Non-Gibbsianness of the reduced SOS-measure
Let \(\rho \) be a Gibbs probability measure on the configuration space \(\Omega ={\mathbb Z}^{{\mathbb Z}^d}\) which corresponds to the pair potential (\(\omega \in \Omega \), \(i,j\in {\mathbb Z}^d\)) \[ \psi_{i,j}(\omega)=J | \omega_i-\omega_j| \;\text{ if }| i-j| =1 \] (and 0 otherwise) with constant \(J>0\). The author shows that the image of \(\rho \) under the mapping \(F:\Omega \to \{-1,1\}^{{\mathbb Z}^d}\), \(F(\omega_i)=1\) if \(\omega_i\geq 0\) and \(F(\omega_i)=-1\) otherwise, is not a Gibbs measure with respect to any translation invariant interaction model on \(\{-1,1\}^{{\mathbb Z}^d}\). The proof uses the concept of relative entropy densities.
scientific article; zbMATH DE number 1364458