Isoradial immersions
| Autor: | Boutillier, Cédric, Cimasoni, David, de Tilière, Béatrice |
|---|---|
| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 47 pages, 27 figures. Added in v2: diagram providing a visualisation of main theorems and the result of Kenyon/Schlenker, Corollary 29 proving that $X_G=Y_G$ if $G$ is minimal, $\mathbb{Z}^2$-periodic. Added in v3: Conj 1 is now Thm 31: a graph without train-tracks self-intersection is minimal iff the set of angle maps satisfying Kasteleyn's condition is not empty. v4: accepted version |
| Databáze: | arXiv |
| Externí odkaz: |
|