Hadwiger Models: Low-Temperature Behavior in a Natural Extension of the Ising Model

All isometrically invariant Markov (strictly local) fields on binary assignments are induced by energy functions that can be represented as linear combinations of area, perimeter, and Euler characteristic. This class of model includes the Ising model, both ferro- and antiferro-magnetic, with and without a field, as well as the “triplet” Ising model We determine the low-temperature behavior for this class of model, and construct a phase diagram of that behavior. In particular, we identify regions with three geometric phases, regions with a single unique phase, and coexistence lines between them.