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The results of this section have been stated in several papers of Bleher and Sinai. However, it seems to us that they did not analyse completely the separation into two pure phases, but rather the evolution under iterates of N of a single phase, as in out Conjecture 6.3. A very nice study of the approach to the thermodynamics limit in the Ising model for temperatures above the critical temperature, i.e. in the single phase region has been given in
G.Gallavotti, H.J.P. KNOPS: Block spin interactions in the Ising model. Commun. Math. Phys. 36, 171 (1974).
G. GALLAVOTTI, A. MARTIN-LÖF: Block spin distribution for short range attractive Ising models. Il Nuovo Cimento 25B, 425 (1975).
In these papers, asymptotic expansions in the inverse volume (virial expansions) are obtained. A very complete control of convergence is also found for another family of interactions, with stronger forces than those found in the Hierarchical Model, in
R.S. ELLIS, C.M. NEWMAN: Limit theorems for sums of dependent random variabls occurring in statistical mechanics (Preprint 1977).
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(1978). Global properties of the flow. In: A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics. Lecture Notes in Physics, vol 74. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017114
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DOI: https://doi.org/10.1007/BFb0017114
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