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Cited by 401

The DOI matches its Crossref record; 128 of the 128 references it cites were found in a registry or an index.

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

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van Enter, Aernout C. D., Fernández, Roberto, Sokal, Alan D.·1993·Journal of Statistical Physics
DOI 10.1007/bf01048183
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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, Gibbsiannessshould not be taken for granted.

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Metadata

Authors
van Enter, A. C. D., Fernández, R., & Sokal, A. D.
Venue
Journal of Statistical Physics
Publisher
Springer Science and Business Media LLC
Published
1993-09
Volume
72
Issue
5-6
Pages
879-1167
ISSN
  • 0022-4715 (ISSN-L)
  • 1572-9613
DOI
10.1007/bf01048183
OpenAlex
W2600289236
Cited by
401
References
128
FWCI
7.30
Type
journal article
Language
en
Open access
green

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