The dimer model is a model of perfect matching whose popularity stems from the fact that it is exactly solvable. It is believed that the large-scale fluctuations of the height function of the dimer model is universal in a certain sense and should not depend on the microscopic properties of the graph. It turns out that in this level of generality, the well-established methods using Kasteleyn matrices become intractable.
We propose a new method for examining the fluctuation of the height function which enables us to obtain a universality result for general graphs with various boundary conditions and even when the underlying surface is a Riemann surface. This provides a new proof of some old results and solves several open questions. Our methods use exact solvability in a weak sense and use some new results in the continuum instead which enables us to get universal results.
Ongoing joint work with Nathanael Berestycki and Benoit Laslier.