The Archimedean limit of random sorting networks

Abstract

A sorting network (also known as a reduced decomposition of the reverse permutation) is a shortest path from12⋯n12 \cdots nton⋯21n \cdots 21in the Cayley graph of the symmetric groupSnS_ngenerated by adjacent transpositions. We prove that in a uniform randomnn-element sorting networkσn\sigma ^n, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-ttpermutation matrix measures ofσn\sigma ^n. As a corollary of these results, we show that ifSnS_nis embedded intoRn\mathbb {R}^nvia the mapτ↦(τ(1),τ(2),…τ(n))\tau \mapsto (\tau (1), \tau (2), \dots \tau (n)), then with high probability, the pathσn\sigma ^nis close to a great circle on a particular(n−2)(n-2)-dimensional sphere inRn\mathbb {R}^n. These results prove conjectures of Angel, Holroyd, Romik, and Virág.