The oriented swap process

Details The oriented swap process The oriented swap process

2008-06-13

arxiv.org/abs/0806.2222v2

Annals of Probability 2009, Vol. 37, No. 5, 1970-1998

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP456 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

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 $n\to\infty$. 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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