The Archimedean limit of random sorting networks

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