Extremal Reversible Measures for the Exclusion Process


The invariant measures \({\mathcal{I}}\) for the exclusion process have long been studied and a complete description is known in many cases. This paper gives characterizations of \({\mathcal{I}}\) for exclusion processes on \({\mathbb{Z}}\) with certain reversible transition kernels. Some examples for which \({\mathcal{I}}\) is given include all finite range kernels that are asymptotically equal to p(x,x+1)=p(x,x−1)=1/2. One tool used in the proofs gives a necessary and sufficient condition for reversible measures to be extremal in the set of invariant measures, which is an interesting result in its own right. One reason that this extremality is interesting is that it provides information concerning the domains of attraction for reversible measures.

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  1. 1.

    E. D. Andjel, Convergence to a nonextremal equilibrium measure in the exclusion process, Probab. Theory Related Fields 73:127–134 (1986).

    Google Scholar 

  2. 2.

    E. D. Andjel, M. D. Bramson, and T. M. Liggett, Shocks in the asymmetric exclusion process, Probab. Theory Related Fields 78:231–247 (1988).

    Google Scholar 

  3. 3.

    M. D. Bramson, T. M. Liggett, and T. S. Mountford, Characterization of stationary measures for one-dimensional exclusion processes, Ann. Probab. 30:1539–1575 (2002).

    Google Scholar 

  4. 4.

    R. Durrett, Probability: Theory and Examples, 2nd ed. (Duxbury Press, Belmont, CA, 1996).

    Google Scholar 

  5. 5.

    S. A. Janowski and J. L. Lebowitz, Exact results for the asymmetric simple exclusion with a blockage, J. Statist. Phys. 77:35–51 (1994).

    Google Scholar 

  6. 6.

    C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes, Comm. Math. Phys. 104:1–19 (1986).

    Google Scholar 

  7. 7.

    T. M. Liggett, Coupling the simple exclusion process, Ann. Probab. 4:339–356 (1976).

    Google Scholar 

  8. 8.

    T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).

    Google Scholar 

  9. 9.

    T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (Springer-Verlag, Berlin/Heidelberg, 1999).

    Google Scholar 

  10. 10.

    S. Sethuraman, On extremal measures for conservative particle systems, Ann. Inst. H. Poincaré 37:139–154 (2001).

    Google Scholar 

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Jung, P. Extremal Reversible Measures for the Exclusion Process. Journal of Statistical Physics 112, 165–191 (2003). https://doi.org/10.1023/A:1023679620839

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  • exclusion process
  • invariant measures
  • extremal invariant measures
  • domains of attraction