Hierarchical ferromagnetic vector spin model possessing the Lee-Yang property. Thermodynamic limit at the critical point and above

Abstract

The hierarchical ferromagneticN-dimensional vector spin model as a sequence of probability measures onRN is considered. The starting element of this sequence is chosen to belong to the Lee-Yang class of measures that is defined in the paper and includes most known examples (ϕ4 measures, Gaussian measures, and so on). For this model, we probe two thermodynamic limit theorems. One of them is just the classical central limit theorem for weakly dependent random vectors. It describes the convergence of classically normed sums of spins when temperature is sufficiently high. The other theorem describes the convergence of “more than normally” normed sums that holds for some fixed temperature. It corresponds to the strong dependence of spins, which appears at the critical point of the model.