Dimers and M-curves: Limit shapes from Riemann surfaces

We present a general approach for the study of dimer model limit shape problems via variational and integrable systems techniques. In particular we deduce the limit shape of the Aztec diamond and the hexagon for quasiperiodic weights through purely variational techniques. Putting an M-curve at the center of the construction allows one to define weights and algebro-geometric structures describing the behavior of the corresponding dimer model. We extend the quasiperiodic setup of our recent paper with Suris to include a diffeomorphism from the spectral data to the liquid region of the dimer. Our novel method of proof is purely variational and exploits a duality between the dimer height function and its dual magnetic tension minimizer and applies to dimers with gas regions. We apply this to the Aztec diamond and hexagon domains to obtain explicit expressions for the complex structure of the liquid region of the dimer as well as the height function and its dual. Finally, we compute the weights and the limit shapes numerically using the Schottky uniformization technique. Simulations and theoretical results match completely.