Discrete Lorentz surfaces and s-embeddings II: maximal surfaces

TL;DR

This paper demonstrates that certain discrete s-embeddings can be lifted to maximal surfaces at the discrete level, providing new insights into the geometric structure of models related to the Ising model.

Contribution

It identifies a class of isothermic s-embeddings that lift to discrete S-maximal surfaces, answering a key open question and extending the geometric framework of s-embeddings.

Findings

Discrete s-embeddings can lift to maximal surfaces before taking limits.

A special class of isothermic s-embeddings corresponds to discrete S-maximal surfaces.

The associated family of s-embeddings maintains constant Ising weights.

Abstract

S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. We answer this question in the positive. In a previous paper we identified a subclass of s-embeddings--isothermic s-embeddings--that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall as a discretization of isothermic surfaces. In this paper we identify a special class of isothermic s-embeddings that correspond to discrete S-maximal surfaces, translating an approach of Bobenko, Hoffmann and Springborn introduced for discrete S-minimal surfaces in Euclidean space. Additionally, each S-maximal surface comes with a 1-parameter family of associated surfaces that are isometric. This enables us to obtain an associated family of s-embeddings for each maximal s-embedding. We show that the Ising weights are constant in the associated family.