119132626
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 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.
122553843
Crystal statistics. I. A two-dimensional model with an order-disorder transition
The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$
119279106
Critical Temperature of Periodic Ising Models
A periodic Ising model has interactions which are invariant under translations of a full-rank sublattice $${\mathfrak{L}}$$ of $${\mathbb{Z}^2}$$. We prove an exact, quantitative characterization of the critical temperature, defined as the supremum of temperatures for which the spontaneous magnetization is strictly positive. For the ferromagnetic model, the critical temperature is the solution of a certain algebraic equation, resulting from the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real zero on the unit torus. With our technique we provide a simple proof for the exponential decay of spin-spin correlations above the critical temperature, as well as the exponential decay of the edge-edge correlations for all non-critical edge weights of the corresponding dimer model on periodic Fisher graphs.
17274971
The critical temperature for the Ising model on planar doubly periodic graphs
We provide a simple characterization of the critical temperature for the Ising model on an arbitrary planar doubly periodic weighted graph. More precisely, the critical inverse temperature $\beta$ for a graph $G$ with coupling constants $(J_e)_{e\in E(G)}$ is obtained as the unique solution of an algebraic equation in the variables $(\tanh(\beta J_e))_{e\in E(G)}$. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices.
121287035
On the Dimer Solution of Planar Ising Models
Derivations of the partition function of the Ising model on a general planar lattice L, which proceed via an associated dimer problem and use Pfaffians, are simplified by constructing a lattice LΔ (the ``terminal lattice'' derived from an ``expanded lattice'' of L) for which (A) the allowed dimer configurations are in one‐one correspondence with allowed Ising polygon configurations on L, and which (B) is planar if L is planar so that Kasteleyn's theorem may be used directly to construct the appropriate Pfaffian. This is in contrast to previous use of nonplanar associated dimer lattices for which the correspondence is not one‐one, so that is has been necessary to prove a somewhat obscure ``cancellation theorem.''
121591928
A combinatorial solution of the two-dimensional Ising model
An elementary method which yields the partition function of a two-dimensional Ising model is described. The method is purely combinatorial and does not involve any of the algebraic apparatus used in this connection by Onsager and Kaufman.
120633388
Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics
It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘Z-invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.
119276324
Phase Transition Free Regions in the Ising Model via the Kac–Ward Operator
We provide an upper bound on the spectral radius of the Kac–Ward transition matrix for a general planar graph. Combined with the Kac–Ward formula for the partition function of the planar Ising model, this allows us to identify regions in the complex plane where the free energy density limits are analytic functions of the inverse temperature. The bound turns out to be optimal in the case of isoradial graphs, i.e., it yields criticality of the self-dual Z-invariant coupling constants.
119307216
The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator
We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs.
54789807
Universality in the 2D Ising model and conformal invariance of fermionic observables
It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof has ever been given, and even physics arguments support (a priori weaker) Möbius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.
123683696
Potential theory on a rhombic lattice
Abstract Of concern are complex valued functions defined on the lattice points of the complex plane. The lattice can be the usual lattice of square blocks but of main interest is an irregular lattice with squares replaced by rhombs. A function is defined to be discrete analytic if the difference quotient across one diagonal of a rhomb equals the difference quotient across the other diagonal. Based on this definition discrete analogs of the following concepts in classical function theory are developed: Laplace equation, Cauchy-Riemann equations, differentiation, contour integration, Morera's theorem, and harmonic polynomials. The theory is more than an analogy because, for a common class of boundary value problems, it proves possible to obtain upper and lower bounds for the classical Dirichlet integral in terms of discrete harmonic functions.
2146930
Discrete Riemann Surfaces and the Ising Model
Abstract: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.
13913815
The critical Z-invariant Ising model via dimers: the periodic case
We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical $${\mathbb{Z}^2}$$ , triangular and honeycomb lattice at the critical temperature. Fisher (J Math Phys 7:1776–1781, 1966) introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.
17575338
The Critical Z-Invariant Ising Model via Dimers: Locality Property
We study a large class of critical two-dimensional Ising models, namely critical Z-invariant Ising models. Fisher (J Math Phys 7:1776–1781, 1966) introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model, consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph. Our main result is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of Kenyon (Invent Math 150(2):409–439, 2002), as a contour integral of the discrete exponential function of Mercat (Discrete period matrices and related topics, 2002) and Kenyon (Invent Math 150(2):409–439, 2002) multiplied by a local function. Using results of Boutillier and de Tilière (Prob Theor Rel Fields 147(3–4):379–413, 2010) and techniques of de Tilière (Prob Th Rel Fields 137(3–4):487–518, 2007) and Kenyon (Invent Math 150(2):409–439, 2002), this yields an explicit local formula for a natural Gibbs measure, and a local formula for the free energy. As a corollary, we recover Baxter’s formula for the free energy of the critical Z-invariant Ising model (Baxter, in Exactly solved models in statistical mechanics, Academic Press, London, 1982), and thus a new proof of it. The latter is equal, up to a constant, to the logarithm of the normalized determinant of the Laplacian obtained in Kenyon (Invent Math 150(2):409–439, 2002).
53602485
The Z-invariant Ising model via dimers
AbstractThe Z-invariant Ising model (Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315–346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When $$k=0$$k=0 the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers (Boutillier and de Tilière in Probab Theory Relat Fields 147:379–413, 2010; Commun Math Phys 301(2):473–516, 2011) to the fullZ-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of Boutillier and de Tilière 
(2011): it involves a local function and the massive discrete exponential function introduced in Boutillier et al. (Invent Math 208(1):109–189, 2017). This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of Boutillier et al. 
(2017), and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model.
15988860
90 90 02 v 1 1 S ep 1 99 9 Discrete Z γ and Painlevé equations
A discrete analogue of the holomorphic map z is studied. It is given by a Schramm’s circle pattern with the combinatorics of the square grid. It is shown that the corresponding immersed circle patterns lead to special separatrix solutions of a discrete Painlevé equation. Global properties of these solutions, as well as of the discrete z are established. Supported by the DAAD grant A/98/38190. Present address: Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany. E-mail: agafonov@sfb288.math.tu-berlin.de Partially supported by SFB 288. E-mail: bobenko@math.tu-berlin.de
8527917
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function
Abstract Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to ℤ d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon’s discrete Green’s function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.
119609036
The Signed Loop Approach to the Ising Model: Foundations and Critical Point
The signed loop approach is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.
119326467
PLANAR ISING MODEL AT CRITICALITY: STATE-OF-THE-ART AND PERSPECTIVES
In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on $\mathbb{Z}^2$ and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.
1474358
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.