Universality of Ising spin correlations on critical doubly-periodic graphs

TL;DR

The paper proves conformal invariance of Ising spin correlations on critical doubly-periodic graphs, showing that their scaling limit matches the critical square lattice and thereby completing the universality picture for periodic lattices. The authors develop a hybrid approach that combines soft discrete complex analysis on \\(s\\)-embeddings with FK random-cluster techniques to overcome the lack of integrability in the periodic setting, including a new boundary treatment that yields a universal half-plane one-arm exponent of \\(1/2\\). A key step is establishing uniform crossing (RSW) bounds and then leveraging CLE$(16/3)$ convergence of FK clusters to translate cluster behavior into spin correlations, with a global scaling factor \\rho(\\delta)=\\delta^{-1/8+o(1)}$. The resulting convergence of spin-correlation ratios to their square-lattice values, together with conformal covariance of the continuous correlation function, provides a rigorous universal description of critical Ising spin correlations on doubly-periodic graphs and strengthens the link between discrete models and their continuum limits.

Abstract

We establish conformal invariance of Ising spin correlations on critical doubly periodic graphs, showing that their scaling limit coincides with that of the critical square lattice, as originally proved by Chelkak, Hongler and Izyurov. To overcome the absence of integrability and quantitative full plane constructions in the periodic setting, we combine discrete analytic tools with random cluster methods. This result completes the universality picture for periodic lattices, whose criticality condition was identified by Cimasoni and Duminil-Copin and whose conformal structure and interface convergence were obtained by Chelkak.

Figures (5)

• Figure 1: (Left) Notation for a quad $z \in \diamondsuit(G)$ with an arbitrary planar embedding. Vertices of the primal graph $G^\bullet$ are indicated as black dots, and vertices of the dual graph $G^\circ$, corresponding to faces of $G$, are shown as white dots. The so-called corners, corresponding to edges of the bipartite graph $\Lambda(G) = G^\bullet \cup G^\circ$, are represented as triangles. This illustration shows a part of the double cover of the corner graph branching around $z$. Corners that are neighbours within this double cover are connected by dashed lines. (Right) A section of the associated $s$-embedding containing the quad $\mathcal{S}^{\diamondsuit}(z)$, tangent to a circle of radius $r_z$ centreed at $\mathcal{S}(z)$. The Ising weight of the edge between $v_0^\bullet$ and $v_1^\bullet$ can be determined from the four angles $\phi_{v,z}$ associated with $\mathcal{S}^{\diamondsuit}(z)$, using the formula in \ref{['eq:theta-from-S']}.
• Figure 2: Left: The wired arc $(b^\delta a^\delta)^\circ$ is drawn in blue while the free arc $(a^\delta b^\delta)^\bullet$ is drawn in red. Right: Boundary values of $H_{X^\delta}$ and $H_{\textrm{comp}}$ used for the comparison principle.
• Figure 3: First line: (Left) Arm event up the boundary of $\Lambda^\delta_R$ with alternating wired boundary conditions on top half of the square and free boundary conditions on the bottom half of the square. (Middle) Arm event up the boundary of $\Lambda^\delta_R$ with alternating free boundary conditions on the bottom segment and wired boundary conditions on the rest of $\partial \Lambda^\delta_R$. (Right) Arm event up the boundary of $\Lambda^\delta_{R/2}$ with alternating free boundary conditions on the bottom segment and wired boundary conditions on the rest of $\partial \Lambda^\delta_R$. Second line: (Left) Arm event up the boundary of $\Lambda^\delta_{R/2}$ with alternating free boundary conditions on the bottom segment and wired boundary conditions on the rest of $\partial \Lambda^\delta_R$, imposing additionally a dual wired circuit separating $\Lambda^\delta_{R/2}$ and $\Lambda^\delta_R$. (Middle) Arm event up the boundary of $\Lambda^\delta_{R/2}$ with an additional a dual wired circuit separating $\Lambda^\delta_{R/2}$ and $\Lambda^\delta_R$. (Right) Arm event up the boundary of $\Lambda^\delta_{R/2}$ in the half-plane with free boundary conditions.
• Figure 4: TOP (Left): Slicing procedure creating some aligned boundary at level $y=0$. (Right): Extension using triangles viewed as tangential quadrilaterals. BOTTOM (Left): Boundary half-quad kite. (Right) $(m,\ell)$-corner with the its kite extension.
• Figure 5: (Left) Notations used within the proof. (Right) Configuration where $\textrm{Uni}_{\varepsilon}(a^\delta,b^\delta)$ inducing two four-arm events (centred respectively at $a^\delta$ and $b^\delta$) between the distances $\sqrt{\varepsilon}$ and $r$. This events are polynomially more unlikely compared with one-arm events at the same scales.

Theorems & Definitions (32)

• Theorem 1.1: Cimasoni-- Duminil-Copincimasoni-duminil
• Lemma 1.2: Lemma 2.3 in Che20
• Theorem 1.3: Chelkak Che20
• Theorem 1.4: Chelkak Che20
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3: Assumption Unif($\delta$)
• Definition 2.4