Histogram of swap locations for a uniformly random 2000-element sorting network:

A uniformly random 2000-element sorting network (selected trajectories shown):

The halfway permutation of a uniform random sorting network:

Trajectories of a half-way sliding window modulo uniform rotation for a 10000-element network (click for larger version)

A 2000-element random sorting network, uniform swap model:

The rhombic tiling corresponding to a uniform random sorting network:

The permutohedron for S

The permutohedron for S