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→∞. 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.
@article{1253539861,
author = {Angel, Omer and Holroyd, Alexander and Romik, Dan},
title = {The oriented swap process},
journal = {Ann. Probab.},
volume = {37},
number = {1},
year = {2009},
pages = { 1970-1998},
language = {en},
url = {http://dml.mathdoc.fr/item/1253539861}
}
Angel, Omer; Holroyd, Alexander; Romik, Dan. The oriented swap process. Ann. Probab., Tome 37 (2009) no. 1, pp. 1970-1998. http://gdmltest.u-ga.fr/item/1253539861/