Journal of Statistical Physics

, Volume 10, Issue 1, pp 11–33 | Cite as

Gibbs and Markov random systems with constraints

  • John Moussouris


This paper concerns random systems made up out of a finite collection of elements. We are interested in how a fixed structure of interactions reflects on the assignment of probabilities to overall states. In particular, we consider two simple models of random systems: one generalizing the notion of “Gibbs ensemble” abstracted from statistical physics; the other, “Markov fields” derived from the idea of a Markov chain. We give background for these two types, review proofs that they are in fact identical for systems with nonzero probabilities, and explore the new behavior that arises with constraints. Finally, we discuss unsolved problems and make suggestions for further work.

Key words

Random system Markov assumption local Markov conditions Gibbs potential Gibbs-Markov equivalence inversion formula for potentials constraints barriers and wells limit representations higher-order equations strongly Markovian systems 


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  1. 1.
    J. M. Hammersley and P. E. Clifford, “Markov fields on finite graphs and lattices,” unpublished (1971).Google Scholar
  2. 2.
    R. L. Dobrushin, “The description of a random field by means of its conditional probabilities, and conditions of its regularity,”Th. Prob. & Appl. [English transl. ofTeoriia Veroiatn.]13:197 (1968).Google Scholar
  3. 3.
    M. B. Averintsev, “On a method of describing complete parameter fields,”Problemy Peredaci Informatsii 6:100 (1970).Google Scholar
  4. 4.
    F. Spitzer, “Markov random fields and Gibbs ensembles,”Am. Math. Month. 78:142 (1971).Google Scholar
  5. 5.
    G. R. Grimmett, “A theorem about random fields,”Bull. London Math. Soc. 5(13):81 (1973).Google Scholar
  6. 6.
    M. Hall,Combinatorial Theory, Blaisdell (1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 1974

Authors and Affiliations

  • John Moussouris
    • 1
  1. 1.The Mathematical Institute and Merton CollegeOxfordEngland

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