在由随机胶合在一起的多边形圆盘形成的表面上
On a surface formed by randomly gluing together polygonal discs
摘要：
从一组$n$定向的多边形圆盘开始，总共有偶数个$N$边，我们通过随机匹配圆盘的边并将它们适当地粘合在一起来生成随机定向的表面。将曲面编码为$[N]$的随机置换$γ$，利用$S_N$上的傅里叶变换，证明了$γ$渐近于交替群$A_N$（$A_N^c$ resp.）如果$N-n$和$N/2$相同（分别相反），平价。我们用它来证明一个局部中心极限定理的顶点数的表面上，从而为它的欧拉特征$χ$。我们还表明，具有高概率的随机表面由一个单一的组件，因此有一个定义良好的属$g=1-χ/2$，这是渐近的高斯随机变量，平均值$(N/2-n-\log N)/2$和方差$(\log N)/2$。
英文摘要：
Starting with a collection of $n$ oriented polygonal discs, with an even number $N$ of sides in total, we generate a random oriented surface by randomly matching the sides of discs and properly gluing them together. Encoding the surface in a random permutation $γ$ of $[N]$, we use the Fourier transform on $S_N$ to show that $γ$ is asymptotic to the permutation distributed uniformly on the alternating group $A_N$ ($A_N^c$ resp.) if $N-n$ and $N/2$ are of the same (opposite resp.) parity. We use this to prove a local central limit theorem for the number of vertices on the surface, whence for its Euler characteristic $χ$. We also show that with high probability the random surface consists of a single component, and thus has a well-defined genus $g=1-χ/2$, which is asymptotic to a Gaussian random variable, with mean $(N/2-n-\log N)/2$ and variance $(\log N)/2$.
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