Height fluctuations in the honeycomb dimer model

We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed "wire frame" boundary condition, as the lattice spacing epsilon -> 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape Sigma(0). In [12], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when Sigma(0) has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about Sigma(0) converge as epsilon -> 0 to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point x are, as conjectured in [3], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of Sigma(0) at x.