周期边界条件下三角晶格上空穴关联的标度极限
The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions
摘要：
我们定义了周期性边界条件下三角形格子上空穴的相关性，并研究了当空穴之间的距离增长到无穷大时其渐近性。我们证明了一个任意集合的格子三角形孔的偶数侧的联合相关满足，为大的孔之间的分离，库仑定律和叠加原理，完美地平行的法律的二维静电，与物理电荷对应的孔，和它们的大小之间的差异，在每个孔的右指向和左指向单位三角形的数量。我们详细说明这一平行表明，作为我们的结果的结果，相对概率找到一个固定的收集孔在给定的相互距离（当采样均匀随机在所有单位菱形平铺的补孔）的方法，为大间距之间的孔，相对概率找到相应的二维物理系统的收费在给定的相互距离。物理温度对应于细化背景三角晶格的参数。我们给出了一个等价的措辞，我们的结果的覆盖面给定的完整性。从这个角度来看，二维静电通过对覆盖表面的所有可能的离散几何形状进行平均而产生。
英文摘要：
We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of lattice-triangular holes of even sides satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of right-pointing and left-pointing unit triangles in each hole. We detail this parallel by indicating that, as a consequence of our result, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approaches, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. We give an equivalent phrasing of our result in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatics arises by averaging over all possible discrete geometries of the covering surfaces.
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