Abstract:A sorting network (also known as a reduced decomposition of the reverse permutation) is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma ^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $\sigma ^n$. As a corollary of these results, we show that if $S_n$ is embedded into $\mathbb {R}^n$ via the map $\tau \mapsto (\tau (1), \tau (2), \dots \tau (n))$, then with high probability, the path $\sigma ^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere in $\mathbb {R}^n$. These results prove conjectures of Angel, Holroyd, Romik, and Virág.

---

中文翻译：

---

随机排序网络的阿基米德极限

摘要：排序网络（也称为反向排列的约简分解）是相邻转置生成的对称群$S_n$的Cayley图中从$12\cdots n$到$n\cdots 21$的最短路径。我们证明了在均匀随机$n$-元素排序网络$\sigma ^n$中，所有粒子轨迹都以高概率接近正弦曲线。我们还发现了 $\sigma ^n$ 的时间-$t$ 置换矩阵度量的弱极限。作为这些结果的推论，我们证明如果 $S_n$ 通过映射 $\tau \mapsto (\tau (1), \tau (2), \dots \ tau (n))$，则路径 $\sigma ^n$ 很可能接近 $\mathbb {R}^n$ 中特定 $(n-2)$ 维球体上的大圆。这些结果证明了 Angel、Holroyd、Romik 和 Virág 的猜想。

---