Limit Theorems and Some Applications in Statistical Physics

In this chapter we give without proof some well known facts of probability theory which we will need later. The proofs or references can be found in /7/, /66/ or /102/.

Let (Ω, F, P) be a probability space. There exist many-various definitions of the measures of dependence between б-fields Ol₁, Ol₂ ⊂ F. Denote

The estimates and inequalities given below play an important role in the theory of limit theorems for dependent random variables.

At present the methods for proving the limit theorems for dependent random variables can be subdivided into two types. The methods of the first type use the approximation with random variables, the asymptotic behaviour of which being known, for instance, the approximation with independent random variables (Bernstein’s method) or with ergodic martingale-differences (Gordin’s method). The methods of the second type use the direct approximation (Stein’s method, the classical methods of moments). In this book we are interested in a number of problems connected with c.l.th. and therefore we will demonstrate the methods using this example, though the applications of these methods are not exhausted by it.

In this chapter for a s.r.p. ξ_t, t ∊ ℤⁱ, satisfying the above mentioned mixing conditions we will study a limit behaviour of the sums.

In /29/ Dobrushin has introduced some weak dependence conditions which represent a natural generalization of the mixing conditions for random fields. He suggested also that the c.l.th. which contains well-known results as a special case under these generalized mixing conditions holds. In this chapter a theorem of such type will be proved for s.r.p. with generalized α-mixing condition. Note that similar results may be obtained both for s.r.p. with generalized φ-mixing condition and for random fields. In the conclusion some generalizations of the c.l. th. will be given in case of so-called non-commutative probability theory.

Limit theorems for random fields have their specific properties caused by the parameter dimensionality. As it was mentioned in Chapter 2 one should be careful to a certain degree while introducing the weak dependence conditions for random field, since it is possible that the random fields classes determined by these conditions practically do not differ from the set of independent random variables. Another difference from the theory of limit theorems for the s.r.p. consists in the wide choice of the passage to infinite volume. We will be interested in the random fields satisfying the conditions of α- and φ-mixing in a sense of Chapter 2. As for the applications to statistical physics let’s note that the limit theorems for the random fields with φ-mixing condition turned out to be interesting, since Gibbs random fields possess this property.

The way of determining a random process by means of its conditional probabilities has been used in probability theory quite for a long time, for example, the definition of Markov chain with the help of its transition matrix. However, it was only in connection with the demands of statistical physics, in particular, with those of the strict definition of Gibbs random field the wide possibilities of this approach were cleared up. Below we will describe some facts of this theory using the results of fundamental works /29/, /32/. As it turned out in contrast to defining a random field by means of its finite-dimensional distributions the uniqueness of a random field with the given conditional distribution does not come from its existence here. As it will be shown in this Chapter for the uniqueness to hold here it is necessary to require that the field should possess some properties of decay of correlations, for example, those of mixing. In doing this it is possible to indicate the estimates of the mixing coefficient in terms of conditional probabilities. At the end of this chapter c.l.th. will be presented for the random field under some requirements for its conditional distribution.

The concept of Gibbs random field arisen in statistical physics with a view to describe a physical system in infinite volume and first was given by Dobrushin /29/, /30/. Lanford and Ruelle /76/. Gibbs random fields are those probability measures which have the given conditional distributions. These conditional probabilities have a specific form involving potential. In this chapter we state the basic facts of the theory of Gibbs random fields as well as some applications of the results of previous chapters.