## Abstract

We consider the n-componentΦ^{4} spin model on ℤ^{4}, for all n ≥ 1 with small coupling constant.We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent(Formala presented)for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for n = 1, 2, 3; double logarithmic scaling for n = 4; and is bounded when n > 4 In addition, for the model defined on the 4-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility.

Original language | English (US) |
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Pages (from-to) | 692-742 |

Number of pages | 51 |

Journal | Journal of Statistical Physics |

Volume | 157 |

Issue number | 4-5 |

DOIs | |

State | Published - Oct 18 2014 |

## Keywords

- Critical phenomena
- Logarithmic corrections
- Renormalisation group
- Scaling limit
- Specific heat
- Susceptibility

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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