Discrete double-porosity models for spin systems

We consider spin systems between a finite number $N$ of "species" or "phases" partitioning a cubic lattice $\mathbb{Z}^d$. We suppose that interactions between points of the same phase are coercive, while between point of different phases (or, possibly, between points of an additional "weak phase") are of lower order. Following a discrete-to-continuum approach we characterize the limit as a continuum energy defined on $N$-tuples of sets (corresponding to the $N$ strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part which describes the combined effect of lower-order terms, weak interactions between phases, and possible oscillations in the weak phase.