论文标题
自由产品的随机排列的本地统计数据
Local Statistics of Random Permutations from Free Products
论文作者
论文摘要
令$α$和$β$分别为$ 2 $和$ 3 $的订单均匀随机排列，分别为$ s_ {n} $，然后考虑置换$αβαβ^{ - 1} $。这个随机排列平均有多少个固定点？当前的论文研究此类问题，将它们与自由群体中的元素的拓扑和代数不变性有关。 正式地，令$γ= g_ {1}*\ ldots*g_ {k} $是$ g_1，\ ldots，g_k $的组的免费产物，既有限，有限生成免费，或者是可定向的双曲线表面组。对于固定元素$γ\inγ$，对称组$ s_ {n} $中的$γ$ - 随机排列是$γ$的图像，通过均匀的随机同构$γ\ to s_ {n} $。在本文中，我们研究了$γ$ - 随机排列的本地统计数据及其渐近学的生长。我们首先考虑$ \ mathbb {e} \ left [\ mathrm {fix}_γ\ left（n \ right）\ right] $，$ s_ {n} $中的$γ$ - random presce in the $γ$ - random press中的预期固定点。我们表明，除非$γ$具有有限订单，否则$ \ mathbb {e} \ weft [\ mathrm {fix {fix}_γ\ left（n \ right）\ right] $ as $ n \ to \ infty $是一个无聊的$，是一个无聊的$，并且等于$ h \leγ$ con的数量$ h \ cong c_ {2}*c_ {2} $。同等地，这是包含$γ$的子组$ h \leγ$的数量，并具有（理性的）欧拉特征零。我们还证明，$ \ mathbb {e} \ left [\ mathrm {fix}_γ\ left（n \ right）\ right] $的渐近扩展，并确定固定点的限制分布为$ n \ to \ to \ infty $。然后将这些结果推广到所有固定长度周期的统计数据。
Let $α$ and $β$ be uniformly random permutations of orders $2$ and $3$, respectively, in $S_{N}$, and consider, say, the permutation $αβαβ^{-1}$. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let $Γ=G_{1}*\ldots*G_{k}$ be a free product of groups where each of $G_1,\ldots,G_k$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $γ\inΓ$, a $γ$-random permutation in the symmetric group $S_{N}$ is the image of $γ$ through a uniformly random homomorphism $Γ\to S_{N}$. In this paper we study local statistics of $γ$-random permutations and their asymptotics as $N$ grows. We first consider $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$, the expected number of fixed points in a $γ$-random permutation in $S_{N}$. We show that unless $γ$ has finite order, the limit of $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$ as $N\to\infty$ is an integer, and is equal to the number of subgroups $H\leΓ$ containing $γ$ such that $H\cong\mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\leΓ$ containing $γ$ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\mathbb{E}\left[\mathrm{fix}_γ\left(N\right)\right]$ and determine the limit distribution of the number of fixed points as $N\to\infty$. These results are then generalized to all statistics of cycles of fixed lengths.
