On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form

Linda Khachatryan () and Boris S. Nahapetian ()
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Linda Khachatryan: National Academy of Science of the Republic of Armenia
Boris S. Nahapetian: National Academy of Science of the Republic of Armenia

Journal of Theoretical Probability, 2023, vol. 36, issue 3, 1743-1761

Abstract: Abstract In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness of a finite random field with a given system of one-point conditional distributions. Using the axiomatic (without the notion of potential) definition of Hamiltonian, we show that any finite random field is Gibbsian. We also apply the proposed approach to Markov random fields.

Keywords: Random field; Conditional distribution; Gibbs distribution; Transition energy field; Hamiltonian; 60G60; 60E05; 60J99; 62H99; 82B03 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01209-6

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