An Importance Sampling Scheme for Models in a Strong External Field

We propose Monte Carlo methods to estimate the partition function of the two-dimensional Ising model in the presence of an external magnetic field. The estimation is done in the dual of the Forney factor graph representing the model. The proposed methods can efficiently compute an estimate of the partition function in a wide range of model parameters. As an example, we consider models that are in a strong external field.

💡 Research Summary

The paper addresses the problem of estimating the partition function Z of a finite‑size two‑dimensional ferromagnetic Ising model subjected to a consistent external magnetic field. The authors exploit the dual representation of the Forney factor graph (FG) associated with the model. By applying discrete Fourier transforms to the original edge and field factors and replacing equality constraints with XOR factors, they obtain a dual FG in which low‑temperature (large coupling J) configurations become high‑temperature in the dual domain, facilitating Monte‑Carlo sampling.

A key methodological contribution is the partition of the dual variables ˜X into two subsets: ˜X_A (the “bond” variables attached to the γ factors) and ˜X_B (the “field” variables attached to the λ factors). Because each XOR factor enforces a linear relation, ˜X_B can be expressed as a deterministic linear combination of ˜X_A. Consequently, the authors define an auxiliary probability mass function q(˜X_A) proportional to the product of γ factors; its normalizing constant Z_q admits a closed‑form expression (Z_q = 2^{|B|} exp(∑_k J_k)). Sampling from q is straightforward: each component is drawn independently with probability ½(1+e^{−2J_k}). After a sample ˜X_A is generated, ˜X_B follows automatically via the XOR constraints.

Using this construction, an importance‑sampling estimator for the dual partition function Z_d is introduced: \