Critical properties of Dyson's hierarchical model. II. Essential singularities of the borderline Ising case

Abstract

The borderline case of Dyson's hierarchical model where a first-order transition is known rigorously to occur is investigated numerically. The susceptibility is found to behave as chi approximately exp(a( beta c - beta )-1) as beta to beta c - and infinite for beta > beta c. The spontaneous magnetisation and correlation functions are also investigated. In terms of renormalisation group theory, the essentially singular behavior results from the relevant operator becoming marginal in the borderline case.