, then we havee^{−(i/2)W(γ)}∈R,|ϕ(γ)|=|γ|+1 and e−(i/2)W(ϕ(γ))=e^−πi/4e^{−(i/2)W(γ)}.

(2) Ifh /∈γ and ˜h∈γ, we havee^{−(i/2)W(γ)}∈R,|ϕ(γ)|=|γ|and e−(i/2)W(ϕ(γ))=e^−3πi/4e^{−(i/2)W(γ)};

there are a number of subcases, as shown in Figure 2.2, for which these relations are satisfied.

(3) Ifh∈γ and ˜h∈γ, we havee^{−(i/2)W(γ)}∈iR,|ϕ(γ)|=|γ|−1 and e−(i/2)W(ϕ(γ))=e^−πi/4e^{−(i/2)W(γ)}:

in this case, we can always choose an admissible walk onγ that is like in Figure2.2.

(4) Ifh∈γ and ˜h /∈γ, we havee^{−(i/2)W(γ)}∈iR,|ϕ(γ)|=|γ|and e−(i/2)W(ϕ(γ))=e^πi/4e^{−(i/2)W(γ)}.

In all the four cases, it is then straightforward to check that equation (2.1) is satisfied.

By the definition offΩ_δ (§1.5), we finally deduce that P_e−3πi/8R[f_Ωδ(a, z)] = 1

ZΩ_δ

X

γ∈C_Ωδ(a,z)

P_e−3πi/8R[α^|γ|e^{−(i/2)W(γ)}]

= 1 Z_Ωδ

X

eγ∈CΩδ(a,w)

P_e−3πi/8

R[α^|γ|ee^{−(i/2)W(γ)e} ] =P_e−3πi/8

R[f_Ωδ(a, w)], which is the s-holomorphicity equation.

The full-plane discrete correlator is also s-holomorphic.

Proposition 2.6. The function f_Cδ(a,·)is s-holomorphic on V_CM δ \{a}.

Proof. This follows directly from the definition of f_Cδ (Definition 1.6) and from LemmaB.1.

w z

w z

w z

w z

w z

w z

w z

w z

w z

w z

w z

w z

w z

w z (1)

(2)

(2)

(2)

(2)

(3)

(4)

Figure 2.2. The four possible cases in the proof of Proposition2.5: how a configuration is changed after two half-edges betweenzandware removed or added.

2.2. Singularity

Near the medial vertex a, the functions f_Ωδ(a,·) and f_Cδ(a,·) are not s-holomorphic:

they both have a discrete singularity, but of the same nature, and consequently the difference (f_Ωδ−f_Cδ)(a,·) is s-holomorphic onVΩ^m_δ , including the pointa. Rather than defining the notion of discrete singularity, let us simply describe the relations that these functions satisfy neara. Forx∈{±1±i}, let us denote bya_x:=a+¹₂xδ∈V_ΩM

δ the medial vertex adjacent toa, by e_x∈E_ΩM

δ the medial edge between aanda_x, and by`_xthe line

`(e_x) (as in Definition2.1). Let e∈E_Ω^h

δ be the horizontal edge with midpointa. Recall thatfΩ_δ(a, a) is defined asZ^{e_Ω^+}

δ /ZΩ_δ (see Definition1.5).

Proposition 2.7. Near a, the function fΩ_δ(a,·)satisfies the relations P`1+i[fΩ<sub>δ</sub>(a, a)] =P`1+i[fΩ_δ(a, a1+i)],

P_{`1−i</sub>[f<sub>Ωδ</sub>(a, a)] =P<sub>`1−i}[f_Ωδ(a, a_1−i)], P_{`−1+i</sub>[f<sub>Ωδ</sub>(a, a)−1] =P<sub>`−1+i}[f_Ωδ(a, a_−1+i)], P_{`−1−i</sub>[f<sub>Ωδ</sub>(a, a)−1] =P<sub>`−1−i}[f_Ωδ(a, a_−1−i)].

Proof. The first two relations are the s-holomorphicity relations and they are ob- tained in exactly the same way as the s-holomorphicity relations away from a. Indeed, let us use the same notation as in the proof of Proposition 2.5. First recall that for γ∈CΩ_δ(a, a1±i) the half-edge ofγ starting atagoes toa1. Now consider the involutions ϕ1+i:C_Ω^{e+}

δ !CΩ_δ(a, a1+i) andϕ1−i:C^{e_Ω^+}

δ !CΩ_δ(a, a1−i) defined by

ϕ1+i(γ) :=γ⊕ha, a1i⊕ha1, a1+ii and ϕ_1−i(γ) :=γ⊕ha, a1i⊕ha1, a_1−ii, respectively: as in the proof of Proposition2.7, we have that these involutions preserve the projections on`<sub>1+i</sub> and `_1−i respectively (a configurationγ∈C_Ω^{e+}

δ is interpreted as a configuration with winding number 0).

For the last two relations, we have that the involutionsϕ_−1+i:C_Ω^{e−}

δ !C_Ωδ(a, a_−1+i) andϕ_−1−i:C^{e_Ω^−}

δ !CΩδ(a, a_−1−i), respectively defined by

ϕ_−1+i(γ) :=γ⊕ha, a₋₁i⊕ha₋₁, a_−1+ii and ϕ_−1−i(γ) :=γ⊕ha, a₋₁i⊕ha₋₁, a_−1−ii, are such that for anyγ∈C_Ω^{e−}

δ we have

−P`<sub>−1+i</sub>[α<sup>|γ|</sup>] =P`_−1+i[α^|γ|e^{−(i/2)W(γ)}] =P`_−1+i[α^{|ϕ−1+i(γ)|}e^{−(i/2)W(ϕ−1+i(γ))}],

−P`<sub>−1−i</sub>[α<sup>|γ|</sup>] =P`_−1−i[α^|γ|e^{−(i/2)W(γ)}] =P`_−1−i[α^{|ϕ−1−i(γ)|}e^{−(i/2)W(ϕ−1−i(γ))}], where γ is interpreted as a configuration with a path from a to a that makes a loop, with winding number ±2π. This follows from the same considerations as in the proof of Proposition2.5. Hence, since Z^{e_Ω^−}

δ /Z_Ωδ=1−fΩδ(a, a), we obtain, by summing the above equations over allγ∈C_Ω^{e−}

δ , the last two identities of Proposition 2.7.

The functionf_Cδ(a,·) has the same type of discrete singularity asfΩ_δ(a,·).

Proposition 2.8. Near a, the function f_Cδ(a,·)satisfies exactly the same projec- tion relations as the ones satisfied by the function f_Ωδ(a,·),given by Proposition 2.7.

Proof. See PropositionB.2in Appendix B.

From Propositions2.5–2.8, we readily deduce the following result.

Proposition2.9. The function(fΩ_δ−f_Cδ)(a,·):V_ΩM

δ !Cis s-holomorphic on V_ΩM δ .

2.3. Boundary values

A crucial piece of information to understand the effect of the geometry of the discrete domain Ωδon the average energy density ata∈V_ΩM

δ is the boundary behavior offΩ_δ(a,·).

On the set of boundary medial vertices∂0V_ΩM

δ , which link a vertex of Ωδ and a vertex of

∂Ωδ, the argument offΩ_δ(a,·) is determined moduloπ. For eachz∈∂0V_ΩM

δ , withzbeing the midpoint of an edgee∈∂EΩ_δ between a vertexx∈VΩ_δ and a vertexy∈∂VΩ_δ, denote by νout(z)∈∂ ~EΩ_δ the oriented outward-pointing edge at z, identified with the number y−x: it is a discrete analogue of the outward-pointing normal to the domain.

Proposition2.10. On∂0V_ΩM

δ ,the argument of the value of fΩ_δ(a,·)is determined (modulo π): for each z∈∂0V_ΩM

δ , we have

Im(f_Ωδ(a, z)ν_out^1/2(z)) = 0.

Proof. From topological considerations, we have that ifz∈∂0V_ΩM

δ andγ∈CΩ_δ(a, z), then Im(e^{−(i/2)W(γ)}ν_out^1/2(z))=0: the winding number of any admissible walk from a to z is determined modulo 2π(see Figure 2.3) and it is easy to check thate^{−(i/2)W(γ)}is a real multiple ofν_out^−1/2(z). Hence, the result follows from the definition offΩ_δ.

Remark 2.11. This is the same kind of Riemann-type boundary conditions as in [S2]

and [CS2]. Notice that in these papers, the argument of the function on the boundary is fully determined (not only moduloπ).

2.4. Discrete integration

An essential tool that we will use for deriving the convergence off_Ωδ(a,·) is the possi- bility to define a discrete version of the antiderivative of the square of an s-holomorphic function, cf. [S2].