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 circle pattern embedding of a dimer planar graph using its Kasteleyn weights. This embedding is the generalization of the isoradial embedding and it is closely related to the T-graph embedding.
“Dimers and Circles” joint with R. Kenyon, W. Lam, S. Ramassamy;
“Holomorphic functions on t-embeddings of planar graphs” joint with D. Chelkak, B. Laslier.