The critical Z-invariant Ising model on an isoradial graph GG maps, via Fisher's correspondence, to a dimer model on a decorated Fisher graph GD\mathcal{G}^D. All dimer, hence Ising, correlations are Pfaffians of the inverse Kasteleyn matrix K−1K^{-1}, and two independent exact formulas exist for its coefficients: a Fourier double-integral formula, valid when GG is Z2\mathbb{Z}^2-periodic (Boutillier–de Tilière 2010), and a local contour-integral formula requiring no periodicity (Boutillier–de Tilière 2011). Uniqueness of the decaying inverse guarantees that the two coincide, but the authors state that they have 'not yet been able to understand the identity by an explicit computation', except on the square lattice. This problem asks for a direct, coefficient-level identification of the two formulas for every Z2\mathbb{Z}^2-periodic isoradial graph. The analogous Fourier-to-local passage has since been carried out explicitly for the massive Laplacian Green function (2017) and for bipartite elliptic dimers on minimal graphs (2023), supplying the technique, but the non-bipartite Fisher-graph case remains open.

Background

The Ising model on a finite graph GG assigns spins σv∈{−1,+1}\sigma_v\in\{-1,+1\} to vertices with probability proportional to exp⁡(∑e=uvJeσuσv)\exp(\sum_{e=uv}J_e\sigma_u\sigma_v), where JeJ_e are coupling constants. A planar graph is isoradial if it is embedded so that every face is inscribed in a circle of common radius, with circumcenters inside the faces; joining each circumcenter to the vertices of its face yields the diamond graph, all of whose faces are rhombi of side length one, and each edge ee of GG is the diagonal of exactly one rhombus, whose half-angle θe\theta_e is attached to ee. Requiring the Ising coupling constants to be invariant under star-triangle (Yang–Baxter) transformations of the underlying graph (Z-invariance) and to satisfy a generalized self-duality forces the critical values J(θe)=12log⁡1+sin⁡θecos⁡θeJ(\theta_e)=\frac{1}{2}\log\frac{1+\sin\theta_e}{\cos\theta_e} (equivalently sinh⁡(2Je)=tan⁡θe\sinh(2J_e)=\tan\theta_e); the resulting critical Z-invariant Ising model, introduced by Baxter, contains the critical square, triangular, and honeycomb-lattice models as special cases.

Fisher's correspondence maps the Ising model on GG to a dimer model on a decorated graph GD\mathcal{G}^D (the Fisher graph), obtained by replacing each vertex of GG by a ring of triangles; perfect matchings (dimer configurations) of GD\mathcal{G}^D correspond to polygonal contour configurations of the Ising model. Dimer statistics are encoded in the Kasteleyn matrix KK: a weighted, oriented adjacency matrix of GD\mathcal{G}^D whose orientation is clockwise odd around every face, with edge weights νe\nu_e explicit functions of the angles θe\theta_e. Correlations (and hence Ising observables such as spin and energy correlations) are Pfaffians of submatrices of an inverse K−1K^{-1} — a Pfaffian being a signed sum over pairings of indices, computable as a square root of a determinant — so explicit formulas for the coefficients of K−1K^{-1} are the central computational object of the theory.

Two independent routes to K−1K^{-1} exist. When GG is Z2\mathbb{Z}^2-periodic, Boutillier–de Tilière computed K−1K^{-1} by Fourier transform: writing K^(z,w)\widehat K(z,w) for the finite matrix of the Kasteleyn operator restricted to (z,w)(z,w)-quasi-periodic functions over the fundamental domain and P(z,w)=det⁡K^(z,w)P(z,w)=\det\widehat K(z,w) for the characteristic polynomial (its zero set P(z,w)=0P(z,w)=0 being the spectral curve of the periodic dimer model), the inverse is given by the double integral over the unit torus T2\mathbb{T}^2,

K(v,x,y),(v′,x′,y′)−1=1(2πi)2∬T2Cof⁡(K^(z,w))v′,vP(z,w) zx′−xwy′−y dzzdww,K^{-1}_{(v,x,y),(v',x',y')}=\frac{1}{(2\pi i)^2}\iint_{\mathbb{T}^2}\frac{\operatorname{Cof}(\widehat K(z,w))_{v',v}}{P(z,w)}\,z^{x'-x}w^{y'-y}\,\frac{dz}{z}\frac{dw}{w},

where v,v′v,v' index vertices of the fundamental domain and (x,y),(x′,y′)∈Z2(x,y),(x',y')\in\mathbb{Z}^2 are translation labels; the same paper proves that the inverse Kasteleyn matrix whose coefficients decrease at infinity exists and is unique (its Proposition 5). Independently, with no periodicity assumption, Boutillier–de Tilière proved that K−1K^{-1} is given by the local formula

Kx,y−1=1(2π)2∮Cx,yfx(λ) fy(−λ) ex,y(λ) log⁡λ dλ+Cx,y,K^{-1}_{x,y}=\frac{1}{(2\pi)^2}\oint_{\mathcal{C}_{x,y}}f_x(\lambda)\,f_y(-\lambda)\,e_{x,y}(\lambda)\,\log\lambda\,d\lambda+C_{x,y},

where ex,ye_{x,y} is the discrete exponential function introduced by Mercat and Kenyon (defined from the angles of the rhombi crossed by a path of the diamond graph from xx to yy), fx,fyf_x,f_y depend only on the local geometry at xx and yy, Cx,y\mathcal{C}_{x,y} is a contour separating the poles of the integrand and avoiding a half-line determined by the pair (x,y)(x,y), and the additive constant Cx,yC_{x,y} equals ±14\pm\frac{1}{4} for nearby vertices and 00 otherwise. The formula is local: Kx,y−1K^{-1}_{x,y} depends only on an edge-path of GG between the vertices naturally associated to xx and yy, so changing the graph away from this path does not change the value. This extends Kenyon's local formulas for the critical Green function and for critical dimers on bipartite isoradial graphs, which do not apply here because GD\mathcal{G}^D is neither bipartite nor isoradial.

In the periodic case both formulas are available and, by the uniqueness result, they coincide. Yet, as the authors state in the introduction of the locality paper, 'the proofs are nevertheless totally different in spirit, and we have not yet been able to understand the identity by an explicit computation, except in the case where G=Z2G=\mathbb{Z}^2.' One structural input is known: the dimer characteristic polynomial is an explicit constant multiple of the critical Laplacian characteristic polynomial, P(z,w)=2∣V1∣∏e∈E(G1)(cot⁡2θe2−1) PΔ(z,w)P(z,w)=2^{|V_1|}\prod_{e\in E(G_1)}(\cot^2\frac{\theta_e}{2}-1)\,P_\Delta(z,w), so the spectral data on the two sides is the same. Later work in the same research line made the analogous Fourier-to-local passage explicit in two neighboring settings: for the massive Laplacian Green function (Boutillier–de Tilière–Raschel 2017, by residue evaluation plus a change of variable through the elliptic parametrization of the spectral curve, the authors noting that this relation 'was not understood' in the earlier critical papers, including the locality paper), and for bipartite elliptic dimers on minimal graphs (Boutillier–Cimasoni–de Tilière 2023, again by residues plus a change of variable). For the Ising Fisher graph itself, the massive successor paper (Boutillier–de Tilière–Raschel 2019) still identifies its local and Fourier inverses only through the uniqueness argument, and a February 2026 classification paper by de Tilière–Rey does not address the question. The explicit, coefficient-level identification for the non-bipartite Fisher graph therefore remains to be written down.

Problem Statement

Prove, by explicit direct computation rather than by the uniqueness of the decaying inverse, that the local and Fourier formulas for the inverse Kasteleyn matrix of the critical Z-invariant Ising model coincide on every periodic isoradial graph.

Precisely: let GG be an infinite Z2\mathbb{Z}^2-periodic isoradial graph with critical coupling constants J(θ)=12log⁡1+sin⁡θcos⁡θJ(\theta)=\frac{1}{2}\log\frac{1+\sin\theta}{\cos\theta}, let GD\mathcal{G}^D be its Fisher graph with the critical dimer weight function, and let KK be the Kasteleyn matrix of GD\mathcal{G}^D. Show, for every pair of vertices x,yx,y of GD\mathcal{G}^D (covering all decoration vertex types and all translate pairs), that the coefficient Kx,y−1K^{-1}_{x,y} given by the local contour-integral formula (Theorem 1 in the introduction, stated precisely as Theorem 5 in Section 4.2.4, equation (5), of Boutillier–de Tilière, 'The critical Z-invariant Ising model via dimers: locality property') equals the coefficient given by the Fourier double-integral formula (Equation (8) in Section 4.2.2 of Boutillier–de Tilière, 'The critical Z-invariant Ising model via dimers: the periodic case'), where the two papers' conventions — Kasteleyn orientations, weight normalizations, branches of log⁡λ\log\lambda, and integration contours — are matched explicitly.

The identification must be established by direct evaluation, for example by computing one of the two integrals of the Fourier formula by residues and performing an explicit change of variable in the remaining one, along the parametrization of the spectral curve P(z,w)=2∣V1∣∏e∈E(G1)(cot⁡2θe2−1) PΔ(z,w)P(z,w)=2^{|V_1|}\prod_{e\in E(G_1)}(\cot^2\frac{\theta_e}{2}-1)\,P_\Delta(z,w), so that the mechanism by which the locality property emerges from the spectral data is made transparent. The uniqueness statement (Proposition 5 of the periodic-case paper) may be invoked only as a final consistency check, never as the step that identifies the two matrices. A proof of the corresponding identity for the massive (elliptic) Z-invariant Ising model of Boutillier–de Tilière–Raschel, followed by an explicit and controlled limit as the elliptic modulus tends to zero, also constitutes a valid solution.

References

  1. Cédric Boutillier and Béatrice de Tilière, The critical Z-invariant Ising model via dimers: locality property, Communications in Mathematical Physics 301 (2011) 473–516. DOI: 10.1007/s00220-010-1151-3, arXiv:0902.1882. https://arxiv.org/abs/0902.1882
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  3. Richard Kenyon, The Laplacian and Dirac operators on critical planar graphs, Inventiones mathematicae 150 (2002) 409–439. DOI: 10.1007/s00222-002-0249-4. https://doi.org/10.1007/s00222-002-0249-4
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  6. Béatrice de Tilière, The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model, Electronic Journal of Probability 26 (2021). DOI: 10.1214/21-EJP601, arXiv:1801.00207. https://arxiv.org/abs/1801.00207
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  8. Béatrice de Tilière and Lucas Rey, Classification of (non)-frustrated 2D Ising models in genus 1 on isoradial graphs, preprint (2026), arXiv:2602.13526. https://arxiv.org/abs/2602.13526
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