The oriented swap process

TL;DR

This paper studies the asymptotic behavior of an oriented swap process where neighboring particles swap in increasing order, revealing convergence of trajectories, permutation measures, and fluctuation distributions as the number of particles grows large.

Contribution

It provides a detailed analysis of the process's asymptotics, including convergence of particle trajectories, permutation measures, and fluctuation limits, which was previously unexplored.

Findings

Particle trajectories converge to random curves with non-differentiability points.

Permutation matrices converge to a deterministic measure with mixed parts.

Finishing times follow Tracy--Widom distribution asymptotically.

Abstract

Particles labelled $1,...,n$ are initially arranged in increasing order. Subsequently, each pair of neighboring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behavior of this process as $n\to\infty$. We prove that the space--time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time $(2+o(1))n$. The finishing times of individual particles converge to deterministic limits, with fluctuations asymptotically governed by the Tracy--Widom distribution.