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 [Formula presented]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≥1. We also provide the formulae for various thermodynamic properties corresponding to the given model.