From the planar Ising model to quasiconformal mappings

TL;DR

This paper establishes the scaling limits of full-plane Kadanoff-Ceva fermions in the Ising model on generic embeddings, linking them to solutions of conjugate Beltrami equations and revealing a connection to quasiconformal mappings and complex analysis.

Contribution

It introduces a broad framework for understanding the scaling limits of the Ising model on non-degenerate s-embeddings, connecting them to conjugate Beltrami equations and quasiconformal mappings.

Findings

Scaling limits described by conjugate Beltrami equations with singularities.

Conformal covariance of energy density in critical doubly periodic graphs.

Scaling factors are local and depend on the embedding geometry.

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.