The Villain model is a planar spin model whose periodic edge weight is a sum of Gaussian weights rather than
the exponential of a cosine. For angles
modulo 
and inverse temperature 
,
its edge weight is
![W_b(theta)=sum_(k in Z)exp[-b(theta+2pik)^2/2].](/images/equations/VillainModel/NumberedEquation1.svg) |
The finite-lattice Gibbs measure multiplies these weights over nearest-neighbor angle differences and uses uniform angle reference measure. The model is related by Fourier expansion to a discrete Gaussian height model.
Coulomb Gas and Maximum of the Integer-Valued Gaussian Free Field." Proc.
London Math. Soc. 127, 653-708, 2023.