  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 Virag.
