### Abstract

Results from percolation theory are used to study phase transitions in one-dimensional Ising and q-state Potts models with couplings of the asymptotic form J_{x,y}≈ const/|x-y|^{2}. For translation-invariant systems with well-defined lim_{x→∞}x^{2}J_{x}=J^{+} (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperatures β with βJ^{+}≤1. (2) If βJ^{+}>q, then by sufficiently increasing J_{1} the spontaneous magnetization M is made positive. (3) In models with 0<J^{+}<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeys M(β_{c})≥1/(β_{c}J^{+})^{1/2}. (4) For Ising (q=2) models with J^{+}<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x-y|^{2} whenever β<β_{c}. Points 1-3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values of q.

Original language | English (US) |
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Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Journal of Statistical Physics |

Volume | 50 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 1988 |

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### Keywords

- 1/r interactions one dimension
- Fortuin-Kasteleyn representation
- Ising model
- Potts models
- Thouless effect
- discontinuous transition
- percolation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

^{2}Ising and Potts models.

*Journal of Statistical Physics*,

*50*(1-2), 1-40. https://doi.org/10.1007/BF01022985