Abstract
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 ϵ → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape Σ0. 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 Σ0 has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about Σ0 converge as ϵ → 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 Σ0 at x.
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Communicated by H. Spohn
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Kenyon, R. Height Fluctuations in the Honeycomb Dimer Model. Commun. Math. Phys. 281, 675–709 (2008). https://doi.org/10.1007/s00220-008-0511-8
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DOI: https://doi.org/10.1007/s00220-008-0511-8