Isoradial immersions

Cédric Boutillier (1, 2) , David Cimasoni (3) , Béatrice de Tilière (4, 2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 IUF - Institut universitaire de France
3 Section de mathématiques [Genève]
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: isoradial graphs; dimers; planar graphs

Mots clés: graphes planaires; graphes isoradiaux; dimères

Domaines: Combinatoire [math.CO]

Cédric Boutillier, David Cimasoni, Béatrice de Tilière. Isoradial immersions. Journal of Graph Theory, In press, 99 (4), pp.715-757. ⟨10.1002/jgt.22761⟩. ⟨hal-02423791⟩