Dimer model and holomorphic functions on t-embeddings of planar graphs

Dmitry Chelkak (1) , Benoît Laslier (2, 3) , Marianna Russkikh (4)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 ICJ - Institut Camille Jordan
3 PSPM - Probabilités, statistique, physique mathématique
4 UNIGE - Université de Genève = University of Geneva

We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent work arXiv:1810.05616. We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.

Domaines: Probabilités [math.PR]; Variables complexes [math.CV]; Physique mathématique [math-ph]

Dmitry Chelkak, Benoît Laslier, Marianna Russkikh. Dimer model and holomorphic functions on t-embeddings of planar graphs. Proceedings of the London Mathematical Society, In press, ⟨10.1112/plms.12516⟩. ⟨hal-02613085⟩