随机覆盖曲面的渐近统计
The Asymptotic Statistics of Random Covering Surfaces
AI总结：
本文针对亏格≥2的闭曲面基本群到对称群的表示空间，提出新的积分方法，得到Wilson环特例的不动点计数期望的渐近公式与任意阶多项式逼近结果，推进了随机覆盖曲面的渐近统计研究。
摘要：
设$Γ_{g}$为亏格$g\geq2$的闭连通可定向曲面的基本群。我们提出了一种在表示空间$\mathbb{X}_{g,n}=\mathrm{Hom}(Γ_{g},S_{n})$上积分的新方法，其中$S_{n}$是集合$\{1,\ldots,n\}$上的置换对称群。等价地，该空间是亏格为$g$的闭曲面的所有顶点标记的$n$叶覆盖空间构成的空间。\n给定$ϕ\in\mathbb{X}_{g,n}$和$γ\inΓ_{g}$，我们定义$\mathsf{fix}_γ(ϕ)$为置换$ϕ(γ)$的不动点个数。函数$\mathsf{fix}_γ$是$\mathbb{X}_{g,n}$上一类被称为Wilson环（威尔逊环）的自然函数族的特例。我们的新方法给出了当$n\to\infty$时，$\mathsf{fix}_γ$关于$\mathbb{X}_{g,n}$上均匀概率测度的期望（记为$\mathbb{E}_{g,n}[\mathsf{fix}_γ]$）的渐近公式。我们证明：若$γ\inΓ_{g}$不是单位元，且$q$是满足$γ$为$Γ_{g}$中某个元素的$q$次幂的最大整数，则当$n\to\infty$时，$\mathbb{E}_{g,n}[\mathsf{fix}_γ]=d(q)+O(n^{-1})$，其中$d(q)$是$q$的约数个数。即便较弱的推论“当$n\to\infty$时$\mathbb{E}_{g,n}[\mathsf{fix}_γ]=o(n)$”也是本文的新结果。我们还证明：若$γ$不是单位元，则$\mathbb{E}_{g,n}[\mathsf{fix}_γ]$可以被$n^{-1}$的多项式逼近到任意阶$O(n^{-M})$。
英文摘要：
Let $Γ_{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We develop a new method for integrating over the representation space $\mathbb{X}_{g,n}=\mathrm{Hom}(Γ_{g},S_{n})$ where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots,n\}$. Equivalently, this is the space of all vertex-labeled, $n$-sheeted covering spaces of the the closed surface of genus $g$. Given $ϕ\in\mathbb{X}_{g,n}$ and $γ\inΓ_{g}$, we let $\mathsf{fix}_γ(ϕ)$ be the number of fixed points of the permutation $ϕ(γ)$. The function $\mathsf{fix}_γ$ is a special case of a natural family of functions on $\mathbb{X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to\infty$, for the expectation of $\mathsf{fix}_γ$ with respect to the uniform probability measure on $\mathbb{X}_{g,n}$, which is denoted by $\mathbb{E}_{g,n}[\mathsf{fix}_γ]$. We prove that if $γ\inΓ_{g}$ is not the identity, and $q$ is maximal such that $γ$ is a $q$th power in $Γ_{g}$, then \[ \mathbb{E}_{g,n}[\mathsf{fix}_γ]=d(q)+O(n^{-1}) \] as $n\to\infty$, where $d\left(q\right)$ is the number of divisors of $q$. Even the weaker corollary that $\mathbb{E}_{g,n}[\mathsf{fix}_γ]=o(n)$ as $n\to\infty$ is a new result of this paper. We also prove that if $γ$ is not the identity then $\mathbb{E}_{g,n}[\mathsf{fix}_γ]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
