The oriented swap process
Omer Angel, Alexander Holroyd, and Dan Romik
Source: Ann. Probab.
Volume 37, Number 5
(2009), 1970-1998.
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.
Primary Subjects: 82C22, 60K35, 60C05
Keywords: Sorting network; exclusion process; second-class particle; permutahedron; interacting particle system
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1253539861
Digital Object Identifier: doi:10.1214/09-AOP456
Zentralblatt MATH identifier:
05625057
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