Thermodynamic limit and central limit theorem for point random fields in non-ergodic case (Q1823566)

A class of point random fields (\({\mathcal X},{\mathcal B},\mu,F)\) is considered where the phase space \({\mathcal X}\) is the space of locally finite configurations of unlabelled particles in \(R^ d\) with the Borel \(\sigma\)-algebra \({\mathcal B}\), translation invariant probability measure \(\mu\) and F is a real random variable on \({\mathbb{R}}^ d\), E \(F^ 2<\infty\). It is supposed that the measure \(\mu\) on the momentum space has the same structure as every Gibbs measure, and additionally that there exists \(s<2\), \(C>0\) such that the variance of the number of particles in a bounded Borel set \(V\subset {\mathbb{R}}^ d\) does not exceed \(C| V|^ s.\) The main result of the paper states that the central limit theorem holds for any point random field (\({\mathcal X},{\mathcal B},\mu,F)\) with E F\(=0\). Finally, an example of a non-ergodic Gibbs measure \(\mu\) is given such that the central limit theorem does not hold for (\({\mathcal X},{\mathcal B},\mu,1)\), though it holds for (\({\mathcal X},{\mathcal B},\mu,F)\) with E F\(=0\).