Dimers and embeddings

We introduce a concept of ‘t-embeddings’ of weighted bipartite planar graphs. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. We also develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon’s holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.