Critical phenomena and universal exponents in statistical physics. On Dyson's hierarchical model
One of the most interesting, yet most difficult, topics in equilibrium statistical mechanics concerns critical behaviour, i.e. the behaviour of Gibbs states at and near the critical temperature \(T_{cr}\). Its study is also important from a purely probabilistic point of view, since it deals with limit theorems for sums of strongly dependent random variables. More exactly, it concerns the large-scale limit of a random field \(\{X_ n\), \(n\in {\mathbb{Z}}^ d\}\) distributed with respect to a Gibbs state, which is parametrized by the temperature T. One gets this limit by dividing \({\mathbb{Z}}^ d\) into large disjoint cubes of equal size and summing over the indices of the different cubes. In the limit one expects of physical reasons a central limit theorem to hold for temperatures \(T\neq T_{cr}\), but a different normalization with Gaussian or non- Gaussian limits for \(T=T_{cr}\). This behaviour is closely connected with an exponential decrease of the correlation function for \(T\neq T_{cr}\) and polynomial decrease for \(T=T_{cr}.\) Another interesting feature is universality, which means that certain quantities like variance, normalization and magnetization behave like a power of \(| T-T_{cr}|\) near \(T_{cr}\) with an exponent, which only depends on some global characteristics of the corresponding model, like the dimension, but not on its finer structure. As mentioned above, all these properties are suggested to hold, but at present no proofs exist for realistic physical models. Dyson's hierarchical model is a simplified model, for which these conjectures can be treated rigorously and which thus can indicate the behaviour of more realistic models. The study of this model is essentially due to the authors of the present paper, partly in collaboration with Ja. G. Sinai. Here, they summarize the corresponding results with detailed sketches of the proof. They also investigate vector-valued models, where non central limit theorems hold for all temperatures below the critical one. Finally they discuss general translation invariant equilibrium states. They present some recent results and formulate some conjectures, which are based on the results for Dyson's hierarchical model. Since at present only little is known about the important problem of limit behaviour of sums of strongly dependent random variables, this illuminating article is recommended to every probabilist.
translation invariant equilibrium states
scientific article; zbMATH DE number 4022331