  This paper finds the bulk local limit of the swap process of uniformly random
sorting networks. The limit object is defined through a deterministic
procedure, a local version of the Edelman-Greene algorithm, applied to a two
dimensional determinantal point process with explicit kernel. The latter
describes the asymptotic joint law near $0$ of the eigenvalues of the corners
in the antisymmetric Gaussian Unitary Ensemble. In particular, the limiting law
of the first time a given swap appears in a random sorting network is
identified with the limiting distribution of the closest to $0$ eigenvalue in
the antisymmetric GUE. Moreover, the asymptotic gap, in the bulk, between
appearances of a given swap is the Gaudin-Mehta law -- the limiting universal
distribution for gaps between eigenvalues of real symmetric random matrices.
  The proofs rely on the determinantal structure and a double contour integral
representation for the kernel of random Poissonized Young tableaux of arbitrary
shape.
