# Random sorting networks: local statistics via random matrix laws

## TL;DR

This paper establishes the local statistical behavior of random sorting networks using random matrix theory, revealing universal laws governing swap timings and gaps through determinantal processes and eigenvalue distributions.

## Contribution

It introduces a local limit law for the swap process in random sorting networks, connecting it to eigenvalue distributions of the antisymmetric GUE and universal random matrix gap laws.

## Key findings

- The local limit of the swap process is described by a determinantal point process.
- The first swap time distribution matches the closest eigenvalue distribution in antisymmetric GUE.
- Bulk gaps between swaps follow the Gaudin-Mehta law, a universal eigenvalue gap distribution.

## 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.