Mixing length scales of low temperature spin plaquettes models

Plaquette models are short range ferromagnetic spin models that play a key
role in the dynamic facilitation approach to the liquid glass transition. In
this paper we perform a rigorous study of the thermodynamic properties of two
dimensional plaquette models, the square and triangular plaquette models. We
prove that for any positive temperature both models have a unique infinite
volume Gibbs measure with exponentially decaying correlations. We analyse the
scaling of three a priori different static correlation lengths in the small
temperature regime, the mixing, cavity and multispin correlation lengths.
Finally, using the symmetries of the model we determine an exact self
similarity property for the infinite volume Gibbs measure.