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

TL;DR

This paper characterizes the low-temperature behavior of a broad class of isometrically invariant Markov fields, including various Ising models, by constructing a phase diagram with multiple geometric phases and coexistence regions.

Contribution

It extends the understanding of low-temperature phases in isometrically invariant Markov fields, providing a comprehensive phase diagram for this class of models.

Findings

Identified three geometric phases in the model

Mapped regions with unique phases and coexistence lines

Included analysis of both ferromagnetic and antiferromagnetic Ising models

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.