The Archimedean limit of random sorting networks

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.