Scaling limit for a class of gradient fields with nonconvex potentials

Marek Biskup, Herbert Spohn

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


We consider gradient fields (φ{symbol}x: x ∈ Zd) 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 κη2], 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 languageEnglish
Pages (from-to)224-251
Number of pages28
JournalAnnals of Probability
Issue number1
StatePublished - Jan 2011


  • Gaussian free field
  • Gradient fields
  • Scaling limit


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