Circular support in random sorting networks

TL;DR

This paper studies the geometric structure of particle trajectories in random sorting networks, revealing they are supported on Lipschitz paths and confined within specific geometric bounds in the limit.

Contribution

It establishes the global limit behavior of trajectories in random sorting networks, including support on Lipschitz paths and elliptical bounds on permutation matrices.

Findings

Trajectories are supported on π-Lipschitz paths in the limit.

Permutation matrices converge within a specific elliptical support.

Trajectories near the edge approximate sine curves within a square root error.

Abstract

A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 2 1$ in the Cayley graph of the symmetric group generated by adjacent transpositions. For a uniform random sorting network, we prove that in the global limit, particle trajectories are supported on $\pi$-Lipschitz paths. We show that the weak limit of the permutation matrix of a random sorting network at any fixed time is supported within a particular ellipse. This is conjectured to be an optimal bound on the support. We also show that in the global limit, trajectories of particles that start within distance $\epsilon$ of the edge are within $\sqrt{2\epsilon}$ of a sine curve in uniform norm.