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Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations | | A.C.D. van Enter
; R. Fernandez
; A.D. Sokal
; | | Date: |
22 Oct 1992 | | Journal: | J. Statist. Phys. 72 (1994) 879 | | Subject: | hep-lat | | Abstract: | We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d ge 3$, these pathologies occur in a full neighborhood $eta > eta_0 ,|h| < epsilon(eta) }$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. | | Source: | arXiv, hep-lat/9210032 | | Services: | Forum | Review | PDF | Favorites | |
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