### Abstract

We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑_{σ′}, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μ_{σ′} be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μ_{σ′} is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36β_{c} and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than β_{c}.

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

Pages (from-to) | 607-637 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 85 |

Issue number | 5-6 |

State | Published - Dec 1996 |

### Fingerprint

### Keywords

- Completely analytic potentials
- Dobrushin uniqueness theorem
- Ising model
- Renormalization group pathologies

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*85*(5-6), 607-637.

**Absence of renormalization group pathologies near the critical temperature. Two examples.** / Haller, Karl; Kennedy, Thomas G.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 85, no. 5-6, pp. 607-637.

}

TY - JOUR

T1 - Absence of renormalization group pathologies near the critical temperature. Two examples

AU - Haller, Karl

AU - Kennedy, Thomas G

PY - 1996/12

Y1 - 1996/12

N2 - We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑σ′, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36βc and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than βc.

AB - We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑σ′, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36βc and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than βc.

KW - Completely analytic potentials

KW - Dobrushin uniqueness theorem

KW - Ising model

KW - Renormalization group pathologies

UR - http://www.scopus.com/inward/record.url?scp=0030501842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030501842&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030501842

VL - 85

SP - 607

EP - 637

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -