Thermodynamic formalism and large deviation principle of multiplicative Ising models

In the paper, we explore the thermodynamics of Ising models in relation to 2-multiple Hamiltonians. We extend the findings of Chazottes and Redig (2014) to $N^d$. We establish the large deviation principle (LDP) for the average $\frac{1}{N}{S}_N^G$, where $S_N^G$ is a 2-multiple sum along a semigroup generated by $k$ co-primes. This extends the previous results by Ban et al. (2022) to a broader class of long-range interactions. Finally, the results are generalized to the multidimensional lattice $N^d$ for $d\ge 1$. We also provide the formulae for various thermodynamic properties corresponding to the given model.