Free boundary dimers: random walk representation and scaling limit

The dimer model, a classical model of statistical mechanics, is the uniform distribution on perfect matchings of a graph. In two dimensions, one can define an associated height function which turns the model into a random surface (with specified boundary conditions). In the 1960s, Kasteleyn and Temperley/Fisher found an exact "solution" to the model, computing the correlations in terms of a matrix called the Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon (2000) who proved that the centred height function converges to the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first proof of conformal invariance in statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched: this is the so-called monomer-dimer model, or dimer model with free boundary conditions. The main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane is the Neumann (or free boundary conditions) Gaussian free field. Key to this result is a somewhat miraculous random walk representation for the inverse Kasteleyn matrix, which I hope to discuss.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).