发表机构
University of Philadelphia, PA（费城大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对3d边单连通多边形区域的均匀随机菱形铺砌，证实其中心化高度函数涨落收敛于高斯自由场，引入铺砌作用函数并构造逆Kasteleyn矩阵近似，验证了Kenyon与Okounkov2007年的预测。
AI 中文摘要
我们针对具有3d条边、方向循环于三个晶格方向的单连通多边形区域上的均匀随机菱形铺砌，建立了高斯自由场涨落理论。更确切地说，在假设液体区域连通且边界数据不会强制任何内点处高度的前提下，我们证明了中心化高度函数的涨落在液体区域中收敛于高斯自由场，从而证实了Kenyon与Okounkov在2007年提出的预测。我们引入了一种铺砌作用函数，该函数通过其临界点编码极限形状的几何结构：在液体区域中具有一对复共轭临界点，在北极边界上具有重实临界点，在冻结区域中具有不同的实临界点。利用该铺砌作用函数，我们通过显式的单轮廓与双轮廓积分构造了逆Kasteleyn矩阵的近似，并证明该近似在整个多边形区域上是一致的，随后通过标准核计算得出其收敛于高斯自由场的结论。
英文摘要
We establish Gaussian free field fluctuations for uniformly random lozenge tilings of simply connected polygonal domains with $3d$ sides whose directions cycle through the three lattice directions. More precisely, assuming that the liquid region is connected and that the boundary data do not force the height at any interior point, we prove that the fluctuations of centered height function converge to the Gaussian free field in the liquid region, confirming a prediction of Kenyon and Okounkov from 2007. We introduce a tiling action function that encodes the geometry of the limit shape through its critical points. The action function has a complex conjugate pair of critical points in the liquid region, repeated real critical points on the arctic boundary, and distinct real critical points in the frozen region. Using this tiling action function, we construct an approximation to the inverse Kasteleyn matrix in terms of explicit single-contour and double-contour integrals and prove that the approximation is uniform throughout the polygonal domain. The convergence to the Gaussian free field then follows from standard kernel computations.
Comments279 pages, 81 figures
