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

Abstract

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.

Figures (9)

• Figure 1: Converting binary assignments on a lattice to polyconvex subsets on the faces of the dual lattice. $\sigma^*$ is illustrated by the union of the black hexagons on the right.
• Figure 2: Energy assigned by each term to the three non-empty vertex states
• Figure 3: Representative subsets of each of the configurations with exactly 1 vertex state
• Figure 4: At $T=0$ the behavior is determined only by states with minimal energy. Along three of the degeneracy lines where multiple vertex states have minimal energy there is nonzero entropy, meaning infinitely many global states are minimal. When both $C$ and $H$ vertex states are minimal the allowed configurations are those of the $T=0$ antiferromagnetic Ising model, the same as those of the dimer model BH82. When both $E$ and $C$ vertex states are minimal the allowed configurations are those of the hard hexagon model with the even distribution on allowed configurations B89. The $F$ and $H$ line is symmetric to the $E$ and $C$ under a spin flip. "Landmark points" are marked, including the pure Euler, Perimeter, and Area terms. The line containing "Pure Ferro Ising," "Pure Area", and "Pure Anti-Ferro Ising" corresponds to the Ising model with external field.
• Figure 5: Phase diagram showing the regions where given configurations dominate. Dark regions surrounding the non-Peierls degeneracy lines indicate areas where Pirogov--Sinai techniques are not applicable. This region shrinks with decreasing temperature, but is never empty. Outside of those areas each color, including white, represents a region where a particular set of configurations dominates. On the boundaries between these regions multiple Gibbs states not related by a symmetry dominate. On some of the non-Peierls lines we use disagreement percolation to prove no domination at any temperature.

Theorems & Definitions (29)

• Theorem 1: Hadwiger's TheoremH56
• Corollary 2
• Definition 1: Hadwiger Model
• Proposition 1
• Definition 2
• Definition 3
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6