From the planar Ising model to quasiconformal mappings

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 connecting (near-)critical planar Ising models to Green kernels of uniformly elliptic operators 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 the convergence of the energy-energy correlations for smooth limiting structures $(z,\vartheta)$ viewed as spacelike surfaces the Minkowski space $\mathbb{R}^{(2,1)}$. This includes explicit formulae when $(z,\vartheta)$ is maximal, generalizing conformal covariance to new setups involving an additional change of metric. 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 in a substantially richer conformal structure than the classical Euclidean one.