Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter q. These measures are related to the usual Gibbs measure on $Z^2$ and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter q tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model. This measure turns out to be interesting also because it is the stationary measure of a class of Probabilistic Cellular Automata (PCA). Such PCA can be used to obtain a fast sample of the Ising measures on 2-d lattices.