Abstract
We introduce the renormalisation group map for the hierarchical model. The renormalisation group map has two components: a perturbative and a nonperturbative coordinate. The analysis of the renormalisation group map occupies the remainder of the book. An advantage of the hierarchical model is that the analysis can be reduced to individual blocks; this is not the case in the Euclidean setting. We explain this reduction. We discuss progressive integration; irrelevant, marginal, and relevant polynomials; and localisation. The renormalisation group map involves the notion of flow of coupling constants (u j, g j, ν j), as well as the flow of an infinite-dimensional non-perturbative coordinate K j. The flow of coupling constants is given to leading order by perturbation theory, which is derived in this chapter.
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References
R. Bauerschmidt, D.C. Brydges, G. Slade, Scaling limits and critical behaviour of the 4-dimensional n-component |φ|4 spin model. J. Stat. Phys. 157, 692–742 (2014)
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Bauerschmidt, R., Brydges, D.C., Slade, G. (2019). The Renormalisation Group Map. 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_5
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DOI: https://doi.org/10.1007/978-981-32-9593-3_5
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Publisher Name: Springer, Singapore
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Online ISBN: 978-981-32-9593-3
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