Journal of Statistical Physics

, Volume 10, Issue 1, pp 11-33

First online:

Gibbs and Markov random systems with constraints

  • John MoussourisAffiliated withThe Mathematical Institute and Merton College

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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