论文标题
随机覆盖表面的渐近统计
The Asymptotic Statistics of Random Covering Surfaces
论文作者
论文摘要
令$γ_{g} $为$ g \ geq2 $的封闭连接方向表面的基本组。我们开发了一种在表示空间$ \ mathbb {x} _ {g，n} = \ mathrm {hom}（γ_{g}，s_ {n}）$的新方法，其中$ s_ {n} $是$ \ \ {1，\ ldots，n \} $的对称性组。同等地，这是所有顶点标记的，$ n $表的空间，覆盖属于$ g $的封闭表面的空间。 给定的$ ϕ \ in \ mathbb {x} _ {g，n} $和$γ\inγ_{g} $，我们让$ \ mathsf {fix {fix}_γ（ϕ）$是置换率$ ϕ（γ）$的固定点的数量。函数$ \ MATHSF {fix}_γ$是$ \ mathbb {x} _ {g，n} $称为Wilson Loops上的自然函数家族的特殊情况。我们的新方法导致渐近公式为$ n \ to \ infty $，对于$ \ mathsf {fix}_γ$的期望，就$ \ mathbb {x} _ {g，n} $的统一概率度量而言$ \ mathbb {e} _ {g，n} [\ mathsf {fix}_γ] $。我们证明，如果$γ\inγ_{g} $不是身份，而$ q $是最大的，使得$γ$是$ q $ t的$ q $ th power in $γ_{g} $，那么\ [\ m athbb {e} _ {g，n} $ n \ to \ infty $，其中$ d \ left（q \右）$是$ q $的除数的数量。即使是$ \ mathbb {e} _ {g，n} [\ mathsf {fix}_γ] = o（n）$作为$ n \ to \ infty $的弱推论。我们还证明，如果$γ$不是身份，那么$ \ mathbb {e} _ {g，n} [\ mathsf {fix}_γ] $可以通过$ n^{ - 1} $中的多项式。
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 $n^{-1}$.
