Height Fluctuations in the Honeycomb Dimer Model
Height Fluctuations in the Honeycomb Dimer Model
复制标题
蜂窝二聚体模型中的高度波动
DOI：
10.1007/s00220-008-0511-8
复制
发表时间：
2004
影响因子：
2.4
通讯作者：
R. Kenyon
中科院分区：
文献类型：
--
作者：
R. Kenyon
我们研究了蜂窝晶格上的二聚体模型中产生的随机表面模型。对于一个固定的“线框”边界条件，当网格间距λ → 0时，Cohn，Kenyon和Propp [3]证明了随机曲面几乎必然收敛到非随机极限形状λ 0。在[12]中，Okounkov和作者展示了如何用解析函数来参数化极限形状，特别是在它们上构造一个自然的共形结构。我们在这里表明，当λ 0没有小平面时，对于一族近似线框的边界条件，关于λ 0的大尺度表面起伏（高度起伏）在λ → 0时收敛到上述共形结构的高斯自由场。我们还证明了在给定点x附近的涨落的局部统计量，如[3]所示，是由唯一的遍历Gibbs测度（在平面构形上）给出的，其斜率是在x处的切平面的斜率。
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.