### Abstract

We present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple. Our analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary. For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1 -p, the critical temperature scales as exp[-J/(kT_{c}(p))]∼|p -p_{c}|, implying that the crossover exponent is Φ = l. If the constant couplings are replaced by a distribution F(J) with mass near J = 0, quite different crossover behaviour is observed. For example, if F(J)∼J ^{α} then, for p near p_{c}, T_{c}(p)∼|p- p_{c}|^{1/α}

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

Journal | Journal of Physics A: Mathematical and General |

Volume | 20 |

Issue number | 5 |

DOIs | |

State | Published - Apr 1 1987 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: Mathematical and General*,

*20*(5). https://doi.org/10.1088/0305-4470/20/5/010

**The phase boundary in dilute and random Ising and Potts ferromagnets.** / Aizenmant, M.; Chayes, J. T.; Chayes, L.; Newman, C. M.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 20, no. 5. https://doi.org/10.1088/0305-4470/20/5/010

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TY - JOUR

T1 - The phase boundary in dilute and random Ising and Potts ferromagnets

AU - Aizenmant, M.

AU - Chayes, J. T.

AU - Chayes, L.

AU - Newman, C. M.

PY - 1987/4/1

Y1 - 1987/4/1

N2 - We present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple. Our analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary. For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1 -p, the critical temperature scales as exp[-J/(kTc(p))]∼|p -pc|, implying that the crossover exponent is Φ = l. If the constant couplings are replaced by a distribution F(J) with mass near J = 0, quite different crossover behaviour is observed. For example, if F(J)∼J α then, for p near pc, Tc(p)∼|p- pc|1/α

AB - We present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple. Our analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary. For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1 -p, the critical temperature scales as exp[-J/(kTc(p))]∼|p -pc|, implying that the crossover exponent is Φ = l. If the constant couplings are replaced by a distribution F(J) with mass near J = 0, quite different crossover behaviour is observed. For example, if F(J)∼J α then, for p near pc, Tc(p)∼|p- pc|1/α

UR - http://www.scopus.com/inward/record.url?scp=0001937199&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001937199&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/20/5/010

DO - 10.1088/0305-4470/20/5/010

M3 - Article

VL - 20

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 5

ER -