The Annals of Probability

The oriented swap process

Omer Angel, Alexander Holroyd, and Dan Romik

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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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Ann. Probab., Volume 37, Number 5 (2009), 1970-1998.

First available in Project Euclid: 21 September 2009

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Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60C05: Combinatorial probability

Sorting network exclusion process second-class particle permutahedron interacting particle system


Angel, Omer; Holroyd, Alexander; Romik, Dan. The oriented swap process. Ann. Probab. 37 (2009), no. 5, 1970--1998. doi:10.1214/09-AOP456.

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