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

Abstract In two dimensions, all isometrically invariant Markov random 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 antiferromagnetic, with and without a field, as well as the Baxter-Wu model. On the hexagonal lattice, we determine the low-temperature behavior for this class of model, and construct a phase diagram of said behavior. In particular, we identify regions with three geometric phases, regions with a single unique phase, and coexistence curves between them. We also characterize the behavior along two non-Peierls lines, where entropy fails to vanish the as temperature goes to zero.