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Poissonian approximation for the tagged particle in asymmetric simple exclusion

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

P. A. Ferrari  and
L. R. G. Fontes
P. A. Ferrari*
Affiliation:
Universidade de São Paulo
L. R. G. Fontes*
Affiliation:
Universidade de São Paulo
*
∗Postal address: Instituto de Matemática e Estatística, Universidade de São Paulo, Cx. Postal 66.281, 05389–970 Sao Paulo SP, Brasil.
∗Postal address: Instituto de Matemática e Estatística, Universidade de São Paulo, Cx. Postal 66.281, 05389–970 Sao Paulo SP, Brasil.
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Abstract

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt = Nt − Bt + B0, where {Nt} is a Poisson process of parameter (p – q)(1– λ) and {Bt} is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt} converges to Brownian motion. A previous result says that in the scale t1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.

Keywords

ASYMMETRIC SIMPLE EXCLUSIONTAGGED PARTICLEPOISSONIAN APPROXIMATIONZERO-RANGE PROCESSSYSTEM OF QUEUES IN SERIES

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K25: Queueing theory
Secondary: 90B22: Queues and service 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 411 - 419
Copyright
Copyright © Applied Probability Trust 1996 

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