Circle patterns and critical Ising models

CIRCLE PATTERNS AND CRITICAL ISING MODELS

arXiv:1712.08736v1 [math-ph] 23 Dec 2017

MARCIN LIS

Abstract. A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a new family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.

1. Introduction The exact value of critical parameters is known only for a limited number of two-dimensional models of statistical mechanics. One of the most prominent examples is the Ising model whose critical temperature on the square lattice was famously computed by Kramers and Wannier [20] under a uniqueness hypothesis on the critical point. A confirmation of this condition came later as a corollary to the groundbreaking solution of the model provided by Onsager [25]. The methods available at that time yielded also the critical temperature of inhomogeneous Ising models on the square lattice with different vertical and horizontal coupling constants. A natural generalization of such models is the setting of arbitrary biperiodic graphs, i.e., weighted planar graphs whose group of symmetries includes Z2 . The critical point in the case of the square lattice with periodic coupling constants was first computed by Li [21], and the result was later extended to all biperiodic graphs by Cimasoni and Duminil-Copin [12]. Both approaches go through the Fourier analysis of periodic matrices that arise from the combinatorial solutions of the Ising model due to Fisher [15], and Kac and Ward [18] respectively. In particular, it is known that criticality in this setting is equivalent to the existence of nontrivial functions in the kernel of the associated Kac–Ward matrix [11]. So far the only other class of planar graphs where critical parameters for the Ising model were explicitly known are the isoradial graphs defined by the condition that each face is inscribed in a circle with a common radius. The critical Z-invariant coupling constants of Baxter [3] arise then as a solution to a system of equations requiring that the Ising model is invariant under a startriangle transformation (which preserves isoradiality of the graph). Criticality in the sense of statistical mechanics of the associated Ising model was proved in the periodic case in [12], and in the general case in [22]. Moreover, it is known that also in this setting the associated Kac–Ward has a non-trivial kernel [23]. Isoradial graphs form the most general family of graphs where the critical Ising model was shown to be conformally invariant in the scaling limit [10]. The proof of Chelkak and Smirnov uses the fact that differential operators admit well Date: December 27, 2017. 2010 Mathematics Subject Classification. 82B20, 60C05, 05C50. Key words and phrases. Ising model, circle patterns, criticality. 1

CIRCLE PATTERNS AND CRITICAL ISING MODELS

2

behaved discretizations on isoradial graphs [13,24]. One should also mention that dimer models on isoradial graphs related to the Ising model were studied in [6–8]. A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle of an arbitrary radius. Circle patterns have been extensively studied in relation to discretely holomorphic functions [1, 4, 5, 26]. In this article we introduce coupling constants for the Ising model defined on the dual graph of a circle pattern, and prove that the resulting model is critical. Our construction, when restricted to isoradial graphs, recovers the critical Z-invariant coupling constants of Baxter. Examples of new graphs where a critical Ising model can be defined include, among many others, arbitrary trivalent graphs whose dual is a triangulation with acute angles. Unlike the previous proofs of criticality [12, 21, 22], we do not invoke duality arguments (which are unavailable for general circle patterns). Instead, we obtain exponential decay of correlations in the high-temperature regime using bounds on the operator norm of the Kac–Ward transition matrix [19, 22]. To establish magnetic order in the low-temperature regime, we construct a non-trivial vector in the kernel of the critical Kac–Ward matrix (which readily implies infinite susceptibility), and then use known correlation inequalities to conclude nonvanishing of magnetization at lower temperatures. Clearly, many of the natural questions about the model we introduce here remain open, and one of the most interesting is perhaps that of existence, conformal invariance and universality (among circle patterns) of the scaling limit. The foundation for the conformal invariance result on isoradial graphs is the strong form of discrete holomorphicity that the fermionic observable associated with the critical Ising model satisfies. We provide the corresponding relations satisfied by the observable in the case of a circle pattern. These are only “half” of the relevant relations in the sense that the isoradial observable satisfies them on both the dual and primal graph, whereas the generic observable only on the dual graph. During the preparation of this article, a preprint by Chelkak [9] appeared where Ising models on graphs called s-embeddings are considered. Although circle patterns are not explicitly mentioned there, it is straightforward to check that our model is a special case of that in [9] when all the tangential quadrilaterals appearing in the construction are kites. However, no questions about criticality are asked in [9] and these are the main focus of the present article. This article is organized as follows. In the next section we introduce the setup and state our main theorems. In Sect. 3 we recall relevant results on the relation between the Ising model and the Kac–Ward matrix. In Sect. 4 we provide the proofs of our results, and in Sect. 5 we briefly discuss discrete holomorphicity properties of the associated fermionic observable. 2. Main results Let G = (V, E) and G∗ = (V ∗ , E ∗ ) be infinite, mutually dual, planar graphs embedded in the complex plane in such a way that each face of G∗ is inscribed in a circle whose center is inside the closure of the face, and the vertices of G lie at the centers of the circles. We identify both G and G∗ with their embedding and we say that G∗ is a circle pattern. (Note that we include the condition about the center being inside the face in the definition of a circle pattern.) For

CIRCLE PATTERNS AND CRITICAL ISING MODELS

3

u θe

θ−e v

Figure 1. Local geometry of a circle pattern G∗ and its dual G with dashed and solid edges respectively. Here, ~e = (u, v) and −~e = (v, u) each e = {u, v} ∈ E, the dual edge e∗ is a common chord of the circles centered at u and v. We denote by θ(u,v) and θ(v,u) half of the respective central angles given by the chord (see Fig. 1). The ordered pairs (u, v) and (v, u) represent the two opposite directed versions of the undirected edge {u, v}. Note that for each v ∈V, X (2.1) θ(v,u) = π, u∼v

where the sum is over all vertices u adjacent to v. We will often assume the bounded angle property of G∗ by requiring that there exists ε > 0 such that (2.2)

ε ≤ θ(u,v) ≤

π 2

−ε

for all directed edges (u, v). This in particular implies that G is of bounded degree. Let J = (Je )e∈E ∈ (0, ∞]E be coupling constants defined by q θ θ(v,u) (2.3) tanh J{u,v} = tan (u,v) 2 tan 2 . One can check that that (2.2) implies that kJk∞ < ∞. We note that (2.3) is equivalent to equation (6.3) of [9] if the associated quadrilateral is a kite (which is the case in our setting). We will study Ising models on finite connected subgraphs G = (VG , EG ) of G with coupling constants J. To this end, let ∂VG = {v ∈ VG : ∃u ∈ V \ VG , {u, v} ∈ E} be the boundary, and let ΩG = {−1, +1}VG be the space of spin configurations. The Ising model [17] at inverse temperature β > 0 with free or ‘+’ boundary conditions conditions  ∈ {f, +} is a probability measure on ΩG

CIRCLE PATTERNS AND CRITICAL ISING MODELS

given for σ ∈ ΩG , by  1  PG,β (σ) =  exp β ZG,β

X

J{u,v} σu σv + 1{=+} β

{u,v}∈EG

X

4



J{u,v} σv ,

v∈∂VG u∈V / G

 where β > 0 is the inverse temperature, and ZG,β is the normalizing constant  called the partition function. We write h·iG,β for the expectation with respect to P G,β . By the second Griffiths inequality, we can define the infinite volume limits of correlation functions by

Y 

Y  σu G,β , σu G,β = lim u∈A

G%G

u∈A

where A ⊂ V is finite, and where the limit is taken over any increasing family of finite connected subgraphs G containing A and exhausting G. Let d(·, ·) be the graph distance on G. The following is our main result identifying a phase transition at βc = 1: Theorem 2.1. Let G∗ be a circle pattern satisfying the bounded angle property, and consider the Ising model on G with coupling constants as in (2.3). Then (i) for every β < 1, there exists Cβ > 0 such that for all u, v ∈ V , hσu σv ifG,β ≤ e−Cβ d(u,v) , (ii) if the radii of circles are uniformly bounded from above, then for every v ∈ V , there exists sv > 0 such that for all β > 1, β sv − 1 hσv i+ ≥ . G,β β sv + 1 Moreover, if the radii of circles are uniformly bounded from below, then one can choose one such s = sv for all v. We also show that the magnetic susceptibility diverges at criticality: Theorem 2.2. Let G∗ be as in Theorem 2.1, and assume that the radii of circles are uniformly bounded from below. Then for all v ∈ V , X χG,β (v) = hσu σv ifG,β < ∞ u∈V

if and only if β < 1. Our main tool is the Kac–Ward transition matrix Λ associated with the Ising model [18]. The exponential decay of correlations for β < 1 will follow from our previous result [22] which says that the Euclidean operator norm of Λ is strictly smaller than 1. This part does not actually require the faces of G∗ to be cyclic polygons and holds true for an arbitrary choice of angles θ(u,v) satisfying (2.1). The complementary lower bound on the two-point function which identifies a phase transition at βc = 1 will follow from a construction of an eigenvector of Λ with eigenvalue 1 combined with known correlation inequalities. This part crucially relies on the geometric properties of the embedding. Note that if all the circumscribed circles have the same radius, then G and G∗ are isoradial, θ(u,v) = θ(v,u) , and (2.3) defines the critical Z-invariant coupling constants of Baxter [3]. The following are examples of circle patterns.

CIRCLE PATTERNS AND CRITICAL ISING MODELS

5

Figure 2. A piece of a circle packing and its dual. The quadrangulation Q from Example 2 is the graph whose vertices are the regions bounded by dashed lines. Its dual Q∗ is a circle pattern Example 1. Any acute triangulation G∗ of the plane is a circle pattern. In this case, G is a trivalent graph. Example 2. A circle packing is a representation of a planar graph M where the vertices are the centers of interior-disjoint disks in the plane, and two vertices are adjacent if the respective discs are tangent. Consider a circle packing of an infinite planar graph M such that each face is convex. Then the dual graph M ∗ can be simultaneously circle-packed in such a way that the circles centered at the endpoints of an edge e and its dual edge e∗ meet orthogonally at one point (see Fig. 2). Let Q be the quadrangulation whose vertices are the vertices and faces of M , and whose edges are of the form {u, u∗ }, where u is a vertex of M and u∗ is one of the faces incident on u. Then Q∗ is the graph with vertices given by the meeting points of the circles and edges connecting every pair of consecutive vertices around every circle. By construction, Q∗ is a circle pattern. Example 3. Let M and M ∗ be as in Example 2. Then each pair of dual edges e and e∗ meets at a right angle. Let Q be a quadrangulation whose vertices are the vertices, faces and edges of M , and whose edges are of the form {e, v} where e is an edge of M and v is either a vertex or a face incident on e. Note that each face of Q has at least two right interior angles, and hence Q is a circle pattern. The dual graph Q∗ is the 4-regular graph that is obtained from M by replacing each edge of M by a quadrilateral and each vertex of degree d by a d-gon face. Example 4. Let G = Z2 and fix 0 < a ≤ b < ∞. For the i-th column of vertical edges, choose a coupling constant Ji ∈ [a, b]. To each horizontal edge between column i and i + 1 assign a coupling constant Ji,i+1 satisfying tanh Ji,i+1 = e−Ji −Ji+1 . The resulting Ising model is critical as these coupling constants are computed using (2.3) applied to the dual lattice G∗ ' Z2 with appropriately stretched or

CIRCLE PATTERNS AND CRITICAL ISING MODELS

6

squeezed columns. Note that if Ji are constant, the classical√anisotropic critical Ising model is recovered, and upon setting Ji = − 12 log( 2 − 1) we get the homogenous critical model. Acknowledgments The author thanks Hugo Duminil-Copin for drawing his attention to the inequality of Lemma 4.2. This work was funded by EPSRC grants EP/I03372X/1 and EP/L018896/1. 3. The Kac–Ward operator and the fermionic observable In this section, we define the Kac–Ward matrix and state a result which relates its iverse to the spin fermionic observable of Smirnov [11, 23]. We also recall the exact value of the operator norm of the Kac–Ward transition matrix obtained in [22], and construct an eigenvector of eigenvalue 1 of the critical (β = 1) transition matrix defined on the dual of a circle pattern. ~ be the set of directed edges of G. For ~e = (u, v) ∈ E, ~ To this end, let E we write t~e = u for its tail, h~e = v its head, −e = (v, u) its reversal, and e = {u, v} ∈ E for its undirected version. Let ~x = (~x~e )~e∈E~ be real weights on the directed edges, and let x = (xe )e∈E be weights on the undirected edges given by xe = ~x~e ~x−~e . We denote the relation between ~x and x by writing ~x ∼ x. ~ given The Kac–Ward transition matrix Λ(~x) on G is a matrix indexed by E by ( i ~x−~e ~x~g e 2 ∠(~e,~g) if h~e = t~g and ~g 6= −~e; Λ(~x)~e,~g = 0 otherwise,
h −t where ∠(~e, ~g ) = Arg hge −tge ∈ (−π, π] is the turning angle from ~e to ~g . The Kac–Ward matrix is given by T (~x) = Id − Λ(~x), where Id is the identity matrix. Consider a finite graph G = (VG , EG ) drawn in the plane with possible edge crossings. We will write ΛG (~x) and TG (~x) for the corresponding matrices indexed ~ G of G. Let EG be the collection of even subgraphs of by the directed edges E G, i.e. subsets ω of EG such that all vertices in VG have even degree in (the subgraph induced by) ω, and let X Y ZG (x) = (−1)C(ω) xe , ω∈EG

e∈ω

where C(ω) is the number of pairs of edges in ω that cross. The seminal identity of Kac and Ward [18] reads p ZG (x) = detTG (~x), (3.1) where ~x ∼ x. If G has no edge crossings, then the heigh-temperature expansion f of the Ising partition function ZG,β is equal, up to an explicit constant, to ZG (x) with xe = tanh βJe , and hence (3.1) establishes an intrinsic relation between the Ising model and the Kac–Ward operator. It turns out that one can go further and also express the inverse of TG (~x) in terms of related partition functions which, in addition to an even subgraph, involve a path weighted by a complex factor. We now define these objects

CIRCLE PATTERNS AND CRITICAL ISING MODELS

7

which are forms of the fermionic observable introduced by Smirnov [27]. For ~ let m~e = (t~e + h~e )/2 be its midpoint. For ~e, ~g ∈ E ~ G, a directed edge ~e ∈ E, we define a modified graph G~e,~g with vertex set VG ∪ {m~e , m~g } and edge set (EG \ {e, g}) ∪ {{m~e , h~e }, {t~g , m~g }}, and define the weights of the undirected half-edges to be x{m~e ,h~e } = ~x~e and x{t~g ,m~g } = ~x−~g . Let EG (~e, ~g ) be the collection of sets of edges ω of G~e,~g containing {m~e , h~e } and {t~g , m~g }, and such that all vertices in VG have even degree in ω. (Note that we do not require that m~e and m~g have even degree.) From standard parity arguments, it follows that each ω ∈ EG (~e, ~g ) contains a self-avoiding path starting at m~e and ending at m~g . We denote by γω the left-most such path. We also write α(γω ) for the total turning angle of γω , i.e., the sum of turning angles between consecutive steps of γω . ~ G given by Let FG (~x) be a matrix indexed by E Y X i 1 e− 2 α(γω ) xe , FG (~x)~e,~g = Id~e,~g + ZG (x) e∈ω ω∈EG (~e,~g )

where ~x ∼ x. For a graph G with no edge crossings, the following identity: (3.2)

FG (~x) = TG−1 (~x)

which gives foundations for our results was proved in [11, 23], and is valid for any weight vector x. The main new contribution of this section is the following construction of a nontrivial vector in the kernel of the critical Kac–Ward matrix defined on the dual of a circle pattern. Theorem 3.1. Let G∗ be aqcircle pattern, and let the weights on the directed edges of G be given by ~x~e = tan θ2~e . For an edge e∗ of G∗ , let |e∗ | be its length, p and define ρ~e = ρ−~e = |e∗ |, where ~e and −~e are the directed edges crossing e∗ . Then Λ(~x)ρ = ρ. This result will directly follow from the next lemma. For v ∈ V , let Inv = ~ : h~e = v} and Outv = {~e ∈ E ~ : t~e = v}. Let J be the involutive {~e ∈ E ~ ˜ x) = JΛ(~x). As automorphism of CE induced by the map ~e 7→ −~e, and let Λ(~ was noted in [22], and is easily seen from the definition of the transition matrix, ˜ x) is a Hermitian, block-diagonal matrix with blocks Λ(~ ˜ x)v , v ∈ V , acting on Λ(~ the linear subspace indexed by Outv , and given by ( i ~x~e ~x~g e 2 ∠(−~e,~g) if t~e = t~g = v; v ˜ Λ(~x)~e,~g = 0 otherwise. Let ρv be the restriction of ρ to the subspace indexed by Outv . Note that Jρ = ρ, and hence to prove Theorem 3.1, it is enough to show the following: Lemma 3.2. Let ~x be as in Theorem 3.1. Then for all v ∈ V , ˜ x)v ρv = ρv . Λ(~ Proof. Fix ~e ∈ Outv . Let ~e1 , . . . , ~en be a clockwise ordering of the edges in Outv \ ~e that satisfy ∠(−~e, ~ei ) ≥ 0, i = 1, . . . , k, and let ~g1 , . . . , ~gm be a counterclockwise

CIRCLE PATTERNS AND CRITICAL ISING MODELS

8

ordering of the remaining edges in Outv \ ~e. Then ∠(−~e, ~ek ) = π − θ~e − 2

k−1 X

θ~ej − θ~ek ,

for k = 1, . . . , n,

θ~gj − θ~gk ,

for k = 1, . . . , m.

j=1

−∠(−~e, ~gk ) = π − θ~e − 2

k−1 X j=1

We can p assume that the radius of the circle centered at v is equal to 1, and hence ρ~g = 2 sin θ~g for all ~g ∈ Outv . We have ˜ x)v ρv ]~e [Λ(~ = ~x~e =

i

X

ρ~g ~x~g e 2 ∠(−~e,~g)

~g ∈Outv \~e

√

2

q i θ sin θ~g tan 2~g e 2 ∠(−~e,~g)

X ~g ∈Outv \~e

X

=2

sin

θ~g i ∠(−~e,~g ) 2 2 e

~g ∈Outv \~e n X


i i i e 2 θ~ek − e− 2 θ~ek e 2

= −i

−i

k=1 m X

e

i θgk 2 ~

− 2i θ~gk

−e




e

π−θ~e −2

Pk−1

− 2i π−θ~e −2

j=1

θ~ej −θ~ek

Pk−1 j=1




θ~gj −θ~gk




k=1


i h i P i π−θ~e −2 n θ~ej (π−θ ) j=1 ~ e 2 −e = −i e 2
i h P − 2i π−θ~e −2 m − 2i (π−θ~e ) gj j=1 θ~ −i −e +e
= 2 sin π2 − θ2~e = 2 cos θ2~e , where the P fifth equality Pfollows from a telescopic sum, and the sixth holds true since θ~e + nj=1 θ~ej + m j=1 θ~gj = π. Hence, q p ˜ x)v ρv ]~e = 2 cos θ~e ~x~e = 2 cos θ~e tan θ~e = 2 sin θ~e = ρv . [Λ(~  ~e 2 2 2 We finish this section with an upper bound on the operator norm of the transition matrix which we will later use to prove exponential decay of correlations in the high-temperature regime. q Lemma 3.3. Let (θ~e )~e∈E~ satisfy (2.1). If |~y~e | ≤ s tan θ2~e , for some s > 0 and ~ then the induced operator norm of Λ(~y ) acting on `2 (E) ~ satisfies all ~e ∈ E, (3.3)

kΛ(~y )k ≤ s.

˜ x)k = kΛ(~x)k, and since Λ(~ ˜ x) is block diagonal with blocks Proof. Note that kΛ(~ ˜ x)v , (3.3) is a direct consequence of condition (2.1) and Lemma 2.4 of [22] Λ(~ ˜ x)v acting on the deg(v)-dimensional which says that the operator norm of Λ(~

CIRCLE PATTERNS AND CRITICAL ISING MODELS

9

Euclidean complex vector space indexed by Outv is the positive solution s of  ~x2  π X arctan ~e = . (3.4)  s 2 ~e∈Outv

4. Proofs of main results In this section we prove our main theorem. In preparation for the proof, we bound the Ising two-point correlation functions from both below and above by the entries of the inverse Kac–Ward matrix. We then recall a general correlation inequality due to Duminil-Copin and Tassion. Finally, we use the eigenvector ρ from Theorem 3.1 to obtain a lower bound on the two-point functions at β = 1. For u, v ∈ VG , let EG (u, v) be the collection of subsets ω of EG such that each vertex in VG \ ({u}∆{v}) (resp. in {u}∆{v}) has even (resp. odd) degree in ω, where ∆ is the symmetric difference. Let X Y xe . ZG (x)u,v = ω∈EG (u,v) e∈ω

The classical high-temperature expansion of the Ising correlation functions yields hσu σv ifG,β =

(4.1)

ZG (x)u,v ZG (x)

for xe = tanh(βJe ), and all u, v ∈ VG . Using (3.2), we can now prove two-sided bounds on the two-point correlation functions in terms of the entries of (possibly signed) inverse Kac–Ward matrices. Lemma 4.1. Let β > 0, Je ≥ 0 be arbitrary positive numbers, and let xe = ~ tanh(βJe ). Then for all u, v ∈ VG , u 6= v, there exists τ ∈ {−1, 1}E such that for all ~x ∼ x, ~x ≥ 0, max

~e∈Inu ,~g ∈Outv ~e6=~g

|TG−1 (~x)~e,~g | ≤ hσu σv ifG,β ≤ ~x~e ~x−~g

X

~x−~e ~x~g |TG−1 (~xτ )~e,~g |,

~e∈Inu ,~g ∈Outv

where ~x~τe = τ~e ~x~e . Proof. We first prove the lower bound. By (3.2), TG−1 (~x) = FG (~x). By the definition of EG (~e, ~g ), removing the two half edges from ω ∈ EG (~e, ~g ) (carrying weights ~x~e and ~x−~g ) yields a configuration in EG (u, v). It is now enough to use (4.1) and the fact that all terms in the definition of ZG (x)u,v are positive, whereas the corresponding terms in FG (~x)~e,~g carry a complex factor. To obtain the upper bound, we construct an augmented graph Gγ by adding to G a simple path γ connecting u and v, in such a way that γ crosses each edge of G at most once and does not pass through any vertex of G. Without loss of ~ generality, we can assume that γ is the single edge {u, v}. We fix a τ ∈ {−1, 1}E satisfying ( −1 if e is crossed by γ, τ~e τ−~e = +1 otherwise, and, when needed, we extend the weights to Gγ by setting τ~γ = τ−~γ = 1.

CIRCLE PATTERNS AND CRITICAL ISING MODELS

10

We chose an orientation ~γ = (u, v) of γ, and for ~xτ ∼ xτ , we have ∂ 1 ZGγ (xτ ) hσu σv ifG,β = ZG (x) ∂~x~γ ~ x~γ =0,~ x−~γ =1   p 1 ∂ τ τ γ = detTGγ (~xτ )Tr TG−1 T (~ x ) (~ x ) γ G ZG (x) ∂~x~γ ~ x~γ =0,~ x−~γ =1   τ ZG (x ) ∂ = TGγ (~xτ ) Tr TG−1 xτ ) γ (~ ZG (x) ∂~x~γ ~ x~γ =0,~ x−~γ =1 τ X i ZG (x ) = e 2 (∠(~γ ,~e)+∠(~g,~γ )) ~x−~e ~x~g TG−1 (~xτ )~e,~g ZG (x) ~e∈Outv ,~g ∈Inu X ~x−~e ~x~g |TG−1 (~xτ )~e,~g |. ≤ ~e∈Outv ,~g ∈Inu

The first equality follows from the high temperature expansion, and the fact that the signs of xτ cancel out the signs from the definition of ZGγ (xτ ) assigned to even subgraphs containing γ. The second equality is a consequence of the Kac–Ward formula for Gγ and Jacobi’s formula for the derivative of a determinant. The third equality follows from the Kac–Ward formula again, and the fourth from the definition of TG (~xτ ). The last inequality holds true since ZGγ (xτ ) counts even subgraphs with additional signs, whereas all terms in ZG (x) are positive.  Let S ⊂ V be such that the subgraph of G induced by S is connected. We will denote by h·i S,β the expectation with respect to the Ising model on this induced subgraph. We define for v ∈ V , XX
ϕS,β (v) = tanh βJ{u,w} hσv σu ifS,β . u∈S w∈S /

The following crucial inequality is due to Duminil-Copin and Tassion. We give its proof for completeness in Appendix A. Lemma 4.2. For any finite subgraph G = (VG , EG ), any v ∈ VG , and β > 0,
d + 2 1 hσv i+ ≥ inf ϕ (v) 1 − (hσ i ) . v S,β G,β G,β β S3v dβ S⊂VG

We will next use the existence of the eigenvector ρ to get a uniform lower bound on ϕS,β (v). Before stating the result, we need to introduce additional notation. Let G = (VG , EG ) be a finite subgraph of G induced by the vertex set VG . Consider the edges of G whose one endpoint is in VG and the other one in V \ VG . Each such edge splits into two half-edges, one of which is incident to VG . We add the incident half-edges to the edge set EG , and we assign them the weights of the corresponding full edges in G. We also add their endpoints (which are the midpoints of the corresponding edges of G) to the vertex set VG ¯ = (V ¯ , E ¯ ). Note that by construction, all and we call the resulting graph G G G ¯ meaning that they have the same degree in G ¯ vertices in VG are interior in G ¯ as in G. Let ~n(∂G) be the set of the directed versions of the half-edges in EG which point outside G. Lemma 4.3. Let G∗ be a circle pattern satisfying the bounded angle property, and such that the radii of all circles are uniformly bounded from above by R < ∞.

CIRCLE PATTERNS AND CRITICAL ISING MODELS

11

Then for every v ∈ V and every finite S ⊂ V containing v, r r tan 2ε , ϕS,1 (v) ≥ R where r is the maximal radius of a circle centered at a neihbor of v, and ε is as in (2.2). Proof of Theorem 2.1. For d(v, V \ S) = 1, the bound easily follows from the definition of ϕS,1 (v) and the bounded angle property. We can therefore assume that d(v, V \q S) > 1.

Let ~x~e = tan θ2~e and let ρ be the eigenvector of Λ(~x) as defined in Theo¯ (here the half-edges are identified rem 3.1 restricted to the directed edges of G with their counterparts in G). Let ζ = TG¯ (~x)ρ, and note that, by the definition of the Kac–Ward matrix, ζ~e = ρ~e for all ~e ∈ ~n(∂G), and ζ~e = 0 otherwise. Therefore, by Lemma 4.1 and the bounded angle property, for any ~e = (u, v) ∈ Inv , p p 2ru sin ε ≤ 2ru sin θ~e = ρ~e   x)ζ ~e = TG−1 ¯ (~ X = x)~e,~g ρ~g TG−1 ¯ (~ ~g ∈~ n(∂G)

≤ ≤

√

√

2R cos ε

X

~x~e ~x−~g hσv σw ifG,1

~g =(w,z)∈~ n(∂G)

2R cos ε

X

~x−~g hσv σw ifG,1

~g =(w,z)∈~ n(∂G)

r cos ε ϕS,1 (v), ≤ 2R tan 2ε where ru is the radius of the circle centered p at u, and where in the last inequality we used that tanh Jg = xg = x~g x−~g ≥ tan 2ε ~x−~g by the bounded angle property. We finish the proof by maximizing over the neighbors u of v.  The last technical statement that we need is a consequence of Lemma 3.3. Lemma 4.4. Let G∗ be a circle pattern satisfying the bounded angle property. ~ let For β > 0, let xe (β) = tanh(βJe ) where Je is as in (2.3). For ~e ∈ E, q θ~e ~x~e (β) = xxee(β) (1) tan 2 , where e is the undirected version of ~e. Then there exists a constant c > 0 such ~ that for all τ ∈ {±1}E and β ∈ [0, 1], kΛ(~xτ (β))k ≤ 1 − c(1 − β), where ~x~τe = τ~e ~x~e . Proof. By the bounded angle property, M := supe∈E Je < ∞. Define s(β, Je ) = xe (β) tanh βJe xe (1) = tanh Je . For β ∈ [0, 1], we have Je 2M ∂ 2Je s(β, Je ) = ≥ ≥ := c, ∂β sinh 2Je sinh 2M tanh Je cosh2 βJe

CIRCLE PATTERNS AND CRITICAL ISING MODELS

12

and since s(1, Je ) = 1, we get s(β, Je ) ≤ 1 − c(1 − β). We finish the proof by applying (3.3).  We can now prove our main results. Proof of Theorem 2.1. Let D be the maximal degree of G, and let ~x(β) be as in Lemma 4.4. Note that ~x(β) ∼ x(β), where xe (β) = tanh βJe . We use the upper bound from Lemma 4.1 and Lemma 4.4, to get for u 6= v, X ~x−~e ~x~g |TG−1 (~xτ (β))~e,~g | hσu σv ifG,β ≤ ~e∈Inu ,~g ∈Outv

≤ D2 ≤ D2

max

~e∈Inu ,~g ∈Outv

|TG−1 (~xτ (β))~e,~g |

kΛG (~xτ (β))kd(u,v) 1 − kΛG (~xτ (β))k

D2 d(u,v)  , 1− where  = 1 − c(1 − β) < 1 with c as in Lemma 4.4. This completes the proof of part (i). To prove part (ii), we use Lemma 4.2 together with Lemma 4.3, and the monotonicity of the magnetization in β, to get for all β ≥ 1, sv d 2 hσv i+ ≥ β1 inf ϕS,β (v)(1 − (hσv i+ )2 ) ≥ (1 − (hσv i+ G,β G,β G,β ) ), S3v dβ 2β p where s2v = Rr tan 2ε is as in Lemma 4.3. We finish the proof by integrating the differential inequality and then taking the limit G % G.  ≤

Proof of Theorem 2.2. The inequality χG,β (v) < ∞ for β < 1 follows from the exponential decay of correlations from Theorem 2.1 (i), and the quadratic growth of balls in the graph distance (due to the fact that the circles have a minimal diameter). The fact that χG,β (v) = ∞ for β ≥ 1 follows directly from Lemma 4.3 and the second Griffiths inequality.  5. Discrete holomorphicity of fermionic observables We finish with a brief discussion of discretely holomorphic properties of the fermionic observable given by the inverse Kac–Ward operator. We note that, even though the observable in the general setting of circle patterns satisfies less constraints than its isoradial version and hence the tools of [10] are not fully applicable, the results of this section point in the direction of conformal invariance of the scaling limit. We first need to generalize the notion of s-holomorhicity introduced by Smirnov [27] for the square lattice, and developed by Chelkak and Smirnov [10], to the setting of circle patterns. We assume that G is a finite subgraph of G such that G∗ is a circle pattern. Let Proj(z; `) be the orthogonal projection of the complex number z onto the complex line `. We say that a function f : EG → C is s-holomorphic at an interior vertex v ∈ VG if for every dual vertex v ∗ adjacent to v, (5.1)

1

1

Proj(f (e1 ); (v − v ∗ )− 2 R) = Proj(f (e2 ); (v − v ∗ )− 2 R),

CIRCLE PATTERNS AND CRITICAL ISING MODELS

13

where e1 , e2 are the two edges incident on both v and v ∗ . −1

Let η~e = h~e − t~e , and `~e = η~e 2 R. We define L to be the real linear space of ~ → C satisfying ϕ~e ∈ `~e for all ~e. One can easily check that L functions ϕ : E is invariant under the action of the Kac–Ward operator. Let S be an operator mapping complex functions on E to functions in L given by Sf (~e) = ρ~e Proj(f (e); `~e ), p where ρ~e = |e∗ | as before. Note that since `~e and `−~e are orthogonal, we have for ϕ ∈ L,
S −1 ϕ(e) = ρ~−1 ϕ(~ e ) + ϕ(−~ e ) . e q Let ~x~e = tan θ2~e be the critical weights. Theorem 5.1 (s-holomorphicity and the Kac–Ward matrix). A function f is s-holomorphic at an interior vertex v of G if and only if TG (~x)Sf (~e) = 0 for all ~e ∈ Inv . p Proof. Note that ρ(v,u) = 2rv sin θ(v,u) , where rv is the radius of the circle centered at v. This implies that the row of TG (~x)S indexed by ~e ∈ Inv is equal √ to the corresponding row of the matrix T S from [23] scaled by 2 rv ~x~−1 e , and hence the result directly follows from Theorem 2.1 of [23]. (Note that the matrix ~ T from [23] is our matrix T (~x) conjugated by Diag{~x~e : ~e ∈ E}).  This in particular implies that s-holomorphic functions can be uniquely recov¯ be as defined before Lemma 4.3, ered from their boundary values. Let G and G ¯ and satisfying and let ϕ ∈ L be a function defined on the directed edges of G ϕ(~e) ∈ l~e

for ~e ∈ ~n(∂G),

and

ϕ(~e) = 0

otherwise.

Following Smirnov [27], we say that f : EG¯ → C solves the discrete Riemann– Hilbert boundary value problem for the pair (G, ϕ) if f is s-holomorphic at all v ∈ VG and Sf (~e) = ϕ(~e)

for all ~e ∈ ~n(G).

Proposition 5.2. Let G and ϕ be as above and let TG¯ (~x) be the critical Kac– ¯ where the half-edges of G ¯ inherit weights from the Ward operator defined on G, corresponding edges of G. Then the discrete Riemann–Hilbert boundary value problem for (G, ϕ) has a unique solution f = S −1 TG−1 x)ϕ. ¯ (~ Proof. Suppose that f is a solution to the discrete Riemann–Hilbert boundary ¯ is not formally a subgraph of G since it contains value problem. Note that G half-edges. However, these half-edges are parallel to the corresponding edges of G, and therefore we can use Theorem 5.1 to conclude that TG¯ (~x)Sf (~e) = 0 for all ~e ∈ / ~n(G). Moreover if ~e ∈ ~n(G), then h(~e) 6= t(~g ) for all ~g 6= −~e. Hence by the definition of the Kac–Ward operator, TG¯ (~x)Sf (~e) = IdSf (~e) = ϕ(~e) for ~e ∈ ~n(G). This means that TG¯ (~x)Sf (~e) = ϕ(~e) for all directed edges ~e, and the claim follows. 

CIRCLE PATTERNS AND CRITICAL ISING MODELS

14

Finally, we wish to prove that certain discrete line integrals related to sholomorphic functions are well defined. To this end, we first list several related basic results. Fix f : E → C, and let ϕ = Sf ∈ L. Hence, we have f (e) = ρ~−1 e) + ϕ(−~e)). Define D~e = η~e /|η~e |. An elementary computation yields e (ϕ(~ (5.2)

Re(η~e∗ f 2 (e)) = 2D~e∗ ϕ(~e)ϕ(−~e),

(5.3)

Im(η~e∗ f 2 (e)) = |ϕ(~e)|2 − |ϕ(−~e)|2 .

Fix a vertex v, and let ϕin = (ϕ(−~e))~e∈Outv and ϕout = (ϕ(~e))~e∈Outv . If f is sholomorphic, then by Theorem 5.1, TG (~x)ϕ(~e) = 0 for all ~e ∈ Inv . Equivalently, ˜ out = ϕin , Λϕ

(5.4)

˜ = Λ(~ ˜ x)v is the block matrix from Sect. 3 acting on the linear subspace where Λ  indexed by Outv . Let D = Diag D~e : ~e ∈ Outv , and let ∠(~e) = Arg(D~e ) ∈ ˜ is skew-symmetric. Indeed, for ~e, ~g ∈ Outv , ~e 6= ~g , (−π, π]. We claim that DΛ we have

˜ ~e,~g 1 1 [DΛ] = ei ∠(~e)−∠(~g)+ 2 ∠(−~e,~g)− 2 ∠(−~g,~e) = ei ∠(~e)−∠(~g)+∠(−~e,~g) = e±iπ = −1. ˜ ~g,~e [DΛ] For a directed edge ~e of G, let ~e∗ be the directed edge of G∗ crossing ~e and such that ∠(~e, ~e∗ ) > 0. Recall that η~e = h~e − t~e . The next lemma says that the discrete line integral of an s-holomorphic function is well defined on G∗ . Lemma 5.3. If f is s-holomorphic at an interior vertex v, then X η~e∗ f (e) = 0. ~e∈Outv

p Proof. Let f = (f (e))~e∈Outv . Recall that ρ = ( |e∗ |)~e∈Outv is an eigenvector of ˜ of eigenvalue 1, and let ρ2 = (|e∗ |)~e∈Out . The desired equality can be now Λ v ˜ is skew symmetric, we have DΛ ˜ = −Λ ˜ T D. rewritten as i(ρ2 )T Df = 0. Since DΛ Hence, by (5.4) we get (ρ2 )T Df = ρT D(ϕout + ϕin ) ˜ out = ρT Dϕout + ρT DΛϕ ˜ T Dϕout = ρT Dϕout − ρT Λ ˜ T Dϕout = ρT Dϕout − (Λρ) = 0, which completes the proof.



We finish with a result saying that if f is s-holomorphic, then the real part of the discrete line integral of f 2 is well defined on G∗ , and moreover, the imaginary part of the integral of f 2 over any closed counterclockwise contour on G∗ is positive. This is the counterpart in the setting of circle patterns of the fundamental result Smirnov [27], and Chelkak and Smirnov [10], stating the existence of R of 2 Re( f ) simultaneously on both G∗ and G, and its sub- and superharmonicity on the respective isoradial graphs.

CIRCLE PATTERNS AND CRITICAL ISING MODELS

15

Lemma 5.4. If f is s-holomorphic at an interior vertex v, then    X  X η~e∗ f 2 (e) ≥ 0. η~e∗ f 2 (e) = 0, and Im Re ~e∈Outv

~e∈Outv

Proof. By (5.2) and (5.4), we have   X ˜ out = 0, η~e∗ f 2 (e) = 2i(ϕout )T Dϕin = 2i(ϕout )T DΛϕ Re ~e∈Outv

˜ is skew-symmetric. where in the last equality we used that DΛ ˜ is bounded Moreover, by (5.3), (5.4) and the fact that the operator norm of Λ by 1 (3.4), we get   X ˜ out k2 ≥ 0, η~e∗ f 2 (e) = kϕout k2 − kϕin k2 = kϕout k2 − kΛϕ Im ~e∈Outv

which completes the proof.



Remark 1. Note that the last two lemmas follow directly from the corresponding results of Chelkak and Smirnov for isoradial graphs [10]. Indeed, both the definition of s-holomorphicity (5.1) and the desired relations depend only on the geometry of the graph G∗ in the immediate neighborhood of v, which is indistinguishable from the one of an isoradial graph. However, we included the concise proofs that use the Kac–Ward matrix as they shed a different light on these relations.

CIRCLE PATTERNS AND CRITICAL ISING MODELS

16

Appendix A. The following proof of Lemma 4.2 is due to Duminil-Copin and Tassion, and (up to small modifications) is contained in an unpublished version of [14]. We include it here for completeness. The graphs considered in this section are not assumed to be planar. Let G = (VG , EG ) be a finite graph, and let ∂VG ⊂ VG be a fixed set of vertices called the boundary. We consider an augmented graph G• = (V• , E• ) where a ghost (or a boundary) vertex • is added to the vertex set, and edges of the form {•, v}, v ∈ ∂VG , are added to the edge set. We will consider Ising models on subgraphs of G• as defined in Sect. 2 with arbitrary positive coupling constants P (Je )e∈E• (to recover the exact setting of Sect. 2, one needs to set Jv,• = u∈V / G J{u,v} for v ∈ ∂VG ). One can easily check that the following relation between boundary conditions holds true for any v ∈ VG , f hσv i+ G,β = hσv σ• iG• ,β .

For S ⊂ V• , we will write h·ifS,β for the expectation of the Ising model with free boundary conditions defined on the subgraph of G• induced by S. The notion of a current will play a crucial role in the proof. A current is a function n : E• → N, whose weight is given by Y (βJe )ne . w(n) = (ne )! e∈E•

P A vertex v is called a source of a current n if u∼v n{v,u} is odd. We denote by ∂n the set of all sources of n. The classical random current representation of correlation functions due to Griffiths, Hurst and Sherman [16] yields for any A ⊂ S ⊂ V• , P

Y f w(n) , σv S,β = P∂n=A w(n) ∂n=∅ v∈A

where both sums are over currents that are zero outside the subgraph induced n by S. For u, v ∈ V• and a current n, we will write u ↔ v if u is connected to v via n

a path of edges in E• with nonzero values of n, and u 6↔ v if it is not connected. The last tool that we will need is the celebrated switching lemma introduced in [16] and developed by Aizenman in [2]. It says that if A ∪ {u, v} ⊂ S ⊂ V• , then X X n w(n1 )w(n2 ) = w(n1 )w(n2 )1[u ↔ v], ∂n1 =A ∂n2 =∅

∂n1 =A∆{u,v} ∂n2 ={u,v}

where n = n1 + n2 , ∆ is the symmetric difference, and where again the sums are taken over currents n1 , n2 that are zero outside the subgraph induced by S. Proof of Lemma 4.2. Fix a vertex 0 ∈ V . From the definition of the Ising model, we have X
d d f hσ0 i+ = hσ σ i = J{u,v} hσ0 σ• σu σv ifG• ,β −hσ0 σ• ifG• ,β hσu σv ifG• ,β . 0 • G ,β G,β • dβ dβ {u,v}∈E•

CIRCLE PATTERNS AND CRITICAL ISING MODELS

17

P Let ZG• ,β = ∂n=∅ w(n) be the partition function of currents on G• with no sources. Using the random current representation of correlation functions and the switching lemma for S = V• , we obtain X n 1 w(n1 )w(n2 )1[0 6↔ •], hσ0 σ• σu σv ifG• ,β −hσ0 σ• ifG• ,β hσu σv ifG• ,β = 2 ZG• ,β ∂n1 ={0,•}∆{u,v} ∂n2 =∅

n

where n = n1 + n2 . Note that if n1 = {0, •}∆{u, v}, ∂n2 = ∅ and 0 6↔ •, then n n n n either 0 ↔ u and v ↔ •, or 0 ↔ v and u ↔ •. Since the second case is the same as the first one with u and v exchanged, we get d 1 X X (A.1) hσ0 i+ = J{u,v} δu,v , G,β 2 dβ ZG • ,β u∈V• v∈V•

where n

X

δu,v =

n

n

w(n1 )w(n2 )1[0 ↔ u, v ↔ •, 0 6↔ •].

∂n1 ={0,•}∆{u,v} ∂n2 =∅

For z ∈ {0, •}, define Sz to be the set of vertices in V• that are not connected to z in n. Let us compute δu,v by summing over all possible S0 : X X n n n w(n1 )w(n2 )1[S0 = S, 0 ↔ u, v ↔ •, 0 6↔ •] δu,v = S⊂V• ∂n1 ={0,•}∆{u,v} ∂n2 =∅

=

X

n

X

w(n1 )w(n2 )1[S0 = S, v ↔ •].

S⊂V• ∂n1 ={0,•}∆{u,v} s.t. v,•∈S ∂n2 =∅ and 0,u∈S /

Note that when S0 = S, n1 and n2 vanish on every {x, y} with x ∈ S and V \S y∈ / S. Thus, for i = 1, 2, we can decompose ni as ni = nSi + ni • , where nA i denotes the current with zero values outside the subgraph induced by A, and with sources ∂nA i = A ∩ ∂ni . Together with the last equality and the random current representation of correlation functions, this gives X X δu,v = w(n1 )w(n2 )hσv σ• ifS,β 1[S0 = S]. S⊂V• ∂n1 ={0}∆{u} s.t. v,•∈S ∂n2 =∅ and 0,u∈S /

Since 1 ≥ hσv σ• ifS,β , we get X X δu,v ≥

w(n1 )w(n2 )(hσv σ• ifS,β )2 1[S0 = S]

S⊂V• ∂n1 ={0}∆{u} s.t. v,•∈S ∂n2 =∅ and 0,u∈S /

=

X

X

w(n1 )w(n2 )1[S0 = S]

S⊂V• ∂n1 ={0}∆{u}∆{v,•} s.t. v,•∈S ∂n2 ={v,•} and 0,u∈S /

=

X

X

S⊂V• ∂n1 ={0}∆{u} s.t. v,•∈S ∂n2 =∅ and 0,u∈S /

n

w(n1 )w(n2 )1[S0 = S, v ↔ •]

CIRCLE PATTERNS AND CRITICAL ISING MODELS

X

=

n

18

n

w(n1 )w(n2 )1[v ↔ •, 0 6↔ •],

∂n1 ={0}∆{u} ∂n2 =∅

where in the second line we used the random current representation of correlation functions, and in the third line again the switching lemma. We now sum over all possible S• : X X n n δu,v ≥ w(n1 )w(n2 )1[S• = S, v ↔ •, 0 6↔ •] S⊂VG ∂n1 ={0}∆{u} ∂n2 =∅

=

X

X

w(n1 )w(n2 )1[S• = S]

S⊂VG ∂n1 ={0}∆{u} s.t. 0,u∈S ∂n2 =∅ and v∈V• \S

=

X

X

w(n1 )w(n2 )hσ0 σu ifS,β 1[S• = S].

S⊂VG ∂n1 =∅ s.t. 0,u∈S ∂n2 =∅ and v∈V• \S

The third line follows from the fact that since S• = S, n1 and n2 can be decomV \S posed as ni = nSi + ni • as before. Combining the inequality above with (A.1), we get X X X X 1 d w(n1 )w(n2 )Ju,v hσ0 σu ifS,β 1[S• = S] hσ0 i+ ≥ 2 G,β dβ ZG• ,β S⊂VG u∈S v∈V• \S ∂n1 =∅ S30 ∂n2 =∅

≥

X 1 X 1 ϕS,β (0) 2 w(n1 )w(n2 )1[S• = S] β ZG• ,β S⊂V S30

≥

∂n1 =∅ ∂n2 =∅

1 1 inf ϕS,β (0) 2 β S30 ZG• ,β

X


n w(n1 )w(n2 ) 1 − 1[0 ↔ •]

∂n1 =∂n2 =∅

1 inf ϕS,β (0)(1 − (hσ0 σ• ifG• ,β )2 ) β S30 1 2 = inf ϕS,β (0)(1 − (hσ0 i+ G,β ) ), β S30

=

where in the second inequality we used that βJ{u,v} ≥ tanh(βJ{u,v} ), and in the first equality we used the random current representation of correlations and the switching lemma for the last time.  References [1] S. I. Agafonov and A. I. Bobenko, Discrete Z γ and Painlev´e equations, Internat. Math. Res. Notices 4 (2000), 165–193. MR1747617 [2] M. Aizenman, Geometric analysis of ϕ4 fields and Ising models. I, II, Comm. Math. Phys. 86 (1982), no. 1, 1–48. [3] R. J. Baxter, Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics, Proc. Roy. Soc. London Ser. A 404 (1986), no. 1826, 1–33. MR836281 [4] A. I. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: patterns with constant angles, Duke Math. J. 116 (2003), no. 3, 525–566. MR1958097

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