Abstract
We define the specific norms used to analyse the renormalisation group map, and specify the domain of the map. The choice of norms is based on considerations concerning the typical sizes of the fluctuation and block-spin fields. We state the main estimates on the renormalisation group map in two theorems, and then use these theorems to construct the global renormalisation group flow in the nonperturbative case. The construction requires, in particular, the construction of the critical point. The latter is done via the Bleher–Sinai argument. The results of this chapter reduce our analysis of the 4-dimensional hierarchical model to the proof of the estimates on the renormalisation group map stated in this chapter. The two theorems which state those estimates are proved in the next two chapters.
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P.M. Bleher, Ya.G. Sinai, Investigation of the critical point in models of the type of Dyson’s hierarchical models. Commun. Math. Phys. 33, 23–42 (1973)
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Bauerschmidt, R., Brydges, D.C., Slade, G. (2019). Global Flow: Proof of Theorem 4.2.1 . In: Introduction to a Renormalisation Group Method. Lecture Notes in Mathematics, vol 2242. Springer, Singapore. https://doi.org/10.1007/978-981-32-9593-3_8
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DOI: https://doi.org/10.1007/978-981-32-9593-3_8
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