Circle Patterns and Critical Ising Models

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 large 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.


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 Z 2 . 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 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][7][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.

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  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 , u∼v θ (v,u) = π, (2.1) 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 for all directed edges (u, v). This in particular implies that G is of bounded degree.
Let J = (J e ) e∈E ∈ (0, ∞] E be coupling constants defined by One can check that that (2.2) implies that J ∞ < ∞. 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 = (V G , E G ) of G with coupling constants J. To this end, let ∂V G = {v ∈ V G : ∃u ∈ V \ V G , {u, v} ∈ E} be the boundary, and let Ω G = {−1, +1} V G 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 Ω G given for σ ∈ Ω G , by where β > 0 is the inverse temperature, and Z G,β is the normalizing constant called the partition function. We write · G,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 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 β c = 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 C β > 0 such that for all u, v ∈ V , if the radii of circles are uniformly bounded from above, then for every v ∈ V , there exists s v > 0 such that for all β > 1, Moreover, if the radii of circles are uniformly bounded from below, then one can choose one such s = s v for all v.
We also show that the magnetic susceptibility diverges at criticality: 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 , 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 β c = 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. 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.
The resulting Ising model is critical as these coupling constants are computed using (2.3) applied to the dual lattice G * Z 2 with appropriately stretched or squeezed columns. Note that if J i are constant, the classical anisotropic critical Ising model is recovered, and upon setting J i = − 1 2 log( √ 2 − 1) we get the homogenous critical model.

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.
To this end, let E be the set of directed edges of G. For e = (u, v) ∈ 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 = (x e ) e∈E be weights on the undirected edges given by x e = x e x − e . We denote the relation between x and x by writing x ∼ x.
The Kac-Ward transition matrix Λ( x) on G is a matrix indexed by E given by where ∠( e, g) = Arg hg−tg he−te ∈ (−π, π] is the turning angle from e to g. The Kac-Ward matrix is given by where Id is the identity matrix. Consider a finite graph G = (V G , E G ) drawn in the plane with possible edge crossings. We will write Λ G ( x) and T G ( x) for the corresponding matrices indexed by the directed edges E G of G. Let E G be the collection of even subgraphs of G, i.e. subsets ω of E G such that all vertices in V G have even degree in (the subgraph induced by) ω, and let where C(ω) is the number of pairs of edges in ω that cross. The seminal identity of Kac and Ward [18] reads If G has no edge crossings, then the heigh-temperature expansion of the Ising partition function Z f G,β is equal, up to an explicit constant, to Z G (x) with x e = tanh βJ e , 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 T G ( 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 which are forms of the fermionic observable introduced by Smirnov [27]. For a directed edge e ∈ E, let m e = (t e + h e )/2 be its midpoint. For e, g ∈ E G , we define a modified graph G e, g with vertex set V G ∪ {m e , m g } and edge set (E G \ {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 E G ( 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 V G 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 ∈ E G ( e, g) contains a self-avoiding path starting at m e and ending at m g . We denote by γ ω the left-most such path. We also write α(γ ω ) for the total turning angle of γ ω , i.e., the sum of turning angles between consecutive steps of γ ω .
Let F G ( x) be a matrix indexed by E G given by where x ∼ x. For a graph G with no edge crossings, the following identity: 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. This result will directly follow from the next lemma. For v ∈ V , let In v = { e ∈ E : h e = v} and Out v = { e ∈ E : t e = v}. Let J be the involutive automorphism of C E induced by the map e → − e, and letΛ( x) = JΛ( x). As was noted in [22], and is easily seen from the definition of the transition matrix, Λ( x) is a Hermitian, block-diagonal matrix with blocksΛ( x) v , v ∈ V , acting on the linear subspace indexed by Out v , and given bỹ Let ρ v be the restriction of ρ to the subspace indexed by Out v . Note that Jρ = ρ, and hence to prove Theorem 3.1, it is enough to show the following: Proof. Fix e ∈ Out v . Let e 1 , . . . , e n be a clockwise ordering of the edges in Out v \ e that satisfy ∠(− e, e i ) ≥ 0, i = 1, . . . , k, and let g 1 , . . . , g m be a counterclockwise ordering of the remaining edges in Out v \ e. Then We can 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 ∈ Out v . We have where the fifth equality follows from a telescopic sum, and the sixth holds true since θ e + n j=1 θ e j + m j=1 θ g j = π. Hence, 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.

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 ∈ V G , let E G (u, v) be the collection of subsets ω of E G such that each vertex in V G \ ({u}∆{v}) (resp. in {u}∆{v}) has even (resp. odd) degree in ω, where ∆ is the symmetric difference. Let The classical high-temperature expansion of the Ising correlation functions yields for x e = tanh(βJ e ), and all u, v ∈ V G . 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.
Proof. We first prove the lower bound. By (3.2), T −1 G ( x) = F G ( x). By the definition of E G ( e, g), removing the two half edges from ω ∈ E G ( e, g) (carrying weights x e and x − g ) yields a configuration in E G (u, v). It is now enough to use (4.1) and the fact that all terms in the definition of Z G (x) u,v are positive, whereas the corresponding terms in F G ( x) e, g carry a complex factor.
To obtain the upper bound, we construct an augmented graph G γ 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 τ e τ − e = −1 if e is crossed by γ, +1 otherwise, and, when needed, we extend the weights to G γ by setting τ γ = τ − γ = 1.
We chose an orientation γ = (u, v) of γ, and for x τ ∼ x τ , we have The first equality follows from the high temperature expansion, and the fact that the signs of x τ cancel out the signs from the definition of Z G γ (x τ ) assigned to even subgraphs containing γ. The second equality is a consequence of the Kac-Ward formula for G γ 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 T G ( x τ ). The last inequality holds true since Z G γ (x τ ) counts even subgraphs with additional signs, whereas all terms in Z G (x) are positive.
Let S ⊂ V be such that the subgraph of G induced by S is connected. We will denote by · S,β the expectation with respect to the Ising model on this induced subgraph. We define for v ∈ V , 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
We will next use the existence of the eigenvector ρ to get a uniform lower bound on ϕ S,β (v). Before stating the result, we need to introduce additional notation. Let G = (V G , E G ) be a finite subgraph of G induced by the vertex set V G . Consider the edges of G whose one endpoint is in V G and the other one in V \ V G . Each such edge splits into two half-edges, one of which is incident to V G . We add the incident half-edges to the edge set E G , 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 V G and we call the resulting graphḠ = (VḠ, EḠ). Note that by construction, all vertices in V G are interior inḠ meaning that they have the same degree inḠ as in G. Let n(∂G) be the set of the directed versions of the half-edges inĒ G 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 < ∞.
Then for every v ∈ V and every finite S ⊂ V containing v, 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 ϕ S,1 (v) and the bounded angle property. We can therefore assume that d(v, V \ S) > 1.
Let x e = tan θ e 2 and let ρ be the eigenvector of Λ( x) as defined in Theorem 3.1 restricted to the directed edges ofḠ (here the half-edges are identified with their counterparts in G). Let ζ = TḠ( 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) ∈ In v , where r u is the radius of the circle centered at u, and where in the last inequality we used that tanh J g = x g = x g x − g ≥ tan ε 2 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. 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 x e (β) = tanh βJ e . We use the upper bound from Lemma 4.1 and Lemma 4.4, to get for u = v, 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, where sv 2 = r R tan ε 2 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 χ G,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 χ G,β (v) = ∞ for β ≥ 1 follows directly from Lemma 4.3 and the second Griffiths inequality.

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 : E G → C is s-holomorphic at an interior vertex v ∈ V G if for every dual vertex v * adjacent to v, where e 1 , e 2 are the two edges incident on both v and v * .
Let η e = h e − t e , and e = η − 1 2 e R. We define L to be the real linear space of functions ϕ : E → C satisfying ϕ e ∈ e for all e. One can easily check that L 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 where ρ e = |e * | as before. Note that since e and − e are orthogonal, we have for ϕ ∈ L, .
Let x e = tan θ e 2 be the critical weights. Proof. Note that ρ (v,u) = 2r v sin θ (v,u) , where r v is the radius of the circle centered at v. This implies that the row of T G ( x)S indexed by e ∈ In v is equal to the corresponding row of the matrix T S from [23] scaled by 2 √ r v 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 recovered from their boundary values. Let G andḠ be as defined before Lemma 4.3, and let ϕ ∈ L be a function defined on the directed edges ofḠ and satisfying ϕ( e) ∈ l e for e ∈ n(∂G), and ϕ( e) = 0 otherwise.
Following Smirnov [27], we say that f : EḠ → C solves the discrete Riemann-Hilbert boundary value problem for the pair (G, ϕ) if f is s-holomorphic at all v ∈ V G and Sf ( e) = ϕ( e) for all e ∈ n(G).
Proposition 5.2. Let G and ϕ be as above and let TḠ( x) be the critical Kac-Ward operator defined onḠ, where the half-edges ofḠ inherit weights from the corresponding edges of G. Then the discrete Riemann-Hilbert boundary value problem for (G, ϕ) has a unique solution Proof. Suppose that f is a solution to the discrete Riemann-Hilbert boundary value problem. Note thatḠ is not formally a subgraph of G since it contains 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 TḠ( x)Sf ( e) = 0 for all e / ∈ n(G). Moreover if e ∈ n(G), then h( e) = t( g) for all g = − e. Hence by the definition of the Kac-Ward operator, TḠ( x)Sf ( e) = IdSf ( e) = ϕ( e) for e ∈ n(G). This means that TḠ( x)Sf ( e) = ϕ( e) for all directed edges e, and the claim follows.
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 of Smirnov [27], and Chelkak and Smirnov [10], stating the existence of Re( f 2 ) simultaneously on both G * and G, and its sub-and superharmonicity on the respective isoradial graphs. where in the last equality we used that DΛ is skew-symmetric.
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. The third line follows from the fact that since S • = S, n 1 and n 2 can be decomposed as n i = n S i + n