The global limit of random sorting networks

A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of \(S_n\) generated by adjacent transpositions. An n-element uniform random sorting network displays many striking global properties as n approaches infinity. For example, scaled trajectories of the elements \(1, 2, ... n\) converge to sine curves and the \(1/2\)-way permutation matrix measure converges to the projected surface area measure of the 2-sphere.

In this talk, I will discuss how the local structure of random sorting networks can be used to find a global limit, proving these statements and more.