Dimers and M-curves: Limit shapes from Riemann surfaces.

Alexander Bobenko
Technische Universität Berlin

We develop a general approach to dimer models analogous to Krichever’s scheme in the theory of integrable systems. This leads to dimer models on doubly periodic bipartite graphs with quasiperiodic positive weights. Dimer models with periodic weights and Harnack curves are recovered as a special case. This generalization from Harnack curves to general M-curves, which are in the focus of our approach, leads to transparent algebro-geometric structures. In particular, the Ronkin function and surface tension are expressed as integrals of meromorphic differentials on M-curves. Using variational descriptions, explicit representations for limit shapes are obtained in terms of Abelian integrals. Also, the relation to discrete conformal mappings and to hyperbolic polyhedra is explained. Based on Schottky uniformization of Riemann surfaces, we compute the weights and dimer configurations. The computational results are in complete agreement with the theoretical predictions. The talk is based on joint works with N. Bobenko and Yu. Suris.

Presentation (PDF File)

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