Random sorting networks: local statistics via random matrix laws

Abstract

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.