论文标题
随机方形表面的拓扑和几何形状
The topology and geometry of random square-tiled surfaces
论文作者
论文摘要
方形表面（STS）是标准正方形圆环的分支盖，恰好在一个点上分支。在本文中，我们考虑了STSS的随机模型和对其他简单翻译表面的分支封面的概括，我们称之为Polygon平铺表面。我们获得了该属的局部中央限制定理，随后获得该属的分布在渐近上正常。我们还研究了随机sts上的载体载体（圆锥点之间的欧几里得位移向量）。我们表明，几乎可以肯定的是，随机sts的单位矢量集包含$ \ mathbb {z}^2 $的原始向量集，并且概率接近$ 1/e $，这些集合相等。
A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. In this paper we consider a randomizing model for STSs and generalizations to branched covers of other simple translation surfaces which we call polygon-tiled surfaces. We obtain a local central limit theorem for the genus and subsequently obtain that the distribution of the genus is asymptotically normal. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. We show that asymptotically almost surely the set of holonomy vectors of a random STS contains the set of primitive vectors of $\mathbb{Z}^2$ and with probability approaching $1/e$, these sets are equal.
