## Abstract

We consider gradient fields (φ{symbol}_{x}: x ∈ Z^{d}) whose law takes the Gibbs-Boltzmann form Z^{-1} exp{-∑_{(x,y)} V (φ{symbol}_{y} - φ{symbol}_{x})}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation V (η):=-log ∫ e (dκ) exp[-1/2 κη^{2}], where e is a positive measure with compact support in (0, ∞). Hence, the potential V is symmetric, but nonconvex in general. While for strictly convex V 's, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential V above scales to a Gaussian free field.

Original language | English |
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Pages (from-to) | 224-251 |

Number of pages | 28 |

Journal | Annals of Probability |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

## Keywords

- Gaussian free field
- Gradient fields
- Scaling limit