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Isoradial immersions

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1912.10297.pdf (715.5Kb)
Date
2019
Publisher city
Paris
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-02423791
Dewey
Principes généraux des mathématiques
Sujet
isoradial graphs; dimers; planar graphs; graphes planaires; graphes isoradiaux; dimères
URI
https://basepub.dauphine.fr/handle/123456789/20684
Collections
  • CEREMADE : Publications
Metadata
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Author
Boutillier, Cédric
Cimasoni, David
de Tilière, Béatrice
Type
Document de travail / Working paper
Item number of pages
44
Abstract (EN)
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 has 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. We also give an application of our result to the bipartite dimer model defined on graphs admitting minimal immersions.

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