Conformal Invariance of the FK-Ising Model on Lorentz-Maximal S-Embeddings

TL;DR

This paper proves that the interface curves of the critical FK-Ising model on non-flat s-embeddings converge to an SLE_{16/3} curve, extending discrete complex analysis techniques to new geometric settings.

Contribution

It demonstrates conformal invariance of the FK-Ising model on non-flat s-embeddings using advanced discrete complex analysis methods.

Findings

Convergence of FK-Ising interfaces to SLE_{16/3} on non-flat s-embeddings.

Extension of discrete complex analysis techniques to Minkowski space embeddings.

Identification of a stronger criticality condition via maximal surfaces in Minkowski space.

Abstract

We show on non-flat but critical s-embeddings the celebrated convergence of the interface curves of the critical FK Ising model to an $\operatorname{SLE}_{16/3}$ curve, using discrete complex analytic techniques first used in arXiv:0708.0039, arXiv:1312.0533 and subsequently extended to more lattice settings including isoradial graphs arXiv:0910.2045, circle packings arXiv:1712.08736, and flat s-embeddings arXiv:2006.14559. In our setting, the s-embedding approximates a maximal surface in the Minkowski space $\mathbb R^{2,1}$, an `exact' criticality condition identified in arXiv:2006.14559, which is stronger than the percolation-theoretic `near-critical' setup studied in, e.g., arXiv:2309.08470. The proof relies on a careful discretisation of the Laplace-Beltrami operator on the s-embedding, which is crucial in identifying the limit of the martingale observable.