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Shocks in the asymmetric exclusion process
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  • Published: June 1988

Shocks in the asymmetric exclusion process

  • E. D. Andjel1,
  • M. D. Bramson2 &
  • T. M. Liggett3 

Probability Theory and Related Fields volume 78, pages 231–247 (1988)Cite this article

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Summary

In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density λ at -∞ and ⌕ at +∞. Earlier results described the limiting behavior in all cases except for 0<λ<1/2, λ+⌕=1. Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in 1975 [7]. The convergence proof is based on coupling and symmetry considerations.

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References

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Author information

Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, Jardim Botânico, CEP 22460, Rio de Janeiro RJ, Brasil

    E. D. Andjel

  2. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    M. D. Bramson

  3. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    T. M. Liggett

Authors
  1. E. D. Andjel
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  2. M. D. Bramson
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  3. T. M. Liggett
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Additional information

Research supported in part by NSF Grant DMS 83-1080

Research supported in part by NSF Grant MCS 83-00836

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Cite this article

Andjel, E.D., Bramson, M.D. & Liggett, T.M. Shocks in the asymmetric exclusion process. Probab. Th. Rel. Fields 78, 231–247 (1988). https://doi.org/10.1007/BF00322020

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  • Received: 10 July 1986

  • Issue Date: June 1988

  • DOI: https://doi.org/10.1007/BF00322020

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Keywords

  • Differential Equation
  • Shock Wave
  • Partial Differential Equation
  • Stochastic Process
  • Probability Theory
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