Dimers on a Riemann surface and compactified free field

Mikhail Basok
University of Helsinki

In this talk I will be speaking about the dimer model sampled on a general Riemann surface. In this setup, the dimer height function becomes additively multivalued with a random monodromy. Given a sequence of graphs approximating the conformal structure of the surface in a suitable way, the underlying sequence of height functions is expected to converge to the compactified free field on the surface. Recently, this problem was addressed by Berestycki, Laslier and Ray in the case of Temperley graphs. Using various probabilistic methods, they obtained the following universal result: given that a sequence of graphs satisfies certain set of probabilistic conditions (which link it with the conformal structure of the surface), the limit of height functions exists, is conformally invariant and does not depend on a particular sequence of graphs. However, the identification of the limit with the compactified free field was missing in this result. In my recent work I fill this gap by studying the same problem from the perspective of discrete complex analysis. For this purpose, I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain local geometric conditions. In this setup I obtain an analytic description of the limit which allows to identify it with a suitable version of the compactified free field; I also prove the convergence in some non-Temperlian cases when the surface is generic. A core part of this approach is the regularity theory on t-embeddings recently developed by Chelkak, Laslier and Russkikh.

Presentation (PDF File)

Back to Workshop I: Statistical Mechanics and Discrete Geometry