Gibbsian Description of Gaussian Random Fields
DOI:
https://doi.org/10.52737/18291163-2025.17.13-1-20Keywords:
Gibbs Random Field, Gaussian Random Field, Conditional Densities, Gibbs Form of Conditional Densities, Transition Energy, Hamiltonian, Markov Random FieldAbstract
This paper provides the solution to the problem of Gibbsian description of Gaussian random fields based on the Gibbs scheme currently being developed in the theory of lattice random fields.
References
M.B. Averintsev, One method of describing random fields with a discrete argument. Problems Inform. Transmission, 6 (1970), no. 2, pp. 169-175.
M.B. Averintsev, Description of Markovian random fields by Gibbsian conditional probabilities. Theory Probab. Appl., 17 (1973), no. 1, pp. 20-33 DOI: https://doi.org/10.1137/1117002
S. Dachian and B.S. Nahapetian, Description of specifications by means of probability distributions in small volumes under condition of very week positivity. J. Stat. Phys., 117 (2004), pp. 281-300 . DOI: https://doi.org/10.1023/B:JOSS.0000044069.91072.0b
S. Dachian and B.S. Nahapetian, On Gibbsiannes of random fields, Markov Process. Relat. Fields, 15 (2009), no. 1, pp. 81-104.
S. Dachian and B.S. Nahapetian, On the relationship of energy and probability in models of classical statistical physics. Markov Processes Relat. Fields, 25 (2019), no. 4, pp. 649-681.
R.L. Dobrushin, Gibbs random fields for lattice systems with pair-wise interaction. Funct. Anal. and Appl., 2 (1968), pp. 292-301. DOI: https://doi.org/10.1007/BF01075681
R.L. Dobrushin, Gaussian random fields – Gibbsian point of view. In: Multicomponent random systems, eds. R.L. Dobrushin, Ya. G. Sinai, New York: M. Dekker, 1980, pp. 119-151 (Translation of Multicomponent random systems (in Russian), Nauka, Moscow, 1978)
R. Fernández and G. Maillard, Construction of a specification from its singleton part. ALEA Lat. Am. J. Probab. Math. Stat., 2 (2006), pp. 297-315.
F.R. Gantmacher, The Theory of Matrices. Chelsea Publishing Co., 1960.
H.-O. Georgii, Gibbs measures and phase transitions, De Gruyter, Berlin, 1988. DOI: https://doi.org/10.1515/9783110850147
L.A. Khachatryan, Description of lattice random fields by systems of conditional distributions. Armen. J. Math., 14 (2022), no. 8, pp. 1-40. DOI: https://doi.org/10.52737/18291163-2022.14.8-1-40
L.A. Khachatryan and B.S. Nahapetian, Gibbs scheme in the theory of random fields. Ann. Henri Poincare (2025), 27 pp. DOI: https://doi.org/10.1007/s00023-025-01573-z
O.K. Kozlov, Gibbs description of a system of random variables. Problems Inform. Transmission, 10 (1974), pp. 258-265.
H. Künsch, Gaussian Markov random fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26 (1979), pp. 53-73.
H. Künsch, Reellwertige Zufallsfelder auf einem Gitter: Interpolationsprobleme, Variationsprinzip und statistische Analyse. Dissertation, 1980.
H. Künsch, Thermodynamics and statistical analysis of Gaussian random fields. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 58 (1981), pp. 407-421. DOI: https://doi.org/10.1007/BF00542645
Ya.G. Sinai, Theory of Phase Transitions – Rigorous Results. Pergamon, Oxford, 1982.
W.G. Sullivan, Potentials for almost Markovian random fields. Commun. Math. Phys., 33 (1973), pp. 61-74. DOI: https://doi.org/10.1007/BF01645607
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