The Primal versus the Dual Ising Model

TL;DR

This paper explores the representation of the Ising model using normal factor graphs in primal and dual domains, revealing contrasting dependency structures and demonstrating their implications through Onsager's solution.

Contribution

It introduces a novel graph-based framework for analyzing the Ising model in primal and dual forms, highlighting their different dependency relations and sampling behaviors.

Findings

Primal domain dependencies are along cycles.

Dual domain dependencies are on cutsets.

Sampling estimators behave oppositely in primal and dual models.

Abstract

We represent the Ising model of statistical physics by normal factor graphs in the primal and in the dual domains. By analogy with Kirchhoff's voltage and current laws, we show that in the primal normal factor graphs, the dependency among the variables is along the cycles, whereas in the dual normal factor graphs, the dependency is on the cutsets. In the primal (resp. dual) domain, dependent variables can be computed via their fundamental cycles (resp. fundamental cutsets). Using Onsager's closed form solution, we illustrate the opposite behavior of the uniform sampling estimator for estimating the partition function in the primal and in the dual of the homogeneous Ising model on a two-dimensional torus.