Critical Phenomena and Universal Exponents in Statistical Physics. On Dyson's Hierarchical Model

Abstract

We are interested in the behavior of Gibbs states at or near the critical temperature. From the point of view of classical probability theory this is a problem about the limit distribution of partial sums of strongly dependent random variables. The problem is very hard in the general case, and almost no rigorous results are known, so we will discuss a special case, Dyson's hierarchical model, in detail. This model can be rigorously investigated, and it may help us to understand the behavior of the more general models. We present the most important results with a sketch of their proofs. Vector-valued models are also discussed, since in this case some new interesting phenomena appear. The last section deals with the translation invariant case. Some recent results are presented and some conjectures and open problems are formulated. The last section can be read independently of the previous sections, but the conjectures formulated there are strongly motivated by the previous results.