Description of random fields by systems of conditional distributions

Authors

  • Linda Khachatryan Institute of Mathematics, National Academy of Science of RA

DOI:

https://doi.org/10.52737/18291163-2022.14.8-1-40

Keywords:

Random field, conditional distribution, specification, Markov random field

Abstract

In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions. In the majority of cases, we provide necessary and sufficient conditions for the system to be a conditional distribution of a (unique) random field. The exception is Dobrushin-type systems for which only sufficient conditions are known. Also, we discuss possible applications of the considered systems.

References

Arzumanyan V.A., Nahapetian B.S., Consistent systems of finite dimensional distributions, Armen. J. Math., 7 (2015), no. 2, pp. 146-163.

Dobrushin R.L., The description of a random field by means of conditional probabilities and conditions of its regularity, Theory Probab. Appl., 13 (1968), no. 2, pp. 197-224. https://doi.org/10.1137/1113026

Dobrushin R.L., Gibbs random fields for lattice systems with pair-wise interaction, Funct. Anal. Appl., 2 (1968), pp. 292-301.

Dobrushin R.L., The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Funct. Anal. Appl. 2, (1968), no. 4, pp. 302-312. https://doi.org/10.1007/bf01075682

Dobrushin R.L., Shlosman S.B., Constructive unicity criterion. In: Statistical mechanics and dynamical systems, Fritz, J., Jaffe, A., Szasz, D. (eds.). New York: Birkhauser, 1985.

Dachian S., Nahapetian B.S., An approach towards description of random fields. Preprint Seminary di Probabilita' e Statistica Mathematica, Eduzioni dell' Universita degli Studi di Cassino, 1998, 20 pp.

Dachian S., Nahapetian B.S., Description of random fields by means of one-point conditional distributions and some applications, Markov Processes Relat. Fields, 7 (2001), pp. 193-214.

Dachian S., Nahapetian B.S., 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. https://doi.org/10.1023/b:joss.0000044069.91072.0b

Dachian S., Nahapetian B.S., On Gibbsiannes of random fields, Markov Processes Relat. Fields, 15 (2009), pp. 81-104.

Dachian S., Nahapetian B.S., On the relationship of energy and probability in models of classical statistical physics, Markov Processes Relat. Fields, 25 (2019), pp. 649-681.

Dalalyan A., Nahapetian B.S., Description of random fields by means of one-point finite conditional distribution, J. Contemp. Math. Anal. Arm. Acad. Sci., 46 (2011), no. 2, pp. 113-119. https://doi.org/10.3103/s1068362311020075

Fernandez R., Maillard G., Construction of a specification from its singleton part, ALEA Lat. Am. J. Probab. Math. Stat., 2 (2006), pp. 297-315.

Goldstein S., A note on specifications, Z. Wahrscheinlichkeitstheorie verw Gebiete, 46 (1978), pp. 45-51. https://doi.org/10.1007/BF00535686

Griffeath D., Introduction to Random Fields. In: Denumerable Markov Chains, Graduate Texts in Mathematics 40, Springer, New York, NY, 1976. https://doi.org/10.1007/978-1-4684-9455-6_12

Khachatryan L.A., Nahapetian B.S., On a class of infinite systems of linear equations originating in statistical physics, Lobachevskii J. Math., 40 (2019), no. 8, pp. 1090-1101. https://doi.org/10.1134/s1995080219080146

Khachatryan L.A., Nahapetian B.S., On direct and inverse problems in the description of lattice random fields, Proceedings of the XI international conference Stochastic and Analytic Methods in Mathematical Physics, Lectures in pure and applied mathematics 6, Universitätsverlag Potsdam, 2020, pp. 107-116. https://doi.org/10.25932/publishup-45919

Khachatryan L.A., Nahapetian B.S., Combinatorial approach to the description of random fields, Lobachevskii J. Math., 42 (2021), no. 10, pp. 2337-2347. https://doi.org/10.1134/s1995080221100103

Khachatryan L.A., Nahapetian B.S., On the characterization of a finite random field by conditional distribution and its Gibbs form, arXiv:2201.09229 [math.PR], 2022.

Kolmogorov A.N., Foundations of the theory of probability, Oxford, England: Chelsea Publishing Co., 1950.

Künsch H., Thermodynamics and statistical analysis of Gaussian random fields, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 58 (1981), pp. 407-421. https://doi.org/10.1007/bf00542645

Lanford O.E., Ruelle D., Observables at infinity and states with short range correlations in statistical mechanics, Commun. Math. Phys., 13 (1969), pp. 194-215. https://doi.org/10.1007/bf01645487

Rényi A., On a new axiomatic theory of probability, Acta Mathematica Academiae Scientiarum Hungaricae, 6 (1955), pp. 285-335. https://doi.org/10.1007/bf02024393

Sullivan W.G., Potentials for almost Markovian random fields, Commun. Math. Phys., 33 (1973), pp. 61-74. https://doi.org/10.1007/bf01645607

Downloads

Published

2022-06-03

How to Cite

Khachatryan, L. (2022). Description of random fields by systems of conditional distributions. Armenian Journal of Mathematics, 14(8), 1–40. https://doi.org/10.52737/18291163-2022.14.8-1-40