Abstract
The dimer model represents the distribution of di-atomic molecules on the surface of a crystal. This is modeled by perfect couplings of a planar graph chosen according to the Boltzmann measure. When the graph is periodic, Kenyon, Okounkov and Sheffield show that the phase diagram is given by the spectral curve, which has the remarkable property of being Harnack ; they also establish a correspondence between these curves and dimer models on bipartite graphs. Another important result is the local expression obtained by Kenyon for correlations when the underlying graph is isoradial and the model is critical. In a series of works in collaboration with Cédric Boutillier (Sorbonne University) and David Cimasoni (University of Geneva), we generalize these results in a unified framework. We consider the model on minimal graphs (a family slightly larger than isoradials) and establish an explicit correspondence with Harnack curves; we also obtain a local formula for the correlations for the two-parameter family of Gibbs measures.
