### 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>β_{0}, |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 language | English (US) |
---|---|

Pages (from-to) | 879-1167 |

Number of pages | 289 |

Journal | Journal of Statistical Physics |

Volume | 72 |

Issue number | 5-6 |

DOIs | |

State | Published - Sep 1993 |

### Fingerprint

### 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

*Journal of Statistical Physics*,

*72*(5-6), 879-1167. https://doi.org/10.1007/BF01048183

**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.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 72, no. 5-6, pp. 879-1167. https://doi.org/10.1007/BF01048183

}

TY - JOUR

T1 - Regularity properties and pathologies of position-space renormalization-group transformations

T2 - Scope and limitations of Gibbsian theory

AU - van Enter, Aernout C D

AU - Fernández, Roberto

AU - Sokal, Alan D.

PY - 1993/9

Y1 - 1993/9

N2 - 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.

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.

KW - block-spin transformation

KW - decimation transformation

KW - Fermat's last theorem

KW - Gibbs measure

KW - Griffiths-Pearce pathologies

KW - Kadanoff transformation

KW - large deviations

KW - majority-rule transformation

KW - non-Gibbsian measure

KW - Pirogov-Sinai theory

KW - position-space renormalization

KW - quasilocality

KW - real-space renormalization

KW - relative entropy

KW - Renormalization group

UR - http://www.scopus.com/inward/record.url?scp=34250084304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250084304&partnerID=8YFLogxK

U2 - 10.1007/BF01048183

DO - 10.1007/BF01048183

M3 - Article

AN - SCOPUS:34250084304

VL - 72

SP - 879

EP - 1167

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -