### Abstract

We consider the majority-rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernández, and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero-temperature majority-rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.

Original language | English (US) |
---|---|

Pages (from-to) | 1089-1107 |

Number of pages | 19 |

Journal | Journal of Statistical Physics |

Volume | 86 |

Issue number | 5-6 |

State | Published - Mar 1997 |

### Fingerprint

### Keywords

- Majority-rule renormalization-group transformation
- Non-Gibbsian measures

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*86*(5-6), 1089-1107.

**Majority rule at low temperatures on the square and triangular lattices.** / Kennedy, Thomas G.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 86, no. 5-6, pp. 1089-1107.

}

TY - JOUR

T1 - Majority rule at low temperatures on the square and triangular lattices

AU - Kennedy, Thomas G

PY - 1997/3

Y1 - 1997/3

N2 - We consider the majority-rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernández, and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero-temperature majority-rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.

AB - We consider the majority-rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernández, and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero-temperature majority-rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.

KW - Majority-rule renormalization-group transformation

KW - Non-Gibbsian measures

UR - http://www.scopus.com/inward/record.url?scp=0031092588&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031092588&partnerID=8YFLogxK

M3 - Article

VL - 86

SP - 1089

EP - 1107

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -