Critical phenomena and universal exponents in statistical physics. On Dyson's hierarchical model
type: Item
instance of: scholarly article
title: Critical phenomena and universal exponents in statistical physics. On Dyson's hierarchical model
zbMATH Open document ID: 0628.60101
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 82B05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4022331
zbMATH Keywords: renormalization
zbMATH Keywords: large-scale limit
zbMATH Keywords: universality
zbMATH Keywords: statistical mechanics
zbMATH Keywords: Gibbs states
zbMATH Keywords: hierarchical model
zbMATH Keywords: central limit theorems
zbMATH Keywords: translation invariant equilibrium states
author: Pavel M. Bleher
author: P. Major
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992155
OpenAlex ID: W2054536357
DOI: 10.1214/AOP/1176992155
description: scientific article; zbMATH DE number 4022331