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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 201–229 (Mi tvp839)  

This article is cited in 31 scientific papers (total in 31 papers)

The description of the random field by its conditional distributions and its regularity conditions

R. L. Dobrushin

Moscow

Abstract: Suppose that for any finite set $\{t_1,…,t_n\}\subset T^\nu$, where $T^\nu$ is the $\nu$-dimensional cubic lattice, and for any $x_i$, $x(t)$, conditional probabilities
$$ \mathbf P\{\xi(t_1)=x_1,…,\xi(t_n)=x_n\mid\xi(t)=x(t), t\in T^\nu, t\ne t_i, i=1,…,n\} $$
corresponding to a random field with a finite number of its values $\xi(t)$ and known and have some natural properties of consistency. The problem is to find out if it is possibjle to find absolute probabilities $\mathbf\{\xi(t_1)=x_1,…,\xi(t_n)=x_n\}$, by which the given family of conditional probabilities is generated. It is proved that there exists a solution of this, problem and that in case $\nu=1$ it is unique. For $\nu>1$, the uniqueness can be proved if conditional distributions are close in a certain sense to those of a field of independent variables. Some systems of statistical physics with phase transitions give us examples when the solution is not unique. In more detail this question is considered in [4]. We prove also that the uniqueness is equivalent to one of the forms of regularity conditions of the field.

Full text: PDF file (1760 kB)

English version:
Theory of Probability and its Applications, 1968, 13:2, 197–224

Bibliographic databases:

Received: 20.04.1967

Citation: R. L. Dobrushin, “The description of the random field by its conditional distributions and its regularity conditions”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 201–229; Theory Probab. Appl., 13:2 (1968), 197–224

Citation in format AMSBIB
\Bibitem{Dob68}
\by R.~L.~Dobrushin
\paper The description of the random field by its conditional distributions and its regularity conditions
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 201--229
\mathnet{http://mi.mathnet.ru/tvp839}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=231434}
\zmath{https://zbmath.org/?q=an:0184.40403|0177.45202}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 197--224
\crossref{https://doi.org/10.1137/1113026}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. A. Minlos, “Lectures on statistical physics”, Russian Math. Surveys, 23:1 (1968), 137–196  mathnet  crossref  mathscinet  zmath
    2. R. A. Minlos, Ya. G. Sinai, “Spectra of stochastic operators arising in lattice models of a gas”, Theoret. and Math. Phys., 2:2 (1970), 167–176  mathnet  crossref  mathscinet
    3. R. L. Dobrushin, “Gibbsian random fields for particles without hard core”, Theoret. and Math. Phys., 4:1 (1970), 705–719  mathnet  crossref  mathscinet  zmath
    4. N. Ali, “Gibbsovskie sluchainye polya dlya reshetchatykh modelei”, UMN, 26:5(161) (1971), 222–222  mathnet  mathscinet  zmath
    5. R. L. Dobrushin, “Asymptotic behavior of Gibbsian distributions for lattice systems and its dependence on the form of the volume”, Theoret. and Math. Phys., 12:1 (1972), 699–711  mathnet  crossref  mathscinet
    6. Ya. G. Sinai, “Construction of dynamics in one-dimensional systems of statistical mechanics”, Theoret. and Math. Phys., 11:2 (1972), 487–494  mathnet  crossref  mathscinet
    7. R. L. Dobrushin, “Conditions for the absence of phase transitions in one-dimensional classical systems”, Math. USSR-Sb., 22:1 (1974), 28–48  mathnet  crossref  mathscinet  zmath
    8. R. L. Dobrushin, B. S. Nakhapetian, “Strong convexity of the pressure for lattice systems of classical statistical physics”, Theoret. and Math. Phys., 20:2 (1974), 782–790  mathnet  crossref  mathscinet  zmath
    9. D. G. Martirosyan, “Uniqueness of Gibbs limit distributions for the perturbed Ising model”, Theoret. and Math. Phys., 22:3 (1975), 236–240  mathnet  crossref  zmath
    10. R. A. Minlos, G. M. Natapov, “Uniqueness of the limit Gibbs distribution in one-dimensional classical systems”, Theoret. and Math. Phys., 24:1 (1975), 697–703  mathnet  crossref  mathscinet
    11. Yu. G. Pogorelov, “Cluster property in a classical canonical ensemble”, Theoret. and Math. Phys., 30:3 (1977), 227–232  mathnet  crossref  mathscinet  zmath
    12. I. L. Simyatitskii, “Gibbs states in the case of classical representation of quantum spin systems”, Theoret. and Math. Phys., 39:3 (1979), 562–563  mathnet  crossref  mathscinet  isi
    13. K. M. Khanin, “Absence of phase transitions in one-dimensional long-range spin systems with random Hamiltonian”, Theoret. and Math. Phys., 43:2 (1980), 445–449  mathnet  crossref  mathscinet  isi
    14. V. A. Malyshev, “Cluster expansions in lattice models of statistical physics and the quantum theory of fields”, Russian Math. Surveys, 35:2 (1980), 1–62  mathnet  crossref  mathscinet  adsnasa  isi
    15. V. A. Zagrebnov, “A new proof and generalization of the Bogolyubov–Ruelle theorem”, Theoret. and Math. Phys., 51:3 (1982), 570–579  mathnet  crossref  mathscinet  isi
    16. V. Ya. Golodets, G. N. Zholtkevich, “Markovian KMS states”, Theoret. and Math. Phys., 56:1 (1983), 686–690  mathnet  crossref  mathscinet  isi
    17. V. A. Chulaevskii, “Inverse scattering method in statistical physics”, Funct. Anal. Appl., 17:1 (1983), 40–47  mathnet  crossref  mathscinet  isi
    18. I. G. Brankov, V. A. Zagrebnov, N. S. Tonchev, “Description of limit gibbs states for Curie–Weiss–Ising model”, Theoret. and Math. Phys., 66:1 (1986), 72–80  mathnet  crossref  mathscinet  isi
    19. Ya. G. Sinai, N. I. Chernov, “Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls”, Russian Math. Surveys, 42:3 (1987), 181–207  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    20. N. N. Ganikhodzhaev, “Pure phases of the ferromagnetic Potts model with three states on a second-order Bethe lattice”, Theoret. and Math. Phys., 85:2 (1990), 1125–1134  mathnet  crossref  mathscinet  isi
    21. B. S. Nakhapetian, S. K. Pogosyan, “Estimate of convergence rate in local limit theorem for the particle number in spin systems”, Theoret. and Math. Phys., 95:3 (1993), 738–747  mathnet  crossref  mathscinet  zmath  isi
    22. B. S. Nakhapetian, “Strong convexity of pressure for spin lattice systems”, Russian Math. Surveys, 52:2 (1997), 341–347  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    23. R. A. Minlos, “R. L. Dobrushin – one of the founders of modern mathematical physics”, Russian Math. Surveys, 52:2 (1997), 251–256  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    24. F. M. Mukhamedov, “Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice”, Theoret. and Math. Phys., 123:1 (2000), 489–493  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. V. A. Malyshev, “Gibbs and quantum discrete spaces”, Russian Math. Surveys, 56:5 (2001), 917–972  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. A. M. Rakhmatullaev, U. A. Rozikov, “Gibbs Measures and Markov Random Fields with Association $I$”, Math. Notes, 72:1 (2002), 83–89  mathnet  crossref  crossref  mathscinet  zmath  isi
    27. Bentkus V., Sunklodas J.K., “On normal approximations to strongly mixing random fields”, Publicationes Mathematicae–Debrecen, 70:3–4 (2007), 253–270  mathscinet  zmath  isi
    28. L. Accardi, F. M. Mukhamedov, M. Kh. Saburov, “Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order $2$”, Math. Notes, 90:2 (2011), 162–174  mathnet  crossref  crossref  mathscinet  isi
    29. A. A. Filchenkov, “Mery istinnosti i veroyatnostnye graficheskie modeli dlya predstavleniya znanii s neopredelennostyu”, Tr. SPIIRAN, 23 (2012), 254–295  mathnet  elib
    30. Kifer Yu., “A Nonconventional Invariance Principle for Random Fields”, J. Theor. Probab., 26:2 (2013), 489–513  crossref  isi
    31. A. V. Alpeev, “Announce of an entropy formula for a class of Gibbs measures”, J. Math. Sci. (N. Y.), 224:2 (2017), 171–175  mathnet  crossref  mathscinet
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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