A shifted tableau of staircase shape has row lengths n,n-1,...,2,1 adjusted on the right side and numbers increasing along rows and columns. Let the number in a box represent the height of a point above that box, then we have proved that the points for a uniformly chosen random shifted staircase SYT in the limit converge to a certain surface in three dimensions.  I will present this result and also how this implies, via properties of the Edelman–Greene bijection, results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations.
(Based on joint work with Samu Potka and Robin Sulzgruber.)