Open Access
Translator Disclaimer
September 2009 The oriented swap process
Omer Angel, Alexander Holroyd, Dan Romik
Ann. Probab. 37(5): 1970-1998 (September 2009). DOI: 10.1214/09-AOP456

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→∞. 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.

Citation

Download Citation

Omer Angel. Alexander Holroyd. Dan Romik. "The oriented swap process." Ann. Probab. 37 (5) 1970 - 1998, September 2009. https://doi.org/10.1214/09-AOP456

Information

Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1180.82125
MathSciNet: MR2561438
Digital Object Identifier: 10.1214/09-AOP456

Subjects:
Primary: 60C05 , 60K35 , 82C22

Keywords: Exclusion process , Interacting particle system , permutahedron , Second-class particle , Sorting network

Rights: Copyright © 2009 Institute of Mathematical Statistics

JOURNAL ARTICLE
29 PAGES


SHARE
Vol.37 • No. 5 • September 2009
Back to Top