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

Aernout C D van Enter, Roberto Fernández, Alan D. Sokal

    Research output: Contribution to journalArticle

    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. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). 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 the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. In addition, we discuss critically the concept of Gibbs measure, which is at the heart of present-day classical statistical mechanics. We provide a careful, and, we hope, pedagogical, overview of the theory of Gibbsian measures as well as (the less familiar) non-Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. We give a rather complete catalogue of the known examples of such occurrences. The main message of this paper is that, despite a well-established tradition, Gibbsianness should not be taken for granted.

    Original languageEnglish (US)
    Pages (from-to)879-1167
    Number of pages289
    JournalJournal of Statistical Physics
    Volume72
    Issue number5-6
    DOIs
    StatePublished - Sep 1993

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    Regularity Properties
    pathology
    regularity
    Renormalization Group
    First-order Phase Transition
    Ising model
    Ising Model
    Gibbs Measure
    theorems
    occurrences
    Israel
    Decimation
    Critical region
    Single valued
    Classical Mechanics
    messages
    Interaction
    Theorem
    statistical mechanics
    Statistical Mechanics

    Keywords

    • block-spin transformation
    • decimation transformation
    • Fermat's last theorem
    • Gibbs measure
    • Griffiths-Pearce pathologies
    • Kadanoff transformation
    • large deviations
    • majority-rule transformation
    • non-Gibbsian measure
    • Pirogov-Sinai theory
    • position-space renormalization
    • quasilocality
    • real-space renormalization
    • relative entropy
    • Renormalization group

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Physics and Astronomy(all)
    • Mathematical Physics

    Cite this

    Regularity properties and pathologies of position-space renormalization-group transformations : Scope and limitations of Gibbsian theory. / van Enter, Aernout C D; Fernández, Roberto; Sokal, Alan D.

    In: Journal of Statistical Physics, Vol. 72, No. 5-6, 09.1993, p. 879-1167.

    Research output: Contribution to journalArticle

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    AB - 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. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). 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 the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. In addition, we discuss critically the concept of Gibbs measure, which is at the heart of present-day classical statistical mechanics. We provide a careful, and, we hope, pedagogical, overview of the theory of Gibbsian measures as well as (the less familiar) non-Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. We give a rather complete catalogue of the known examples of such occurrences. The main message of this paper is that, despite a well-established tradition, Gibbsianness should not be taken for granted.

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