The oriented swap process

Particles labelled 1 · · · n are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as n → ∞. 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 TracyWidom distribution.