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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


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.

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

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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
\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
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 197--224

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    This publication is cited in the following articles:
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    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
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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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