From the planar Ising model to quasiconformal mappings

TL;DR

The paper reveals that scaling limits of planar Ising observables on generic $s$-embeddings are governed by conjugate Beltrami equations, linking near-critical Ising behavior to quasiconformal geometry. It introduces a robust framework where primitives of discrete fermions correspond to Green functions on S-graphs, yielding explicit, locally embedded scaling factors determined by the embedding geometry. It shows conformal covariance of energy density in bounded domains for critical doubly-periodic graphs and provides maximal-surface formulas in Minkowski space, illustrating a broader conformal structure than the classical Euclidean one. By establishing subsequential convergence for two- and multi-point fermionic correlators and connecting them to Green-function objects, the work validates Chelkak’s predictions and opens pathways to spin correlations and random-environment Ising models within a unified quasiconformal/Lorentzian geometric viewpoint.

Abstract

We identify the scaling limit of full-plane Kadanoff-Ceva fermions on generic, non-degenerate $s$-embeddings. In this broad setting, the scaling limits are described in terms of solutions to conjugate Beltrami equations with prescribed singularities. For the underlying Ising model, this leads to the scaling limit of the energy-energy correlations and reveals a connection between the scaling limits of (near-)critical planar Ising models and quasiconformal mappings. For grids approximating bounded domains in the complex plane, we establish, in the scaling regime, the conformal covariance of the energy density on critical doubly periodic graphs. We complement this result with an analogous statement in the case where the limiting conformal structure generates a maximal surface $(z,\vartheta)$ in Minkowski space $\mathbb{R}^{(2,1)}$. All scaling factors obtained are local and expressed in terms of the geometry of the embedding, even in situations where they vary drastically from one region to another. These results confirm the predictions of Chelkak and highlight that the scaling limits of generic (near-)critical Ising models naturally live on a substantially richer conformal structure than the classical Euclidean one.

Figures (5)

• Figure 1: (Left) Notation around some given quad $z \in \diamondsuit (G)$ with an arbitrary embedding in the plane. Vertices of the primal graph $G^\bullet$ are represented as black dots while vertices of the dual graph $G^\circ$, corresponding to faces of $G$, are represented as white dots. Moreover, the so called corners, which correspond to the edges of the bipartite graph $\Lambda(G)= G^\bullet \cup G^\circ$, are drawn as triangles. One represents in this picture a piece of the double cover of the corner graph branching around $z$. Corners that are neighbours in that double-cover are linked with dashed segments. (Right) A piece of one associated $s$-embedding containing the quad $\mathcal{S}^{\diamondsuit}(z)$, tangential to a circle of radius $r_z$ centred at $\mathcal{S}(z)$. The Ising weight of the edge attached to the vertices $v_{0}^{\bullet},v_{1}^\bullet$ can be recovered using the four angles $\phi_{v,z}$ attached to the quad $\mathcal{S}^{\diamondsuit}(z)$ using the formula \ref{['eq:theta-from-S']}.
• Figure 2: (Left) Bosonization of the Ising model as a dimer model: each quad $z \in \diamondsuit(G)$ is split into four triangles, colored in a bipartite fashion between black faces $B$ and white faces $W$. The dimer weight $K(w,b)$ of the match between faces $b \in B$ and $w \in W$ is either $\cos(\theta_z)$, $\sin(\theta_z)$, or $1$, depending on whether the faces belong to the same quad or not. To each corner $c \in \Upsilon$, one attaches two neighboring faces $c^\circ \in W$ and $c^\bullet \in B$, which allows computing the associated dimer origami square root$\eta^{\frak{d}}_{c^\bullet} = \eta^{\frak{d}}_{c^{\circ}} := \bar{\varsigma}\eta_c$ using \ref{['eq:def-eta']}. (Right) A piece of the S-graph $\mathcal{S}-i\mathcal{Q}$: the black triangle $(\mathcal{S}(v_{0}^\bullet) \mathcal{S}(v_{1}^{\circ}) \mathcal{S}(z))$ is mapped to the segment $[(\mathcal{S} - i\mathcal{Q})(z), (\mathcal{S} - i\mathcal{Q})(v_1^\circ)]$, which contains the point $(\mathcal{S} - i\mathcal{Q})(v_0^\bullet)$ in its interior. The continuous-time directed random walk $X^{(\alpha)}_{t}$ for $\alpha = \exp(-i\frac{\pi}{4})$ has nonzero jump rates from $(\mathcal{S} - i\mathcal{Q})(v_0^\bullet)$ to $(\mathcal{S} - i\mathcal{Q})(z)$ and $(\mathcal{S} - i\mathcal{Q})(v_1^\circ)$ (see Definition \ref{['def:RW-S-graph']}), as indicated by the dashed oriented arrows. The invariant measure $\mu^{(\alpha)}(v_0^\bullet)$ equals the area of the triangle $(\mathcal{S}(v_{0}^\bullet) \mathcal{S}(v_{1}^{\circ}) \mathcal{S}(z))$.
• Figure 3: (Left) Transitions in a non-degenerated S-graph $\mathcal{S}+\alpha^2\mathcal{Q}$. The allowed jump-rates of the directed random walk $X_t^{(\alpha)}$ are given by the dashed arrows. The triangle containing the $3$ red arrows corresponds to the image of a face $c^\circ$. The walk can move from $v$ to $v^{\pm}$ if and only if $v\in (v^{-};v^+)$. The variance of $X_t^{(\alpha)}$ is normalised so that the expected time for $X_t^{(\alpha)}$ to exit from $v$ is equal to the expected time for one dimensional Brownian motion started at $v$ along the segment $[v^{-},v^{+}]$ to hit on of its extremities. (Right) Transitions in a degenerated S-graph $\mathcal{S}+\alpha^2\mathcal{Q}$: When moving $\alpha \in \mathbb{T}$ up to value $\alpha=i \cdot \overline{\eta^{\frak{d}}_{c^\circ}}$, the image of the white face $c^\circ$ is shrank to a single point. The three red transition rates become infinite as $\alpha$ approaches $i \cdot \overline{\eta^{\frak{d}}_{c^\circ}}$, but their respective ratio remains finite and is given in the limit by ratios $m_k$ in \ref{['eq:jump-rates-degenerate']}. This explains why the 3 outgoing $q^{(\alpha)}(v\to v_k)$ jump-rates from the degenerated vertex $v$ are first multiplied by this limiting ratio then used the non-degenerated geometric formula. The invariant measure $\mu^{(\alpha)}(v)$ of the degenerate vertex $v$ is the sum of the areas of the blacks faces surrounding $c^\circ$.
• Figure 4: (Left) The double cover $\Upsilon^\times$. (Right) The double cover $\Upsilon^\times_{(a)}$. The double covers can be identified with each other away from $a$. The corner $a^+$ is chosen so that the two double-covers have the same branching structure around the quad $z_a^+$. This figure is similar to Cim-universality.
• Figure 5: Integration trick to identify the full-plane fermion. The contour $\Gamma^\delta$ is shown in pink and can be reduced to an integral over $a^\circ$. Since $F^{\delta}_{(a)}$ is not $s$-holomorphic at $a$, the mismatch between $\mathrm{Proj}[F^{\delta}_{(a)}(z_a^+); \eta_a \mathbb{R}]$ and $F^{\bullet}_{(a)}(a^\bullet) = \mathrm{Proj}[F^{\delta}_{(a)}(z_a^-); \eta_a \mathbb{R}]$ allows one to compute the residue.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.1: Theorem 1.2 in Mah23
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3: Assumption Unif($\delta$)
• Definition 2.4