# The Local Limit of Random Sorting Networks

## TL;DR

This paper proves the existence of a local limit process for the positions of transpositions in large random sorting networks, revealing a universal behavior described by a stationary, mixing swap process with specific time-scaling properties.

## Contribution

It introduces a new local limit process for random sorting networks and connects it to staircase-shaped Young tableaux via a bijection, advancing understanding of their asymptotic structure.

## Key findings

- The local limit process $U$ is stationary and mixing.
- Dependence on the parameter $a$ is through a specific time-scaling factor.
- A local limit for staircase-shaped Young tableaux is established.

## Abstract

A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$. To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.