Open Access
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


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.


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Omer Angel. Alexander Holroyd. Dan Romik. "The oriented swap process." Ann. Probab. 37 (5) 1970 - 1998, September 2009.


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

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

Primary: 60C05 , 60K35 , 82C22

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

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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