Triviality of hierarchical Ising model in four dimensions
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.
scientific article; zbMATH DE number 1742838
scientific article