Journal of Statistical Physics

, Volume 72, Issue 5, pp 879–1167

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

DOI: 10.1007/BF01048183

Cite this article as:
van Enter, A.C.D., Fernández, R. & Sokal, A.D. J Stat Phys (1993) 72: 879. doi:10.1007/BF01048183


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, Gibbsiannessshould not be taken for granted.

Key words

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

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Aernout C. D. van Enter
    • 1
  • Roberto Fernández
    • 2
  • Alan D. Sokal
    • 3
  1. 1.Institute for Theoretical PhysicsRijksuniversiteit GroningenAG GroningenThe Netherlands
  2. 2.Institut de Physique ThéoriqueEcole Polytechnique Fédérale de Lausanne, PHB-EcublensLausanneSwitzerland
  3. 3.Department of PhysicsNew York UniversityNew York

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