Isoradial immersions

TL;DR

This paper extends the concept of isoradial embeddings to immersions and minimal immersions, providing combinatorial characterizations and describing the space of such embeddings, with applications to the dimer model.

Contribution

It introduces new classes of embeddings—isoradial and minimal immersions—and characterizes when planar graphs admit them, expanding the understanding of their structure and applications.

Findings

A planar graph admits a flat isoradial immersion iff its train-tracks do not form closed loops.

A bipartite graph has a minimal immersion iff it is minimal.

The paper describes the space of such immersions and applies results to the dimer model.

Abstract

Isoradial embeddings of planar graphs play a crucial role in the study of several models of statistical mechanics, such as the Ising and dimer models. Kenyon and Schlenker give a combinatorial characterization of planar graphs admitting an isoradial embedding, and describe the space of such embeddings. In this paper we prove two results of the same type for generalizations of isoradial embeddings: isoradial immersions and minimal immersions. We show that a planar graph admits a flat isoradial immersion if and only if its train-tracks do not form closed loops, and that a bipartite graph has a minimal immersion if and only if it is minimal. In both cases we describe the space of such immersions. The techniques used are different in both settings, and distinct from those of Kenyon and Schlenker. We also give an application of our results to the dimer model defined on bipartite graphs admitting minimal immersions.