Abstract:

One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits “nice” discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and introduce the definition of discrete holomorphicity on such an embedding. We discuss a concept of `perfect t-embeddings’ of weighted bipartite planar graphs. We believe that these embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model.

Based on:

“Dimers and Circles” joint with R. Kenyon, W. Lam, S. Ramassamy;

“Dimer model and holomorphic functions on t-embeddings of planar graphs” joint with D. Chelkak, B. Laslier.