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

R. L. Dobrushin

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 201-229

Résumé

Suppose that for any finite set $\{t_1,\dots,t_n\}\subset T^\nu$, where $T^\nu$ is the $\nu$-dimensional cubic lattice, and for any $x_i$, $x(t)$, conditional probabilities $$ \mathbf P\{\xi(t_1)=x_1,\dots,\xi(t_n)=x_n\mid\xi(t)=x(t),\ t\in T^\nu,\ t\ne t_i,\ i=1,\dots,n\} $$ 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 $\mathbf\{\xi(t_1)=x_1,\dots,\xi(t_n)=x_n\}$, 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.