### Abstract

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J_{x}=1/|x|^{d+δ} with 0<δ≤d/2, then the exponents {Mathematical expression}, δ, γ and Δ_{4} exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with δ>-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.

Original language | English (US) |
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Pages (from-to) | 39-49 |

Number of pages | 11 |

Journal | Letters in Mathematical Physics |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Letters in Mathematical Physics*,

*16*(1), 39-49. https://doi.org/10.1007/BF00398169