  We study shifted standard Young tableaux (SYT). The limiting surface of
uniformly random shifted SYT of staircase shape is determined, with the
integers in the SYT as heights. This implies via properties of the
Edelman-Greene bijection results about random 132-avoiding sorting networks,
including limit shapes for trajectories and intermediate permutations.
Moreover, the expected number of adjacencies in SYT is considered. It is shown
that on average each row and each column of a shifted SYT of staircase shape
contains precisely one adjacency.
