Abstract
A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.
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Averintsev, M. B.: On a method of describing descrete parameter random fields. Problemy Peradici Informacii6, 100–108 (1970)
Averintsev, M. B.: The description of Markov random field by Gibbs conditional distributions. Teor. Verojatnost i Primenen17, 21–35 (1972)
Dobrushin, R. L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Probability Appl.13, 197–224 (1968)
Gallavotti, G., Miracle-Sole, S.: Statistical mechanics of lattice systems. Commun. math. Phys.5, 317–323 (1967)
Gallavotti, G., Miracle-Sole, S.: Correlation functions of a lattice gas. Commun. math. Phys.7, 274–288 (1968)
Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. math. Phys.23, 87–99 (1971)
Lanford, O. E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. math. Phys.13, 194–215 (1969)
Loève, M.: Probability theory. New York: D. Van Nostrand 1963
Ruelle, D.: Statistical mechanics, rigorous results. New York: Benjamin 1969
Sherman, S.: Markov random fields and Gibbs random fields. To appear in Israel J. Math
Spitzer, F.: Markov random fields and Gibbs ensembles. Amer. Math. Monthly78, 142–154 (1971)
Sullivan, W. G.: Finite range random fields and energy fields. To appear
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Sullivan, W.G. Potentials for almost Markovian random fields. Commun.Math. Phys. 33, 61–74 (1973). https://doi.org/10.1007/BF01645607
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DOI: https://doi.org/10.1007/BF01645607