Partition Function of the Ising Model via Factor Graph Duality

TL;DR

This paper explores the relationship between the partition functions of factor graphs and their duals, applying it to the Ising model to derive known solutions and improve Monte Carlo efficiency at low temperatures.

Contribution

It introduces the use of factor graph duality to analyze the Ising model's partition function, providing new derivations and computational advantages.

Findings

Analytical solution for 1D Ising model derived via duality

Monte Carlo methods are more efficient on dual graphs in 2D at low temperatures

Duality improves computational efficiency in partition function estimation

Abstract

The partition function of a factor graph and the partition function of the dual factor graph are related to each other by the normal factor graph duality theorem. We apply this result to the classical problem of computing the partition function of the Ising model. In the one-dimensional case, we thus obtain an alternative derivation of the (well-known) analytical solution. In the two-dimensional case, we find that Monte Carlo methods are much more efficient on the dual graph than on the original graph, especially at low temperature.