The Archimedean limit of random sorting networks

TL;DR

This paper demonstrates that in random sorting networks, particle trajectories approximate sine curves and the network path closely resembles a great circle on a sphere, confirming several conjectures in the field.

Contribution

It establishes the asymptotic sine curve behavior of trajectories and the spherical approximation of the network path, advancing understanding of random sorting networks.

Findings

Particle trajectories are close to sine curves with high probability.

The time-$t$ permutation measures converge to a weak limit.

The network path approximates a great circle on a sphere in high dimensions.

Abstract

A sorting network (also known as a reduced decomposition of the reverse permutation), is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $\sigma^n$. As a corollary of these results, we show that if $S_n$ is embedded into $\mathbb{R}^n$ via the map $\tau \mapsto (\tau(1), \tau(2), \dots \tau(n))$, then with high probability, the path $\sigma^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere in $\mathbb{R}^n$. These results prove conjectures of Angel, Holroyd, Romik, and Virag.