Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory

1992/10/22 by Aernout C. D. van Enter, A. C. D. van Enter, Roberto Fernández +3 · 12 citations
Materials Science · Mathematics · Physics and Astronomy · #Computer science #Economics #Group (periodic table) #Mathematical physics #Mathematics #Physics #Position (finance) #Pure mathematics #Quantum mechanics #Quasicrystal Structures and Properties #Renormalization group #Scope (computer science) #Space (punctuation) #Spectral Theory in Mathematical Physics #Statistical physics #Theoretical and Computational Physics #Theoretical physics #hep-lat

paper · pdf · doi:10.1007/bf01048183

published as J. Statist. Phys. 72 (1994) 879 · 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.Z

arxiv created 1992/10/22 · openalex publication_date 1993/09/01 · arxiv updated 2015/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 ≥ 3, these pathologies occur in a full neighborhood \ β> β₀ , |h| < ε(β) \ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d ≥ 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 ≥ 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.

Citations

• The renormalization group and the ε expansion
• On Information and Sufficiency
• Mean-field critical behaviour for percolation in high dimensions
• On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems
• Mathematical properties of position-space renormalization-group transformations
• Logarithmic Corrections to the Molecular-Field Behavior of Critical and Tricritical Systems
• On the critical behavior of the magnetization in high-dimensional Ising models
• The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas
• The renormalization group: Critical phenomena and the Kondo problem
• The critical properties of the two-dimensional xy model
• The phase transition in the one-dimensional Ising Model with 1/r 2 interaction energy
• On Ising's model of ferromagnetism
• A tutorial on hidden Markov models and selected applications in speech recognition
• I-Divergence Geometry of Probability Distributions and Minimization Problems
• Topological Vector Spaces
• Asymptotic evaluation of certain markov process expectations for large time. IV

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