UVictoria Dynamics and Probability Seminar: Misha Basok

In this talk I will be speaking about the dimer model sampled on a general Riemann surface. Dimer model on a graph consists of sampling a random perfect matching with the probability proportional to the product of edge weights. In the case when the graph is planar, perfect matchings are in correspondence with their height functions defined on faces of the graph. Given a sequence of graphs approximating a given planar domain in a small mesh size, one can ask whether the underlying sequence of dimer height functions has a scaling limit and how to describe it. The landmark result of Kenyon asserts that in the case of a simply-connected domain approximated by Temperley polygons on the square grid the fluctuations of the height functions around their means converge to the (properly normalized) Gaussian free field in the target domain. The same result in the case of general Temperley graphs was established by Berestycki, Laslier and Ray 15 years later.

The dimer height function can still be defined when the graph is not planar, but is embedded into a general Riemann surface. In this case the height function becomes additively multivalued and is expected to converge to the compactified free field on the surface in the scaling limit. Recently, this problem was studied by Berestycki, Laslier and Ray in the case of general Temperley graphs embedded into the Riemann surface. Using soft probabilistic methods they proved the scaling limit exists, is conformally invariant and does not depend on a particular sequence of graphs. However, its identification with the compactified free field was missing. My goal is to fill this gap by studying the same problem from the perspective of discrete complex analysis. For this I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain geometric assumptions with respect to the local Euclidean structure. Using various analytic methods (both discrete and continuous counterparts are non-trivial) I obtain convergence to the compactified free field when the Riemann surface is chosen generically. Moreover, I am able to prove that the tightness of the underlying height fluctuations always implies their convergence to the compactified free field.