Height fluctuations in the honeycomb dimer model Richard Kenyon

We study a model of random crystalline surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing ǫ → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape Σ0. We show here that when Σ0 has no facets, for a large family of boundary conditions approximating the wire frame, the large-scale surface fluctations (height fluctuations) about Σ0 converge as ǫ → 0 to a Gaussian free field for an appropriate conformal structure determined by Σ0. We also show that the local statistics of the fluctuations near a given point x are given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of Σ0 at x.