Triviality of hierarchical Ising model in four dimensions
type: Item
instance of: scholarly article
title: Triviality of hierarchical Ising model in four dimensions
zbMATH Open document ID: 1001.82044
author: Takashi Hara
author: Tetsuya Hattori
author: Hiroshi Watanabe
published in: Communications in Mathematical Physics
publication date: 2002-05-21T00:00:00Z
review text: The aim of the paper is to study the asymptotical behaviour of the hierarchical Ising model in dimensions \(d\geq 4\). The authors show that the related renormalization group transformation has a ``critical trajectory'' converging to the Gaussian fixed point. The main ingredient of this computer-assisted proof is a certain sufficient condition (in the spirit of \textit{P. Bleher} and \textit{Ya. G. Sinai} [Commun. Math. Phys. 33, 23-42 (1973); ibid. 45, 247-278 (1975)]) for a probability measure to lie in a domain of attraction of the Gaussian law. This condition is formulated in terms of a \textit{finite} number of truncated correlations (semiinvariants) and is subsequently verified using Mathematica and C++ programs. According to the results of computation, in dimension \(d=4\), already 70 iterations of the renorm-group transformation suffice for the initial Ising measure to fall into a vicinity of the Gaussian fixed point. The analysis of convergence to the limit is based on Newman's inequalities for truncated correlations.
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 82B28
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 1742838
zbMATH Keywords: hierarchical Ising model
zbMATH Keywords: renormalization group
zbMATH Keywords: triviality
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: s002200100440
OpenAlex ID: W2121272461
DOI: 10.1007/S002200100440
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description: scientific article; zbMATH DE number 1742838