Circle patterns and critical Ising models

TL;DR

This paper introduces a new family of Ising models on planar graphs with duals that are circle patterns, proving their magnetic criticality and extending the critical isoradial models of Baxter.

Contribution

It defines and proves the magnetic criticality of a novel class of Ising models related to circle patterns, generalizing Baxter's critical isoradial models.

Findings

Established magnetic criticality for the new Ising models

Extended the class of models including Baxter's critical isoradial models

Connected circle patterns with phase transition properties in Ising models

Abstract

A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a new family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.