Abstract
Our implementation of the renormalisation group method relies on a finite-range decomposition of the Gaussian free field to allow progressive integration over scales. This requires an appropriate decomposition of the covariance of the Gaussian free field into a sum of simpler covariances. In this chapter, we provide a self-contained derivation of a finite-range covariance decomposition. This is easy for the case of the continuum, which we consider first. We then consider the lattice case, where the finite-range decomposition is generated by making use of the finite speed of propagation of the discrete wave equation. This then gives rise to a finite-range decomposition on the discrete torus. The finite-range decomposition provides our main motivation for the definition of the hierarchical model in Chap. 4, which is the focus of the book until Chap. 12 where Euclidean models are discussed.
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Bauerschmidt, R., Brydges, D.C., Slade, G. (2019). Finite-Range Decomposition. 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_3
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DOI: https://doi.org/10.1007/978-981-32-9593-3_3
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